
























				Populus!












			RAH!  RAH!  RAH!



















































9     	DDSWGV.GR                                                                       9  	   LPG.GR                                                                          ~C  C   RC.GR                                                                           N  9   DSOQC.GR                                                                        h  o   	3AHHPC.GR                                                                       iu  {   	ODCBOE.GR                                                                       {  
   	WOOZLE.GR                                                                         
   MAIN.PH                                                                         i      MAIN.GH                                                                         Q      ATT.BGI                                                                         A     CGA.BGI                                                                         
     
EGAVGA.BGI                                                                          $ HERC.BGI                                                                        {  <   IBM8514.BGI                                                                       	   
PC3270.BGI                                                                       |   MAIN.H                                                                          <' 4   MAIN.BH                                                                         p( 	   LVC.GH                                                                          y) ;   LVC.PH                                                                          , A   DAS.GH                                                                          .    	GDAMCM.GH                                                                       	0    AL.GH                                                                           1    DSOQC.GH                                                                        3 ~   LPG.GH                                                                          5    
INBREED.GH                                                                      6    AL.PH                                                                           8    DAS.PH                                                                          :     	DDSWGV.PH                                                                       ; !   	DDSWGV.GH                                                                       =    DIPG.PH                                                                         > y   DIPG.GH                                                                         Z@ 7   DSOQC.PH                                                                        B    
INBREED.PH                                                                      mD a    LPG.PH                                                                          D    	WOOZLE.PH                                                                       lH n   	SOASLL.GH                                                                       K     SOTL.GH                                                                         L     SOTL.PH                                                                         GM    PAQG.GH                                                                         aP ^   PAQG.PH                                                                         Q    HERIT.GH                                                                        IT    HERIT.PH                                                                        NX M   
INTRAGS.GH                                                                      Z !   PS.GH                                                                           \    MNP.GH                                                                          _ u   SGFAC.PH                                                                        4a    RC.GH                                                                           d )   RC.PH                                                                           %i    	ODCBOE.GH                                                                       m =   	ODCBOE.PH                                                                       p t   FDSEM.GH                                                                        s     FDSDM.PH                                                                        t M   FDSDM.GH                                                                        u    FDSDM.GR                                                                        |w    APPD1.GR                                                                         l   APPD2.GR                                                                         [   APPD4.GR                                                                        H 
   APPD5.GRR                                                                       4    LVPP.GR                                                                         ˶ n   LVPP.PH                                                                         9    LVPP.GH                                                                         7    TLPP.PH                                                                             TLPP.GH                                                                              APPD7.GRR                                                                       c X   OPTIONS.MEN                                                                         APPD8.PH                                                                        ]    APPD7.PH                                                                            APPD6.PH                                                                            APPD5.PH                                                                            APPD4.PH                                                                         _   APPD3.PH                                                                            APPD2.PH                                                                            APPD1.PH                                                                         G   LVC.GR                                                                           <	   APPD.GR                                                                          M	   MNP.GR                                                                          c A   MNP.PH                                                                           q   KR.PH                                                                                 EP24PIN.BGI                                                                     5# `8  + 
EP9PIN.BGI                                                                      [ 5   	HPLJ2.BGI                                                                        8  # HPPJ.BGI                                                                         :   	PSBGI.BGI                                                                           PSBGI.PS                                                                            
INBREED.GR                                                                      "    SAM.PH                                                                          +    SAM.GH                                                                          (0    KR.GH                                                                           3 B   APPD3.GR                                                                        U6 Q   APPD1.GH                                                                        B ,   APPD2.GH                                                                        F    APPD3.GH                                                                        aI X   APPD4.GH                                                                        K    APPD6.GR                                                                        >N j   APPD6.GH                                                                        Y 3   APPD7.GH                                                                        \ |   APPD8.GH                                                                        W` n   APPD5.GH                                                                        c 
   MAIN.SH                                                                         f ]   DAS.GR                                                                          ,m p   FDSEM.PH                                                                        t    	SOASLL.GR                                                                       lw    	SOASAL.PH                                                                       }    AL.GR                                                                           c    	SOASLL.PH                                                                       ~    GDAMM.GR                                                                         	   
INTERGS.PH                                                                          
INTERGS.GH                                                                       /   KR.GR                                                                           Π    	GDAMCM.PH                                                                           GDAMM.GH                                                                        ?    GDAMM.PH                                                                            PS.PH                                                                            5   SGFAC.GR                                                                            SGFAC.GH                                                                            	2AHHPC.PH                                                                           	2AHHPC.GH                                                                       	    	3AHHPC.GH                                                                           	SOASAL.GR                                                                           
INTERGS.GR                                                                       g
   
INTRAGS.GR                                                                      "    	GDAMCM.GR                                                                           	2AHHPC.GR                                                                        	   	3AHHPC.PH                                                                       =    PAQG.GR                                                                         /    SOTL.GR                                                                         ;    MAIN.ZH                                                                             FDSEM.GR                                                                        ! I   MAIN.IH                                                                         , J   MAIN.OH                                                                         '/    MAIN.DH                                                                         ?5 ,   	SOASAL.GH                                                                       k8    HERIT.GR                                                                        j;    TRIP.CHR                                                                         D B  @ SAM.GR                                                                           W   PS.GR                                                                           F    LITT.CHR                                                                        Ӧ W   TLPP.GR                                                                         *    
INTRAGS.PH                                                                          MAIN.CH                                                                          U   DIPG.GR                                                                          <   APPD8.GR                                                                        ?    ASPG.GH                                                                             ASPG.GR                                                                         x 6
   ASPG.PH                                                                             IE.PH                                                                            c   IE.GH                                                                           / >   IE.GR                                                                           m    ACKNOW.                                                                         9    	PALETTES.                                                                       *    MAINMENU.MEN                                                                    3    SS.GR                                                                           Z; L	   SS.PH                                                                           D    POPULUST.DOC                                                                    I    SS.GH                                                                           i    MAINTITL.GR                                                                     k r   LVM.GR                                                                          om b   IHPI.GR                                                                         t  
   3PM.GR                                                                          ~    READT.ME                                                                        h K
   






14t+3
Density Dependent Selection
with Genetic Variation

	This model shows simultaneous evolution of gene frequency and population size.  It is a hybrid model:  it combines elements of the genetic model of selection on one locus, and the ecological model of logistic growth.  There are assumed to be three genotypes that differ in intrinsic rate of increase (r) and carrying capacity (K).  The r's and K's affect both the growth of the population and the change in gene frequencies.

	A particular population can be represented by a point on an (N,p) plane.  Over time, the point moves on the plane, showing how population size and gene frequency evolve.





	This model was developed by Anderson (1971), Charlesworth (1971), and Roughgarden (1971).  Its basic features are as follows:

	1) The fitness of a genotype depends on its r and K values, and on population density.  In particular,

             nrAA
wAA = 1 + rAA -     N
             nKAA

where N is total population density.

	2) It is often assumed that there is a trade-off between r and K values.  Some genotypes are highly fertile but not very good competitors at high density; these are the "r-strategists".  Other genotypes might be less fertile but produce more robust progeny that are good competitors at high density; these are the "K strategists".









	3) In a stable environment in which all mortality is due to density dependent selection, the K values of the different genotypes determine the ultimate pattern of genetic equilibrium.  If the heterozygote has the highest K, there will be a stable polymorphism; if it has the lowest K there will be an unstable polymorphic equilibrium.  If one of the homozygotes has the highest K then fixation of that type will be a stable equilibrium.

	4) As in the logistic model without genetic variation, the magnitude of the r's determines whether there are oscillations in population size and/or gene frequency.  Generally an r value in the range of 2 will produce some oscillations.










Further Reading:

Anderson, W. W. (1971) Genetic equilibrium and population growth under density-regulated selection.  American Naturalist 105:  489-498.

Charlesworth, B. (1971) Selection in density regulated populations.  Ecology 52:  469-474.

Roughgarden, J. (1971) Density-dependent natural selection.  Ecology 52:  453-468.







t+314
Logistic Population Growth


	These models simulate density-dependent population growth, assuming that negative feedback of population size on per capita growth rate is a simple linear function.  They require specification of a starting population size ("N0"), a maximum sustainable population size or environmental carrying capacity ("K"), an intrinsic growth rate ("r"), and (optionally) a feedback lag ("T").  The program includes continuous, lagged continuous, and discrete simulations.





	Density-dependent models assume that population size affects the per capita growth rate.  While the feedback of density on growth can take many forms, the logistic model imposes a negative linear feedback on the per capita growth rate.  Note that if K is the environmental carrying capacity (in terms of individuals, N), then (K-N) gives a measure of the unused carrying capacity, and (K-N)/K gives the fraction of carrying capacity still remaining.  Thus
 ndNn             
   = r{(K-N)/K}
nNdtn            
If N is near zero, the environment is largely unused, and dN/Ndt is near "r."  If N = K, the environment is totally used or occupied, and dN/Ndt = 0.  In this continuous, differential equation model, "r" is an instantaneous rate, but its numerical value is defined over a finite time period.  To simplify comparison with the analogous discrete model, we set this arbitrary time interval to the average generation length.









	Sometimes the feedback of density on per capita growth rate is not instantaneous.  For example, the effect of malnutrition on population growth might not be strongly evident before malnourished juveniles reach reproductive age.  We can simulate this process by assuming that growth rates are affected by population size in some previous time period.  Thus

 ndNn                 
   = r{(K-N(t-T))/K}
nNdtn                

where T is a time lag measured in units of average generation time.  How does this lag affect the dynamic behavior of the logistic?





	A finite difference equation produces a discrete simulation analogous to the continuous logistic model:

Nt+1 = Ner(1-N/K)

Here, "r" is a finite rate of increase, or multiplicative growth factor per generation.  Because this model simulates growth in discrete generations, density dependent feedback is not instantaneous, and it shows interesting dynamics which are not seen in the continuous model.

References

May, R. M. 1974. Biological populations with non-overlapping generations: stable points, stable cycles and chaos. Science 186:645-7.

Hutchinson, G. E. 1978. An Introduction to Population Ecology. Yale University Press, New Haven. pp. 1 - 40.

Ehrlich, P. R., and J. Roughgarden. 1987. The Science of Ecology. MacMillan Publishing Co. New York. pp. 90-107.t+314



Resource Competition Models

	All organisms are consumers, and most risk being consumed by other organisms.  The mechanisms of consumer-resource interactions may thus be major determinants of the species composition and diversity of natural communities. An understanding of these mechanisms may allow prediction of the dynamics and outcome of interspecific interactions.

	A resource is defined as any substance which is consumed by an organism and which leads to an increased per capita growth rate as its availability is increased. If a species is consuming a single limiting resource, the population may eventually reach an equilibrium in which the growth rate of the population equals its loss rate, and the rate of supply of the resource equals its consumption rate.








	At equilibrium (dNi/Nidt = 0 = dR/dt), population growth equals loss and resource supply equals resource consumption. This equilibrium occurs at one particular concentration of the limiting resource, called R*. R* is the concentration of the resource required for the growth of the species to equal its loss rate. This is the concentration to which the resource would be reduced by the species at equilibrium.

	When several species consume the same limiting resource, the equilibrial outcome of their competition is determined by the R*'s of the species. The species with the lowest R* is predicted to competitively displace all others from the habitat at equilibrium.

	This may be understood by considering a model of consumer-resource interactions. Although there are many such models, one simple version that has been successfully applied to a variety of organisms is:

dNi/Nidt = riR/(R+ki) - mi

dR/dt = a(S-R) - i=1,n[Nici(dNi/Nidt + mi)]

where the subscript i refers to species i,
"N" refers to its population density (expressed either as number of individuals per unit area or biomass per unit area),
"R" is the concentration or availability of the limiting resource,
"r" is the maximal per capita growth rate of the consumer species,
"k" is the resource concentration at which the species attains half of its maximal growth rate, 
"m" is the mortality or loss rate that the consumer experiences,
"S" is the "supply" point for a habitat and is the maximal amount of the resource in the habitat,
"a" is a constant determining the rate at which the resource is converted from an unavailable to an available form,
"c" is the function describing the resource consumption rate of the consumer species.









	When these equations are solved at equilibrium, 

Ri* = kimi/(ri-mi)

	One interesting exercise that can be done for the model of competition for a single limiting resource is to vary k, m, and r for several species, and determine if each species does reduce the resource down to its R* when growing by itself in monoculture, and if as predicted, the species with the lower R* is the superior competitor at equilibrium.








t+214
Competition for two Essential Resources

	This model can be expanded to include competition for two limiting resources. There are many such cases because there are many different ways that a species can respond to two resources. One possible response is that of plants to mineral nutrients and light. These resources are essential. For perfectly essential resources, the growth rate of a plant is determined by the one resource in lowest supply compared to need. Thus, the growth rate of a plant is determined by the concentration of the one resource that leads to the lowest growth rate.



	This can be incorporated into a model of 14Competition for Two Essential Resources:

dNi/Nidt = [MINj=1,2(riRj/(Rj+kij) - mi]

dRj/dt = a(Sj-Rj) - i=1,n[Nicij(dNi/Nidt + mi)

	The MIN function states that the growth rate of a species is determined by the resource that leads to the lower potential growth rate. Two species can stably coexist when consuming two essential resources if each is limited by a different resource and each, relative to the other species, consumes more of the resource which limits it.

	The outcome of competition for two resources depends on the relative availability of the resources. This is determined by resource supply rates and species-specific loss rates in a given habitat. The point (S1, S2) is called the supply point. The nature of competition for two essential resources is most easily understood by determining the outcome of competition associated with various supply points. To visualize this, use the graphs of resource-dependent zero-net-growth isoclines and the trajectory of R1 versus R2.


t+214
Competition for Two Switching Resources

	Although most plant resources are nutritionally essential, most animal resources are nutritionally substitutable. Thus, an animal can survive by consuming just one of the resources. There are many ways that two nutritionally substitutable resources can affect growth rate. An animal could forage for all resources, i.e., be a generalist. This would give straight-line resource-dependent zero-net-growth isoclines. Alternatively, a small animal living in a patchy environment could specialize on one resource and consume only it. This would be favored when there were costs to foraging for both resources simultaneously. Such costs could include the cost of traveling from one patch to the next, the risk of predation during such travel, etc.  If an animal forages so as to specialize on one particular resource at any given instant, then it is said to be foraging in a switching manner. In the extreme case, the animal should consume the one resource that leads to the higher fitness at any given instant. Animals that foraged in such a switching manner could have their growth modeled as follows:

		dNi/Nidt = MAXj=1,2[(riRj/(Rj+kij) - mi]                        

		dRj/dt = a(Sj-Rj)-i=1,n[{ 0, if j is the resource that leads to 
							the lower growth rate for this
							species }
								     or
						  { Nicij(dNi/Nidt + mi), if j is the
							resource that leads to the greater
							growth rate }]

Please note that this model assumes that, at any given instant, each species will consume only one resource, the one that leads to higher growth rate.





References

Tilman, D. 1980. Resources: A graphical-mechanistic approach to competition and predation. American Naturalist 116: 362-393.

Tilman, D. 1982. Resource Competition and Community Structure. Princeton University Press, Princeton, New Jersey. 296 pages.9
t+314
Directional Selection on
Quantitative Characteristics

	This module simulates an artificial selection experiment.  The user specifies genetic effects, population size, and the number of individuals selected to be parents of the next generation.  The program shows the distribution of phenotypes in each generation, and how the mean phenotype changes in the population over generations.4
	Phenotypic selection regimes can be classified into three categories, depending on which phenotypes are favored.  Stabilizing selection refers to selection in favor of the intermediate phenotypes.  Disruptive selection means selection in favor of the extremes.  Directional selection refers to selection in favor of one extreme, e.g., the biggest, tallest, etc.  This module simulates a directional selection experiment in a finite population.

	If there is usable genetic variation (i.e., additive genetic variance), then directional selection changes the mean phenotype in the population.  The rate and magnitude of that change depend on a number of factors, including:

	Allele frequency - alleles at intermediate frequencies contribute more usable genetic variance than alleles at extreme frequencies.

	Population size - In very small populations, drift can cause the loss of usable genetic variation, inhibiting selection response.  Or it can cause unselected alleles to increase, simply by chance.  In general, large populations respond more reliably to selection than small populations.4
	Number of individuals selected - Selecting the few most desirable individuals as parents of the next generation exerts a very strong selection pressure, but also increases the effect of drift.  If you select too few individuals then the usable genetic variation might be lost; if you select too many, then the selection pressure is weak.

	Genotypic values - These are the numbers that tell us what the phenotypes of the three genotypes would be in the absence of environmental effects.  If the genotypic values are very similar, there will not be much progress by selection.  If the genotypic values are very different then selection can usually change the population mean.  Selection progress is also strongly affected by the heterozygote genotypic value:  if the heterozygote is the "best" genotype then the population cannot be changed as much as the case where a homozygote is the best genotype.

	Environmental variance - If the environmental variance is large, then an individual's phenotype is not a very reliable indicator of his genotype.  Environmental effects are not transmitted to progeny, so large environmental variance inhibits selection response.8
	In a very large population, the expected response to one generation of directional selection is

R = h2S

where R is the selection response (measured as the difference between the mean phenotypes of parents and offspring), h2  is narrow sense heritability, and S is the selection differential (measured as the difference between the mean phenotype of all parents and the mean of the selected parents).


Further reading:

Falconer, D.S.  Introduction to Quantitative Genetics.9
t+314
Host-Parasite Coevolution+214
a triallelic haploid model

	This model is similar to the diallelic host-parasite model, with the exception that three different alleles (at a single locus) are present in both host and parasite populations.  Again, type1 parasites are successful on type1 hosts, but suffer a fitness penalty if they encounter type2 or type3 hosts.  Likewise, host1 incurs a fitness penalty when it is attacked by parasite1, but not when it encounters parasite2 or parasite3.  Encounters are assumed to occur in proportion to the genotypic frequencies.

	The transformation equations which are iterated in each new generation for type1 hosts and parasites are:

         h1(1-s)p1 + h1p2 + h1p3
H1' =                                                                 
         h1[(1-s)p1+p2+p3] + h2[(1-s)p2+p1+p3] + h3[(1-s)p3+p1+p2]

         p1[h1 + (1-t)h2 + (1-t)h3)]
P1' =                                                                 
         p1[h1+(1-t)h2+(1-t)h3)] + p2[h2+(1-t)h1+(1-t)h3)] + p3[h3+(1-t)h1+(1-t)h2)]

where	s = the penalty to hosts of being parasitized, and
		t = the penalty to parasites of being rebuffed.

	Equations for the type2 and type 3 hosts and parasites are analogous.

	For a detailed discussion of this host-parasite coevolution model and its implications, see

Seger, J. and W. D. Hamilton. 1988. Parasites and Sex. pp 176-193 in The Evolution of Sex, R. E. Michod and B. R. Levin, eds., Sinauer Associates, Inc.

Sunderland, MA. see especially Figures 1 & 2 and related text.2
t+314
Optimal Diet Choice

	Most organisms are faced with the choice of consuming all or some subset of the food items which they encounter.  This simple fact forms the basis for a large subset of behavioral ecology known as optimal foraging theory.  One of the first situations that ecologists thought about in the early 1970's was a system in which the different food types differ in energy content and the amount of time that it takes to capture and ingest an item of food.  Each food type i is characterized by an abundance, here denoted R, a handling time (denoted H), and an energy or nutrient content (denoted E). It is theoretically possible for the forager to take all, none, or some fraction of the food items of a given type; the proportion of all type i which are consumed is denoted p.  The energy intake rate is given by an expression known as the disc equation (it accurately described the rate at which a blindfolded secretary found paper discs on a table top in a famous set of experiments by the Canadian ecologist, C.S. Holling).  This expression has the summation of pRE in the numerator, and 1 plus the sum of pRH in the denominator:3
     piRiEi
Qn =            
     1 + piRiHi

	In the early 1970's, ecologists worked through the algebra, and found that the optimal diet (that which maximized energy intake rate) had the following characteristics:
(1) The optimal proportion of a given food type which should be taken (attacked) is either 0 or 1; under the simple assumptions of the disc equation model, it is never optimal to consume a fraction of the items encountered.
(2) The foods should be ranked based on the values of E/H, energy content divided by handling time.  The optimal diet consists of the top x ranked types; that is, it can never be optimal to include the food which has the (n+1)th largest E/H value unless the diet also includes the type with the nth largest E/H.
(3) Changing the abundance, R, of foods which are not included in the diet will not lead to their inclusion.  Increasing the abundance of the top ranked foods which are included in the diet will eventually result in lower ranked food types being dropped from the diet.6
	This program is designed to illustrate these principles.  You are asked to enter the number of food types (N) and the E, H, and R values of each type.  The program then performs a number of computations and displays a graph which describes the energy intake rates for all diets consisting of the top n ranked food items (n = 1 to N).  It also tells you the optimal diet (the number of top ranked food types that should be consumed to maximize energy intake rate.  Furthermore, it tells you the percentage reduction in energy intake that results from consuming one too many or one too few food types.  By hitting any key, you can return to the data screen and modify any of the parameters.

	One exercise you may wish to perform is to alter the abundances of either some or all of the food types, and see what effect this has on the optimal diet.  Increasing the abundances of all food types proportionately will eventually result in only the top-ranked (highest E/H) food being included in the diet.13
References

Stephens, D. W. and Krebs, J. R. 1986 Foraging Theory, Princeton University Press.

Pyke, G. H. 1984 Optimal foraging theory:  a critical review, Annual Review of Ecology and Systematics, 15:523-575.

Krebs, J. R. and Davies, N. 1987 An Introduction to Behavioural Ecology 2nd ed., Sinauer.9
t+314Woozleology

	In his book The Blind Watchmaker, Richard Dawkins confronts the old saw analogizing evolution to the chance keystrokes of a monkey who, "sooner or later," will reproduce the works of Shakespeare.  He sketches a computer program which uses cumulative selection to model "evolution" of the phrase "METHINKS IT IS LIKE A WOOZLE" (Hamlet and Polonius thought their cloud looked like a weasel, but 3-year-old Amy Alstad considers Milne the pinacle of English literature).  Since the phrase has 28 characters and spaces, the monkey can be expected to type it correctly once in 2827 attempts.7
	We have written a program like Dawkins' to illustrate the power of cumulative selection.  It works in the following way:

(a) An initial phrase consisting of 28 randomly chosen letters or spaces becomes the first-generation "parent."

(b) The 28 characters are copied to form one or more "offspring."  For each letter the program tests a random number against the user-specified "mutation rate," to determine whether this position in the phrase receives a new, randomly chosen character, or the original parental letter.  Users specify the total "broodsize."

(c) Each of the progeny is compared with the target phrase "METHINKS IT IS LIKE A WOOZLE," and the offspring which matches the target at the largest number of positions becomes the next-generation parent.5
(d) We have also incorporated a recombination process in the program.  If a user-specified recombination fraction exceeds 0, then two parental lines are kept.  For each letter position in the progeny, random numbers are tested against the recombination fraction and the mutation rate.  A draw less than the recombination fraction forces a "crossover" or transcription change from one parent to the other.  If a random draw is less than the mutation rate, that position receives a new random character; otherwise it receives a parental character determined by a second random number.  The best offspring phrases from first and last halves of the total brood become next-generation parents.

	Depending on the parameter values specified, this program usually "evolves" the target phrase in a few dozen or a few hundred generations, demonstrating that cumulative selection is much more efficient than a single-step random process.  As a model of natural selection, woozleology is unrealistic in its unrelenting focus on a fixed future target, but it gives a good intuitive feel for the cumulative, generation-to-generation process.12
References

Dawkins, Richard. 1986. The Blind Watchmaker.  W. W. Norton & Co. New York. 332 pp.

Maynard Smith, John. 1989. Evolutionary Genetics.  Oxford University Press.  Oxford. 325 pp.
	Each model has a help screen describing its parameters.  It suggests practical values for each parameter, and often describes its effect on the model's dynamics.  To see this screen, hit <F1> while viewing the data entry window.
	Each model that produces a graph has a help screen that describes the output and some of the interesting features that may occur.  Press <F1> while looking at a graph (and after it has finished drawing) to obtain the graphs help screen.pkBGI Device Driver (ATT) 2.00 - Mar 21 1988
Copyright (c) 1987,1988 Borland International
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Copyright (c) 1987,1988 Borland International
                              CGA                  U ]CB      s S[o      "''SA  4   r _                Ø         .>z u ˸ ˴$.~ &CC&CC5z<s
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;6t&N>z ut&-OG&-t& &-Fɀ>z utGG&-62àE&-:u&;6$|+6,OGa&-"":t8""
;6}&Fɀ>z ut&-GG&-t
 &-N60E&-:u&;6&s6,GGÀFɀ>z utGG &"":u^>Bu
&Gt
>C t:Du
Ft
? u)":t!urMM^ MMv MM~ MMN ;60snJ&:u&0+,&:u;6,GG&"":t!0+,Q.E&:-u;s6,GGYÉ6KM>OQ
 &GHËÊBt3
؀>z t"tGPR% >z uZЀ>z t u&G[Ë6KM>OQH&G&>z uE2ƴtsE "t"uEÀuSQP2
C"u2XY[mP, >z t, ,$+,@&+(+*XEUEUEUejotPS2 G"[XKu  ÁCuǰÁ  KPCPKZCZà<r$<s SӃ<t4KSӃuZ..[CSӃuZKSӃuP.h[CSӃuP? ?@??  0    @       pkBGI Device Driver (EGAVGA) 2.00 - Mar 21 1988
Copyright (c) 1987,1988 Borland International
                         EGAVGA               U ]CB      _=      ,R{?     r                 ï         ø 2˴ . 
.  $.X f&CC&CC<s
2"tW2e"y@t
2ۈ>Y2Ê2ʸð                                <u &GC72W ؊2sêuQ YË&^؋`Ë?
P!ZË>  t!

  89:;<=>?                    ?<t  > t  U"t<t
> t                          
640 x 200 EGA 
640 x 350 EGA 
640 x 480 VGA 640 X 350 EGA MONO     (#X  ]](#XF  (#X'  ]](#XF#-;4&$YUs2>Y22۴3}<ur<u>Ut2tÀtu
r 2@r  U2pX %>u%VáZ\Z\QRȋӇZ\ ZYË6V>W<r,2ྐ7 <t27           >cuK&"t*SQZ\Y[^>cu\KZC"x-b&c ^ `SQV.Y[ x<t
2؋    ]      (   
.>(t߱..ú 
(  +  ]]
`	 P <
u? <t<u@  &$`u> 2á&  .> tS<r2.[PRBJZXR2.> t Z.>X tPRZXÖV>RQSP 
U BFJBv
+vyFF
FFFFދFVʀB^+^x{;rGA++ފ*.s
#y*P̃ 
sG* A++.s%#yȃ Pz;rFA++ފ*.s
#y*Pă 
sO*.A++.s%#y PHJB*]   ;r;rڋ7+BSދQY+[PVQS^[Êz[Êr]WSW _[?r7>uͺSߊ I Y"ي"&"2&Gð:uQW2_Y&G?@            @8px<iZ-KB$$B3333     "  "                              8l  ||    |~  x00000x |     ||  || x`0x 000000 | l8  l8l f< 0` <00000< `0 <<  8l           00       |~~    |~ ~v   |~ 0|0000   ~v|  8<  8 8<         ||     ~v  np```   || 00|000   ~   l8   l   l8l   v|  0` &"uÈ"u- - ;|>+>;s;|>+>;sSQf _^.V...WP X_.>uE.un&^  .=Àt%"yGuNu<0uO2<Ar<{r1<r"  |  &#t&G#u n  > tA&F &^&N&V ֖MuڽS	ؠ 	&2Ë> t	&&[&CS>U uWU]_& 	6	 		W.> u&$FP_GMuXXô2==&$"u=FP                                                      @                      ++B5
QS_[H.zQRV; ^ZYu.6.5
uSU[ìଊ#t	&%GJ"&%G  
+B5
Sދ+â
@[PVQS^[ .z[ .r ]SQUW  _]Y[u.>.5
u݋S[SM2#t".>
u&%&G嬊Mض.
"*:Z}%R.>
u&%&ZG2"ں.>
u&%&ÃRPXZ  SR.
2.Z[S2.[ 
R&&
Zñ*˺2&="
}> t tÀtNy
O> ty׋ںӻ

0
1
3
xo&

  RMMF +MMF   
MMv 	
 `MM~ Z&"u2([
N EEu
&

Ë~ EEv EE^ EES
P
  Xtu>	
 u s9 ' r/ F 뉡@;
uH9~
@9áH;
t2á
9
s
+>
;
t
>2;6t &"u׊&;6tpFu
2G> t׋ںӻ
6,)V&"u<׊&;6},Fu
2G> t׋ںӻ
"u
 N6&= tÀ> uCR$ 3ɡ

W&= u;sGA_t 2ZRB> tZR&= u;6

s G
 uݺB+
&&%-
&&%UVB+
͋-
͋͋^]ðRZ);6rmFu2G> tgŁt &"tu28*
t80
t8/
t
	
&ŀu׊׊Ћں&"%ӻ
ur
MMF MMv MM~ MMN _

]&= u9RB&= u;sG> t
 uްJZ
É6


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
Ë6


>

> ttà
<tzQR+
3ҋ5A2ڋW>uՊ+
-
2;1
u;3
u/

&*
"
2ЈCuZY
> t RWV
*
"
GGG+
s Cu
^_ZÀ>
t>s]>s  
- 

+
+
RBB
> tZRPBX> t Z
RBZôR
 uBZÀuSQP252
C"u2XY[pkBGI Device Driver (HERC) 2.00 - Mar 21 1988
Copyright (c) 1987,1988 Borland International
                           HERC                 U ]CB      BY   b   $PUo    r                 ;=?AØ         .> u ˸ " t   ˺$
@ ûe &$. 
&CC&CC<s
2"t2"y@uÀu> tÀ> u>&?<u 
&GCË&؋ 	

<t  > u |                               [[(#XL  720 x 348 HERCULES <u<u
2Ê2@<s 2 &$> u02<r<t#6?6AA?3ɶO22ø  K3À> t3"t"tÉó:v:vããQRȋӇ ZYË6>8    <rU,2@
w> t u u >  Ъ ª2ҳs
uÀ> t"t <t3ۃ	w                  >uK&"t*SQEY[>uKC"x=&  SQrZRY[ x<t
2؋	                       ? (  ? (   (  P  [Z  P  ]Z  <r2
؁C7?G9AG3G0W2P Xش>2t
2$ø ø ø@ 0  33 @5-.[WW  3۹ PSQ؎ 2BJY03X󫺸X
@ ûe &Â4t  Á4tðÁ  .3.+3    QuZS$
2.[YQɀuZ    QuPS$
2.[YQɀuP;7w;7w;9w;9vËv>5;rډF^F ;rF쀑+F+ډ^/F^;rF^+؉^+؉^  ^F+ÉF+ÉF  ^V~vF$>s$
F> t@> t fnnE5s t&0'
&"
&S2j"yCCh[Muƀuh֨@uŀuC5s t&0'
&"
&Muƀu֨@uŀuCh5s t&0'
&"
&뽋NV<s <r3р> t"Ŋf"uufv<sF t  ^0<r^㎇<u   U  AEIUQ&&0
&  "&"
&;r;rڋ6> uw+B0S273ދ+â5> t@[PVQS 2^[ .> u.	[ .	].2PSUW.> t.6uV/ ^ .> t w;rw_][Xh.0uSM#t.3GMu.> t.5"*.> u:Z}
.3Gĳ".3Ã.> u? U2BW<s <r3؋ֹ?&"t""uȆ0[&s
t  Àr㎇u                  @8px<iZ-KB$$B3333     "  "                          &
䀈&
$"uÈ
"u
- 
- 
;;|>?+>
;s;=|>A+>
;sSQ _^.
..
PS&V  .
 Ou[8 XuE.
uu7"
7.> u"xCu.
.>s
.h.
.h.
<0uO2<Ar<{r1<r"  |  &#t&G#u n ]
>
 tA&F &^&N&V
 ֖Muڽ
}
> t"t	"8l  ||    |~  x00000x |     ||  || x`0x 000000 | l8  l8l f< 0` <00000< `0 <<  8l           00       |~~    |~ ~v   |~ 0|0000   ~v|  8<  8 8<         ||     ~v  np```   || 00|000   ~   l8   l   l8l   v|  0` 
 
QSQذbY^VQWS;X [SP.
>f "uE&'.h[_GYXX                                                                 ++BQS_[u.> t.	.PRV+ ^ZX.>r.h.h.uԋìଊ#t	&%GJ"&%G   |lpt&0&
& "&"
&+BS2b_ދ+âa> t@[PVQS^[ .> u.	[ .	Z.PSRW _Z[X.h.uSJ2#t._G嬊J.> t.a"*.> u:Z}._G2"._Ã.> uVKFyy0222Ã> t022拄\&"
&Ëڋǃ玅P ڋǃ玅Z 拄|&"
&øZ P RZ 拄|&"
&p ΃A&$Ëǃ玅P ǃ玅Z ΀&$øZ P RZ R蜿P s  Ztr> tr&  ރ	> t"tRMMF 9+MMF   ?MMv  MM~ Zk&"":t2%[N EEu
&Ë~ EEv EE^ EE"SP$x Xtu> u s5 % r+ ; 떡@;uH9~@9áH;t2;sF4;t02&-;6;tN> utO&-"":t8""
;6;t&N> ut&-O&-t& &-Fɀ> utG&-6à&-:u&;6|+6Oa&-"":t8""
;6?}&Fɀ> ut&-G&-t
 &-N6&-:u&;6s6GÀFɀ> utG &"":u^>u
&t
> t:u
t
? u)":t!urMM^ MMv MM~ MMN ;6sn&:u&+&:u;6G&"":t!+Q.&:-u;s6GYÉ6>
 &ËÊt	؀> t"tPR9% > uZwЀ> t u&[Ë6>&&> u2ƴts "t"uÀuSQPwu2
C"u2XY[mP > t ;?+@9+A9+=X%6KPS22ËG[XKu  ÁCuǰÁ  KPCPKZCZà2<r$<s SӃ<t4KSӃuZ.[CSӃuZKSӃuP.[CSӃuP? ?@??  0    @       pkBGI Device Driver (IBM8514) 2.00 - Mar 21 1988
Copyright (c) 1987,1988 Borland International
   i                    i IBM8514              U ]CB      Á/k  H 	tF     !]r                 ')+-Û         ø :  $.
PS[Xȋ           f        f2                   
          	&CC&CC 
xP&CC&CC[X6
.
            "y+@t2䣦êª <u  &GC&؋                                au640 x 480  8514/A AI 1024 x 768 8514/A AI   (#X'  (#X'                                                                      > t%UWVAtS2&^_]UWV  5!ts	^_]    &+-2&
ģ }o	ã0 gd  2 R<u <u25Ê2@/
Q
21%G+G-
 Ìȣ3G% DG
               $$      p H  444  p    p  pp p   pp  $          SQ	 kY[ð
  _ã:t
PCXà      QRȋӇ	 ZYËƋ؀>
u ]MU  >
u^ <tN<t<s{"u<d  ád= r"    d  1 d= r dpÀ> t>d<sf]d_O =rKK&S   /[ UWVS2&^_]  S3ACC&CCu[  <r,2ྰ> <t2> ȣ6,    >                
>uK&$t*SQY[>uKC " ;"x0&>
 tSQY[                 <t  PSQRZY[X  s;r;r@I;wCJ;w]+AM+BU             CE3҈G?A!#PS3ҹZ 3= r [X;tSPZ GXS0 X[) ?APS[X!#CE+>=#u 66QR6WSSPP X X !X #_666!P3>=s)S!#A	26 d- rd dL>G uG?A_V _V ЋË Ëȋ Q<t!2؊ Y ð#t3ҹ "x* B4 B#u  ȣ 2
"y2
"x                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     @8px<iZ-KB$$B3333     "  "         U* F
^FڋF^F ;FFFFuFFuFF^;wd s F܉Vދ^FV3FFFF܋Vދ^^FVs s +FVމFV։~v o F^F^F^;^|
u;Fr l F;F|uF;Fr / F^F^;^
u;Fw5 T F tы] FFFFFFF+FFFFFFFFFFFFFFFFNF)FFFF+FFFFFËVv~F u$ 	  WVv~5r~ ^_ 	  ދƋ ~ u;F|;F~;F};F~
x
x+؋p
x++t3ɇʑËȋrËƋ;? ?@??  0    @        U ?????      &"uÈ"u#x>+;s#x>-;sSQ _^&F UQ Y]Eu        <0uO2<Ar<{rn a> tA&F &^&N&V6 ֖Muڽ6sC2ûf2Ҷ&tr	u	
uu݉6]>_7 22[gYVWUȣea_^Ê22tfQ.F CEuYu
            f    8l  ||    |~  x00000x |     ||  || x`0x 000000 | l8  l8l f< 0` <00000< `0 <<  8l           00       |~~    |~ ~v   |~ 0|0000   ~v|  8<  8 8<         ||     ~v  np```   || 00|000   ~   l8   l   l8l   v|  0`          ;w'a
:l !H#l%')+-/2468:<> @ACEGjI<KMNPFRSUKWXZ\]_`
bycd9fghjYklmnoqrsttuvwGxyyhz
{{2||3}}~l~~L UFZUF   F}V 3ҹh= ~- V =Z ~ @3Ѓ~  t
Ѓ  ]_P2X=pkBGI Device Driver (PC3270) 2.00 - Mar 21 1988
Copyright (c) 1987,1988 Borland International
                         PC3270               U ]CB      ,   2    "q'aA     r _                
Ø         .>z u ˸ ˴$.~ R
&CC&CC
<s
2"t}2K"y@uÀu>z tÀ> u>&?<u `
&GCË&؋ 	

<t  >z u N                               ]](#XQ  720 x 350 3270 PC <u<u
2óÊ2@{z<sz 2~ &$> u{-<u<r<t#663ɶO22ø  K3À>z t3"t"tÉ|ó:v:vã|ãQRȋӇ ZYË6|>.    }<rU,2
Q>z t O O >}  Ъ ª2ҳs
uÀ>z t"t <t3ۃ	Q                  >uK&"t*SQ2Y[>uKC"x-&  SQ|Y[ x<t
2؋                             ? (  ? (   (  P  [Z  P  ]Z  <r
؁G	GG W2P Xش>t
2$ø ø ø@ 0  33 @5-.[WW  3۹ PSQ؎ 2BJY 3X󫺸X
@ ûe &RRbRbRbrx4t  Á4tðÁ  ..+    QuZS$
2.~[YQɀuZ    QuPS$
2.[YQɀuP;w;w;	w;	vËv>;rډF^F ;rF쀑+F+ډ^/F^;rF^+؉^+؉^  ^F+ÉF+ÉF  ^V~vF$>{s$
F>z t@>~ t fnnEs t&0'
&"
&S{2:"yCC8[Muƀu8֨@uŀuCs t&0'
&"
&Muƀu֨@uŀuC8s t&0'
&"
&뽋NV{<s <rр>z t"Ŋfz"uu{fv<sF t  ^ <r^㎇~<u   %  %!&&0
&  "&"
&;r;rڋ>z uQ+B S2ދ+â>z t@[PVQS ^[ .>z u.{	[ .s	].PSUW.>z t.uV/ ^ .>z t Q;rQ_][X8. uSM#t.GMu.>z t."*.>z u:Z}
.Gĳ".Ã.>z u? U2BW{<s <r؋ֹ?&z"t""uȆ [&{s
t  Àr㎇~u                  @8px<iZ-KB$$B3333     "  "                 &u
䀈&p
$"uÈq
"ur
- v
- x
#x>+>v
;s#x>+>x
;sSQ _^.t
..r
PS&V  .q
 Ou[6 XuE.t
ut7"
7.>z u"xCu.s
.>{s
.8.s
.8.s
        <0uO2<Ar<{r<r n 
>u
 tA&F &^&N&V` ֖Muڽ`s
|>z t"t	"8l  ||    |~  x00000x |     ||  || x`0x 000000 | l8  l8l f< 0` <00000< `0 <<  8l           00       |~~    |~ ~v   |~ 0|0000   ~v|  8<  8 8<         ||     ~v  np```   || 00|000   ~   l8   l   l8l   v|  0`                                                       ++BQSc_[u.>z t.{	.PRV+ ^ZX.>{r.8.8.uԋìଊ#t	&%GJ"&%G   &0&
& "&"
&+BS2âދ+â>z t@[PVQS^[ .>z u.{	[ .s	Z.PSRW _Z[X.8.uSJ2#t.G嬊J.>z t."*.>z u:Z}.G2".Ã.>z uN8' 22Ã>z t 2拄&"
&Ëڋǃ玅P ڋǃ玅~Z 拄&"
&øZ P RZ 拄&"
&p ΃A&$Ëǃ玅P ǃ玅~Z ΀&$øZ P RZ R蜿P s  Ztr>z tr&

  ރ	>z t"tRMMF 	+MMF 

  
MMv  MM~ Zk&"":t2%[




N EEu
&
Ë~ EEv EE^ EE"SP$x Xtu> u s5 % r+ ; 떡
@;u
H9
~
@9
á
H;t2;
s
F4;
t
02&-;6tN>z utO&-"":t8""
;6t&N>z ut&-O&-t& &-Fɀ>z utG&-6
à&-:u&;6
|+6
Oa&-"":t8""
;6}&Fɀ>z ut&-G&-t
 &-N6
&-:u&;6
s6
GÀFɀ>z utG &"":u^>u
&t
> t:u
t
? u)":t!urMM^ MMv MM~ MMN ;6
sn
&:u&
+
&:u;6
G&"":t!
+
Q.&:-u;s6
GYÉ6
>
 &ËÊt	؀>z t"tPR	% >z uZQЀ>z t u&[Ë6
>&&>z u2ƴts "t"uÀuSQPQO2
C"u2XY[P
 >z t
 

+
@
	+
	+

X%*Pev/4PS2
G
[XKu  ÁCuǰÁ  KPCPKZCZà<r$<s SӃ<t4KSӃuZ.~[CSӃuZKSӃuP.[CSӃuP? ?@??  0    @       
Use  and  to highlight the model
that you want to run.  Then press
the Enter key to run it.

Esc   - Title screen
F1    - This help screen
F2    - Introduction

Alt-X - Exit

When in a model, press F1 once for
context-sensitive help, and twice
for a help menu.

Press any key to continue.
	Each model is preceeded by several pages of narrative material which introduce the mathematics of the simulation and end with basic references.  You can read through this material using <PgDn> or <PgUp>, or jump directly to the input screen by pressing <Enter>.
	The N vs. T graphs illustrate changes in population density for both species as the competition progresses.  Typically, if N0 is small relative to the carrying capacity, the populations will grow until density dependent effects become important and the competitive outcome begins to develop.  Depending on the parameters chosen, both longterm coexistence and competitive exclusion are possible.

	The phase plane diagram illustrates both species isoclines and the N2 vs N1 tragectory of the interaction.  Note that the isocline for species 2 is given by N2 = K2 - N1, and the slope of the line is -; similarly, the species 1 isocline is N1 = K1 - N2, but the slope is -1/, because both isoclines are plotted on the same coordinate system (i.e., the species 1 isocline is plotted as N2 vs N1, rather than the reverse).
1. The simulation has two output options.  Choose a graph showing the densities of each species vs. time, or a "phase-plane" graph showing N2 vs. N1 isoclines for each species with the tragectory of the competitive interaction.

2. Four parameter values are required for each species.  (a) N0 is the initial population size which can range from 0 to 9,999.  (b) "r," the intrinsic rate of increase, can range from -1 to 5.  (c) "K," the environmental carrying capacity, can range from 0 to 9,999, and (d) "" and "," the competition coefficients, can range from 0 to 100.
	The graph shows how allele frequency changes over time under the combined forces of drift and natural selection. If the simulated population goes to fixation before the specified number of generations, the simulation will halt and display frequencies only to that point.  
	   Gene frequencies for 1 to 6 drifting loci are plotted vs. time in generations.  Plots terminate when one of the two loci drifts to fixation.  If the simulation is run long enough, all of the loci will eventually fix in a finite population, but the rate at which this happens is strongly dependent on population size.  Initial frequencies also play a role.  A population with p = .9 is likely to fix sooner that one beginning with p = .5  
	   The figure shows a plot of Wbar, the average fitness in the population, as a function of gene frequency.  For any particular gene frequency there is a corresponding average fitness.  The dots show how a particular population changes in gene frequency and mean fitness over time.  Note that gene frequency changes caused by natural selection always cause immediate increase in the average fitness in the average fitness in the population. 
	There are two types of graphs that can be viewed each generation. One shows the distribution of phenotypes for all individuals in the population. Only a fraction of these individuals will be selected to be parents of the next generation. 
	The other graph shows how the mean phenotype in the population changes over time. 
	Switch between the graphs by hitting the space bar. 
	The graph shows expected population sizes (on the ordinate) plotted against generation (on the abscissa).  It is particularly interesting to compare the dynamic behavior of the continuous and discrete versions using similar input parameters.  Note that population size is represented in the simulations as a real number, and this might permit N to fall to decimal values 0 < N < 1. 
	The graph shows how F, the inbreeding coefficient, changes over time in a finite population. F varies from 0 to 1, where 0 means no inbreeding and 1 means complete inbreeding. This simulation starts with initial F = 0. 
	The smooth curve is the expected F for the given population size and number of generations of random mating. The jagged curve is the realized F for a particular simulation.  
1. wAA, wAa, and waa are the relative fitnesses of the three genotypes.  Specify a real number ranging from 0 to 1.  By convention, the most fit of the three genotypes is given a relative fitness of 1.  A genotype with wxx = 0.25 will contribute  as many progeny per capita as the best-fit genotype to the next generation.  If wxx = 0, the genotype is lethal; all such individuals fail to survive or reproduce.

2. How many generations should you simulate?  The answer will depend on the strength of the selection process specified by your choice of relative fitnesses.  Reasonable values might range from 10 to 1000 generations.

3. Enter some starting frequency between 0 and 1.
Population size - Enter a positive integer.
Allele frequency - Enter a number on the range 0 - 1.
w's - Enter the relative fitnesses of the three genotypes.  Positive numbers.
Number of generations - Enter a positive integer.
	There are three different genotypes.  You must input one r (intrinsic rate of increase) and one K (carrying capacity) for each genotype.  Both r's and K's must be positive.
	Initial gene frequency p is the frequency of the "A" allele at the beginning of the simulation.  It must be between 0 and 1.
	Initial population size must be positive.  It can be greater than or less than the carrying capacity.
	Number of generations: Most simulations come to equilibrium within 50 generations, but this will vary with fitness parameters selected.
	The graph represents a population by a point on a plane.  The location of the point is determined by the population size (N) and the frequency of the "A" allele (p).  The point moves through time, eventually coming to equilibrium for both population size and gene frequency.
1. N0 is the starting population size. The simulation will accept values ranging from 0 to 1E6.

2. Lambda is the coefficient of geometric increase, or finite growth rate constant for discrete populations.  Values can range from 0.01 to 150.

3. r is the instantaneous growth rate for continuously growing populations.  The input filters will accept values from -5 to 5.
	The screen shows a plot of population density vs time, measured in generations (discrete) or average generation length (continuous).  If this model is allowed to grow for many generations with a positive r or a Lambda > 1, the run will often terminate with an overflow (i.e. population size will exceed the computer's long-integer numerical capacity), and it will appear (because of the scaling) that nothing happens in the first few generations.  Plot the same starting conditions for a smaller number of generations to see the early growth pattern more clearly.
Population Size - Input a positive integer.

Number Selected - The number of individuals chosen as parents for the next generation. Enter a positive integer less than or equal to the population size.

Allele frequency - A real number in the range 0 - 1.

Environmental variance - A positive number that determines the magnitude of environmental effects on phenotypes.

Genotypic values - Average phenotypes for each genotype in the absence of environmental effects.
Population size - Enter a positive integer.
Number of generations - Enter a positive integer.
1. The continuous model employs a differential equation to simulate the growth of a population with overlapping generations and continuous reproduction.  It can be run with or with out a lag in the expression of density-dependent effects.  The discrete model uses a finite difference equation to simulate non-overlapping generations.

2. "N0," the starting population size, can be any positive integer, either above or below K.  Reasonable values might range anywhere from 1 to 10,000.

3. "K," the environmental carrying capacity, must be a positive integer.  Again, reasonable values might range from 1 to 10,000.

4. "r," the intrinsic growth rate of the population, should be a real number greater than zero.  Reasonable values might range from 0 to 5.  We suggest that you begin with 0 < r < 1, and then try larger values.

5. "T" the lag in expression of density dependent effects can take values from 0 to 5.
Woozleology Input Parameters

1. "BroodSize" sets the number of offspring phrases screened in each generation.  Values can range from 2 to 1000.

2. For each position in an offspring phrase, "Mutation Rate" determines whether the child receives a new, randomly chosen letter, or a faithful copy from the parental phrase.  Permissible values range from .0005 to .5

3. The "Recombination Fraction" determines whether transcription shifts from one parental phrase to the other, and is tested for each position in the offspring phrases.  Values can range from 0 to 0.5, where a zero means that transcription proceeds from start to finish on a single parental phrase.

Woozleology Outputs

	Output windows show the target phrase, the current phrase, the current generation number, and a record of the accumulation of agreements between current parents and the target.
	The output shows how frequencies of the "X" allele change in males and females over time.  Notice that frequencies usually become equal or approximately equal in the two sexes.
	"Freq" output means that the frequencies of gametes are plotted as a function of time.  "D" plots linkage disequilibrium over time.  "Wbar" shows how mean fitness changes over time.
	"Freq" output means that the frequencies of gametes will be plotted as a function of time.  "D" plots linkage disequilibrium over time.  Wbar shows how mean fitness changes over time.
	pAB, pAb, paB, and pab are frequencies of the four different gamete types.  Enter numbers between zero and one.  Note that the four values must sum to one, and the program will normalize if your inputs do not meet this condition.
	R is the recombination fraction.  R = 0 means complete linkage; R = 0.5 means unlinked loci.  Enter a number between 0 and 0.5. 
	The w's are fitnesses of nine genotypes in the two-locus system. Enter any positive numbers.
	The number of generations to run to equilibrium will generally depend on the strength of selection.  Usually 100 generations or less will suffice.
Upper left:  The adaptive topography shows how mean fitness (W) changes as a function of allelic frequency (p).

Lower left:  This figure shows how heritability (h) changes over time (generations).

Upper right:  This figure shows how mean fitness changes over time.

Lower right:  This figure shows how allelic frequency changes over time.
	To run this program you must provide three fitnesses for the three genotypes AA, Aa, and aa.  The fitnesses can be any non-negative numbers; the program will scale them to one.  If the numbers are very different (e.g. 0, 100, 5) then the system will attain equilibrium very quickly; only a few generations (<10) will need to be iterated.  But if fitness differences are small (e.g. 20, 21, 19) then you should specify 100 or more generations in order to reach equilibrium.
	The environmental variance (ve) is the non-genetic component of phenotypic variance attributable to variations in the organism's environment.  Specify a value from 0 to 1.
	There are two graphical screens: one shows the theoretical heritability and the other shows the estimate of heritability in a breeding "experiment". Switch between the screens by pressing the spacebar.
	The theoretical screen shows the expected regression of offspring on mid-parent values; the slope of this line is the heritability. The curves that rest on the axes show the distribution of phenotypes in the parental and offspring generations. The upper right hand corner shows parameter values.
	The Monte Carlo screen shows the same information, plus individual data points. Each data point corresponds to the measurement of one pair of parents and their progeny. Note that the accuracy of the estimate depends on population size.
	You can change parameters while watching either graphical screen. To alter allele frequencies press "p" or "q". To increase genotypic values press "1", "2", or "3"; to decrease use "Alt-1", "Alt-2", "Alt-3". To alter environmental variance press "E" (to increase) or "G" (to decrease).
Allele frequency - Enter a number in the range 0 - 1.

Genotypic values - Enter three positive numbers representing the average phenotype measurement for each genotype.

Environmental variance - Enter a positive number representing the magnitude of the random environmental effect.

Increments - When graphical screens are displayed you can alter allele frequency, genotypic values, or environmental variance by these amounts.  See the output help screen for appropriate keyboard controls.

Population Size - Enter a positive number for the Monte Carlo estimate of heritability.
	The graph plots frequencies of the altruistic allele against time, measured in generations.  Red points plot the simple average of frequencies in all 100 trait groups (size N) immediately after their random reformulation from the disperser pool.  Yellow points plot the simple average of frequencies over all trait groups for generations which pass within the subdivisions.  These points show what is happening under individual selection within the groups, but do not present an accurate weighted average of frequencies in the deme at large.
	This simulation produces two output displays.  The first plots gene frequency in each of the 6 demes against time, and is identical to the output produce by our Monte Carlo Drift Model.  The second graph plots Fis, Fst, and Fit.  In runs where there is very little interdemic migration, genetic drift can cause substantial differentiation among demes, which is reflected in a high Fst.  At intermediate levels of migration (0.02 to 0.05), there can still be significant interdemic differentiation,and the introduction of migrants of different frequency produces a negative Fis reflecting a heterozygote excess within the demes.  This is a "Wahlund Effect."  At very high migration rates, the entire population becomes homogenized into a single randomly-breeding unit.
	This model graphs population-wide gene frequency vs. time, showing either a polymorphic equilibrium, or the fixation of one allele.  If you have not specified a starting gene frequency, then the simulation performs a sensitivity analysis, initiating separate runs from six different starting frequencies, to illustrate the presence or absence of polymorphic equilibria.
1. The gradient mode, heterozygote advantage mode, local heterozygote advantage mode, and frequency dependent mode indicate four different ways in which the relative fitness of AA Aa and aa genotypes can vary across the environmental gradient (see intro screen 2).

2. The parameter "s" sets the intensity of selection changes across the gradient, and can take values ranging from 0 to 1.

3. "g" determines the local proportion of individuals that are derived as migrants from adjacent demes in each generation.

4. "h1" is the minimum selective advantage of heterozygotes over homozygotes in the heterozygote advantage mode.  "h2" is the constant advantage held by heterozygotes over whichever homozygote is locally most fit in the local heterozygote advantage mode.

5. The simulation shows starting frequencies in all 50 demes; it then runs for an interval that can be set by the user, or to equilibrium (press <Ctrl-Enter>) before producing a new view.
N vs T:
The population density of the consumer species through time. For competition for two or three resources, this graph should be compared with the screen of R versus R.

R vs T:
The dynamics of resources during time. The level to which the resource concentration is eventually reduced is called R*, and determines the position of the zero net growth isocline.

R vs R:
The right-angle curves are zero-net-growth isoclines for the competitors. A zero-net-growth isocline is a line along which dN/dt = 0. It shows the concentrations of the resources for which there is no net change in the size of the population density of a species. There is one isocline shown for each species. The graph also shows how resource concentrations are reduced during competition. If the interaction was allowed to proceed for sufficient time, this trajectory will end up on an isocline. If this ends up at a point at which two isoclines cross, then these two species will coexist. If it is not at such a point, a single species should displace all others at equilibrium.
	Choose either competition for a single limiting resource, competition for essential resources, or competition for switching resources.

	Next choose to plot N (population size) versus t (time), or R (resource availability) versus t, or R versus R (only if there are two or three limiting resources.

	To specify which species will compete and which resources will be used, it is only necessary to enter the number that is used as the name of the species that you want to have compete.

	You should also specify a, which is the rate constant determining resource recycling time. This is usually set equal to the mortality rate, m, but it need not be.

	The initial concentration of R1 in the habitat, R10, is usually equal to S1, which is the maximum amount of resource 1. R20, S2, R30 and S3 must also be specified.

	For each species, you must specify No = initial population density; r = maximal growth rate; m = mortality rate; k1 = half-saturation constant for growth on R1; k2 = half-saturation constant for growth on R2; c1 = consumption rate of R1, or amount of R1 needed to produce a new individual; c2 = consumption rate of R2, or amount of R2 needed to produce a new individual; and so on.
	The line that is illustrated gives the energy intake rate for various hypothetical diets.  The diets are formed by arranging all of the food types in order of their E/H value, and then considering subsets consisting of (i) the top ranked food only; (ii) the top two ranked foods, (iii) the top three ranked foods; etc.  The optimal diet is that subset which results in the greatest energy intake rate, which is easily seen from the graph.

	If you wish to see the original data values and the ranks of each food type, you may hit any key to return to the data screen.
	Energy content of one item, E must be a positive number.

	The handling time required to consume one item, H, must also be positive.

	The abundance of a food type, R, must be positive or zero.

	In determining values, it is good to keep in mind that the abundance and handling times determine what proportion of the total foraging time is used searching for prey vs. handling prey.  The fraction of time spent searching is 1/(1 + RH).  Thus, if the sum of the RH values for all food types included in the diet is much greater than one, this implies that most of the foraging time is spent handling, rather than searching for prey. Although this does happen in environments with very abundant prey, it is more common for foods in nature to have RH (i.e. the product of R and H) values of 1 or less.  It would be extremely unusual to have foods whose RH was greater than 10.
	The output shows the frequencies of the invader strategy and the resident strategy over 500 generations of selection.  If the resident is an ESS, it cannot be invaded by any other strategy.  Try to find the ESS for the default payoff values.
	For each run the user must provide an initial frequency of the "A" allele, denoted "p", the number of generations to be simulated, and nine fitness parameters, w1 - w9.

	The fitness parameters describe the relative viability of each genotype when interacting with the other genotypes.  Relative fitnesses must be non-negative.
	There are two graphical outputs.  The first shows how allelic frequency "p" changes over time under the operation of frequency dependent selection.

	The second shows how the average fitness of the population w bar changes over time. Note that, in contrast to the classical constant fitness models, with frequency dependence mean fitness can decrease under the operation of selection, though it does not always do so.5t+314
Frequency-Dependent Selection:
Diploid Model

	Frequency dependence means that the fitnesses of genotypes change as the genetic makeup of the population changes.  This is in contrast to classical models of population genetics, which assume that genotypic fitnesses are constant.

	This model simulates a situation in which a large population of diploid organisms is subject to frequency dependent selection.  Individuals are assumed to interact in pairs, in such a way that the fitness of each individual depends not only on his own genotype, but also on the genotype of his "social partner".  The model applies to any situation in which organisms come together in pairs to compete for resources, or to assist each other in obtaining resources.  For simplicity, pairs are assumed to form at random, like particles colliding randomly in a box.7
	The set up of the model is as follows.  There are assumed to be three autosomal genotypes, AA, Aa, and aa.  We must specify nine fitness parameters, one for each genotype interacting with each other genotype:

							   INTERACTING WITH

							 AA		 Aa		 aa

					AA		w1		w2		w3

		FITNESS OF	Aa		w4		w5		w6

					aa		w7		w8		w9

We assume random mating or random union of gametes, so genotypes are initially present in Hardy-Weinberg frequencies, before selection operates.


	What is the fitness of each genotype when they interact randomly in pairs?  The fitness of AA will be w1 if he interacts with another AA, w2 if he interacts with an Aa, and w3 if he interacts with aa.  With randomly chosen "social partners", the probability of interacting with a particular genotype is simply its frequency in the population, so the total fitness of AA is

Fitness (AA) = p2(w1) +  2pq(w2) + q2(w3).

Similar calculations apply to the other genotypes.  Once we have calculated relative fitnesses in this way, the equation for the new allele frequency is the same as for the classical constant fitness model:

         p2(Fitness AA) + pq(Fitness of Aa)
New p =                                   .
         Mean Fitness of the Population

	This model differs from the classical constant-fitness model in some important ways.  Most importantly, mean fitness may not be maximized under frequency dependence.  It is possible to have stable genetic equilibria which are not maxima of the mean fitness surface; this means that the classical concept of adaptive topography does not carry over to frequency dependent selection models.11
Further Reading:

	This model, which is probably the most widely cited diploid frequency dependence model, is fully analyzed by Cockerham, Burrows, Young, and Prout in "Frequency dependent selection in randomly mating populations", American Naturalist 106:493-515, 1972.

	For a more recent and more general general discussion of frequency dependent selection see the Philosophical Transactions of the Royal Society, London, Series B, Vol 319, 1988, which presents a collection of papers on the subject.6t+314
The Nicholson-Bailey Model

	The predator-prey model formulated by Nicholson (1933) and Nicholson and Bailey (1935) offers a basic starting point for comparison with more complex and realistic models.  It is based on two simplifying assumptions: (1) the number of encounters, Ne between Pt parasitoids or predators with host or prey is proportional to host density, Nt, and (2) these encounters are randomly distributed among hosts.  This means that some hosts will be encountered more than once, so Ne can be larger than Nt.  The number of hosts not parasitized is given by the zero term of the Poisson distribution,

p0 = exp(-Ne/Nt)

and the number of hosts actually parasitized is

Na = Nt[1 - exp(-Ne/Nt)].
6
The number of encounters between Pt parasitoids and their hosts Nt, can be restated as

Ne = aNtPt,

where "a" is a proportionality constant called the parasitoid's area of discovery.   It is a measure of searching efficiency, and can be thought of as the proportion of all hosts that will be encountered by an individual parasitoid during its lifetime.  It follows that Ne/Nt = aPt, and the number of parasitized hosts can be rewritten as

Na = Nt[1 - e-aPt].

This expression implies that parasitism will rise as a saturating function of aPt, because parasites encounter fewer and fewer unparasitized hosts as their numbers and searching efficiency increase.  Nicholson called this relationship a competition curve.6
	If the host survival function (the number of hosts remaining unattacked) is

f(Nt,Pt) = e-aPt

then the recursion equations defining host and parasatoid dynamics are

Nt+1 = Nte-aPt

Pt+1 = Nt[1 - e-aPt]

where  is the intrinsic geometric growth rate of the hosts.

	This model is directly analogous to the Lotka-Volterra predator-prey model, save for its formulation as a pair of finite difference equations.  While the continuous Lotka-Volterra version produces a neutrally stable limit cycle, the Nicholson-Bailey dynamics give an unstable, increasing oscillation.  May (1973, 1975) showed that this difference in the models' dynamics is attributable to the built-in time lag associated with the discrete Nicholson-Bailey difference equations.8
	The introduction of density-dependent host growth has a stabilizing influence on Nicholson-Bailey dynamics.  If you choose the density-dependent option, Populus substitutes a host recursion suggested by Beddington, Free and Lawton (1975) as follows:

Nt+1 = Nt er(1-N/K)-aPt

The parasitoid recursion remains as above.  The stability of this density-dependent version is determined by the host reproductive rate (given here as r = ln ) and by a, the parasitoids searching efficiency or area of discovery.  If the parasitoid is extremely efficient (large a) it is able to hold hosts below their carrying capacity and dynamics are determined most strongly by the unstable host-parasitoid interaction.  If the parasitoid is inefficient (small a), host dynamics are determined largely by the density-dependent feedback.6
References

Beddington, J. R., C. A. Free, and J. H. Lawton. 1975. Dynamic complexity in predator-prey models framed in difference equations. Nature 225:58-60.

Hassell, M. P. 1978. The Dynamics of Arthropod Predator-Prey Systems. Monographs in Population Biology, Princeton University Press. Princeton, NJ. pp. 12-27.

May, R. M. 1973. On relationships among various types of population models. American Naturalist 107:46-57.

May, R. M. 1975. Biological populations obeying difference equations: stable points, stable cycles, and chaos. J. Theor. Biol. 49:511-524.

Nicholson, A. J. 1933. The balance of animal populations. J. Anim. Ecol. 2:132-178.

Nicholson, A. J. and V. A. Bailey, 1935. The balance of animal populations. Part *. Proc. Zool.  Soc. Lond. 1935, 551-598.1114t+3
Functional Responses

	As host density increases, the number of hosts parasitized per parasitoid should increase.  Holling (1959) called this the predator's functional response (following Solomon, 1949), and suggested that it might take three forms:

			Type I, prey consumption rises linearly to a plateau
			Type II, consumption rises asymptotically to saturation
			Type III, consumption is a sigmoid function of prey density8	We can simulate a linear functional response (type I, but with insatiable parasitoids) by assuming that our host survival function f(Nt,Pt) = exp(-a'TPt), where Pt is the parasitoid density at time t, a' is the parasitoid's instantaneous search rate, and T is the fixed, constant search time available.  Then

Nt+1 = Nt exp(-a'TPt)

Pt+1 = cNt[1 - exp(-a'TPt)]

where again Nt and Pt are the host and parasitoid population densities at time t,  is the intrinsic growth rate of the hosts, and c is the numerical response indicating the number of parasitoids (often 1) produced per host consumed.  Note that a'T is equivalent to Nicholson's area of discovery, a, if the searching time, T, is one full generation; so this linear functional response formulation is identical to the basic Nicholson-Bailey model.6
	Often some handling time, Th, is required for a parasotoid to oviposit on each host, and search time is consequently reduced.  As host density rises, handling consumes an increasing proportion of the parasitoid's time budget, and the functional response is a saturating, Type II function.  If a' and Th are both constant and parasitoid search is random, the prey survival function f(Nt,Pt) = exp[-(a'TPt)/(1 + a'ThNt)] illustrates this sort of response, giving the difference equations

Nt+1 = Nt exp[-(a'TPt)/(1 + a'ThNt)]

Pt+1 = cNt[1 - exp{-(a'TPt)/(1 + a'ThNt)}],

The dynamics of interactions with a Type II functional response are always less stable than the simple Nicholson-Bailey model, because the hosts escape at high density.  The magnitude of this destabilizing effect is determined by the ratio of Th to total search time T, and is relatively unimportant if Th/T << 1 (Hassell and May, 1973).7	Sigmoid Type III functional responses result when one or more of the components of parasitoid searching activity (a', T, or both) are increasing functions of prey density.  It is often suggested that a type III functional response results when the parasitoids are smart and learn to be more efficient as prey density rises, but in fact responses should be sigmoid whenever payoffs at the lowest host densities are below some threshold required for constant searching activity.  Hassell developed a type III model in which a' varies with host density according to the expression a' = (bNt)/(1+cNt) where b and c are constants (again, c is the numerical response, often 1.0 for parasitoids).  This yields the prey survival function

f(Nt, Pt) = exp[-(bTNtPt)/(1+cNt+bThNt2)]

and the resulting difference equations for a simulation are

Nt+1 = Nt exp[(-bTNtPt)/(1 + cNt + bThNt2)]

Pt+1 = cNt[1 - exp[(-bTNtPt)/(1+ cNt + bThNt2)]
13
	Sigmoid functional responses are potentially stabilizing, because the parasitoids impose a density-dependent effect on hosts at low density.  While this stabilizing influence is observable in a continuous, differential equation formulation like the theta-logistic model, Hassell and Comins (1978) showed that it is too subtle to overcome the instability inherrent in difference equation models with a one-generation delay between changes in predator density and prey mortality.7
References

Hassell, M. P. 1978.  The Dynamics of Arthropod Predator-Prey Systems. Monographs in Population Biology, Princeton University Press. Princeton, NJ. pp. 28-49.

Hassell, M. P. and Comins, H.N. 1978.  Sigmoid functional responses and population stability. Theoretical Population Biology 9:202-221.

Hassell, M. P. and R. M. May. 1973.  Stability in insect host-parasitoid interactions. J. Anim. Ecol. 42:693-736.

Holling, C.S. 1959.  The components of predation as revealed by a study of small mammal predation of the European pine sawfly. Canadian Entomologist 91: 293-320.

Solomon, M.E. 1949.  The natural control of animal populations. Journal of Animal Ecology 18:1-35.t14+3
Interference

	Parasitoids that aggregate in patches of high host density are likely to encounter one another in the course of their foraging activities.  Many species have been shown to display aggressive behavior toward other nearby females; these interactions waste searching time and increase the parasitoids' tendency for dispersal.  The first of two parasitoid interference models included in Populus is attributable to Hassell and Varley (1969).  It assumes that the negative relationship between searching efficiency, a, and parasitoid density is linear on log-log scales

a = QPt-m

where both Q and m are constants.  The interference constant, m, is interpretable as the slope of this decline in search efficiency with parasitoid density.  This leads to the recursions 

Nt+1 = Ntexp(-QPt1-m)

Pt+1 = Nt[1 - exp(-QPt1-m)]

The stability of this model increases as values of m vary from 0 to 1.0, and declines as the intrinsic growth rate of hosts increases (Hassell and May 1973).6	Models with a linear relationship between searching efficiency and parasitoid density are obviously oversimplified; efficiency cannot rise indefinitely as parasitoids become increasingly rare.  We have therefore included a curvilinear model by Beddington (1975) who assumes that parasitoids encounter one another randomly at a rate, b, which is analogous to the rate of their encounters with hosts (a') introduced in the functional response models.  He further assumes that after each encounter between parasitoids there is a period of wasted time, Tw, during which no further searching takes place.  Beddington's recursions are

Nt+1 = Nt exp[-a'TPt/{1+bTwPt-1}]

Pt+1 = cNt[1 - exp(-a'TPt/{1+bTwPt-1})],

In effect, this model allows the interference coefficient, m, to vary between 0 and 1.0 and its stability properties are similar to those of the linear version.  It collapses back to the basic Nicholson-Bailey model if bTw = 0.  Students who are interested in interference are directed to Hassell's book for discussions of its importance at equilibrial densities, and its relation to aggregation and nonrandom foraging.10References

Beddington, J. R. 1975. Mutual interference between parasites or predators and its effect on searching efficiency. J. Anim. Ecol 44:331-340.

Hassell, M. P. 1978. The Dynamics of Arthropod Predator-Prey Systems. Monographs in Population Biology, Princeton University Press. Princeton, NJ. pp. 80-105.

Hassell, M. P. and R. M. May. 1973. Stability in insect host-parasitoid interactions. J. Anim. Ecol. 42:693-736.

Hassell, M. P. and G. C. Varley. 1969. New inductive population model for insect parasites and its bearing on biological control. Nature 223:1133-36.t14+3
Threshold Predator Reproduction

	Simulations appropriate to a predator-prey interaction should differ in several respects from the parasitoid-host models in this set.  Since predators usually consume many prey individuals that are smaller than the consumers themselves, the numerical response, c, will often be less than 1.0; in addition, since a certain amount of energy and resources are likely for the predators to mature and maintain themselves without reproducing, the functional relationship between prey availability and predator reproduction is unlikely to be a simple proportionality.  Hassell's book describes a model by Beddington, Free, and Lawton (1976) that relates a predator's lifetime fecundity to the number of prey consumed as

F = c[(Na/Pt) - ]

where c is the efficiency of prey conversion into predators (the numerical response) and  is the minimum threshold prey consumption required before predators begin to reproduce.  Assuming that prey population growth is density dependent, this assumption leads to the recursions

Nt+1 = Ntexp[r(1-Nt/K) - aPt]

Pt+1 = c[Nt[1 - exp(-aPt)] - Pt]
7
	If  = 0, this model collapses to the density-dependent version of Nicholson-Bailey, but with c > 0 the equilibrial densities of predators and prey are no longer globally stable.  Instead they have a local basin of attraction that shrinks as the c product increases.  One practical implication of this behavior is that the success of such a predator in biological control releases would be sensitive to the initial densities of predator and prey.


References

Beddington, J. R., C. A. Free, and J. H. Lawton. 1976. Concepts of stability and resilience in predator-prey models. J. Anim. Ecol. 45:791-816.

Hassell, M. P. 1978. The Dynamics of Arthropod Predator-Prey Systems. Monographs in Population Biology, Princeton University Press. Princeton, NJ. pp. 106-121.8t+314
Lotka-Volterra
Predator-Prey Dynamics

	Let's assume (1) that except for the presence of predators, prey live in an ideal (density-independent) environment, (2) that the predator's environment is similarly ideal and its population growth is limited only by the availability of prey, (3) that both predators and prey reproduce continuously with ageless populations of identical individuals, and (4) that the predation rate is proportional to the rate of encounter between predators and prey, which is a random function of population density.  These simplifying assumptions underlie a very basic model of predator-prey dynamics, embodied in the Lotka-Volterra predator-prey equations.3
	If N = the number of prey and P = the number of predators, then in the absence of predators, prey should grow exponentially,

dN/dt = r1N,

where r1 is the prey intrinsic growth rate.  Without prey, the predator population will starve,

dP/dt = -r2P,

where -r2 is a measure of the predators' starvation rate.

	The chance of encounter between predator and prey is

encounter rate = (C1)(N)(P)

where C1 is a constant related to prey escape ability and the number of prey a predator takes per unit time.  C1 is often called the "functional response" of the predator; by giving it a constant value, we are assuming that the number of prey taken varies linearly with prey abundance.3
	Bringing the two species together and introducing the encounter rate into both equations, we have

dN/dt = r1N -C1NP

dP/dt = -r2P + C2NP

where C2 is a constant defining the conversion efficiency of prey into predators (called the predator's "numerical response").

	The behavior of this model at equilibrium can be analyzed by setting both

dN/dt = dP/dt = 0, then

P = r1/C1    and    N = r2/C2

These expressions imply that there is a constant number of predators (r1/C1) above which prey densities will decrease and below which they will increase.  Likewise, there is a constant number of prey (r2/C2) above which predator densities will increase and below which they will decrease.3
	You will notice that this model exhibits the peculiar "neutral stability" of a frictionless pendulum.  The Populus package includes a second predator-prey model (the "theta-logistic") which makes fewer simplifying assumptions, introducing nonlinear functional responses and density-dependent prey population growth.  You should find it instructive to compare the dynamic behavior of the two models.
4
References

	The Lotka-Volterra predator-prey model is discussed in most elementary ecology texts.  See for example,

Ricklefs, R. E. 1990. Ecology (3rd edn.) W. H. Freeman & Co. New York pp. 413-416.

Begon, M., Harper, J., and C. R. Townsend. 1986. Ecology: Individuals, Populations, and Communities. Sinauer Associates, Inc. Sunderland, MA. pp.350-353.
1. The simulation produces two different graphical outputs.  One presents population densities of both predator and prey over time, and the other gives a "phase-plane" diagram, plotting P vs. N.

2. For both the predators and prey, it is necessary to specify a starting population density (P0 and N0, with values up to 100,000), intrinsic rates of increase (r1 and r2, with values ranging between 0 and 5), and constants for the functional and numerical responses (C1 and C2, which can take values between 0 to 999).

3. The simulation can be made to run for any number of generations, up to 10,000.  Note however, that this model generates neutral oscillations and its full dynamic behavior is usually evident within a few dozen or a few hundred generations.
	The P,N vs T output screen graphs population densities of both predators and prey vs. Time.  The phase plane output graphs P vs N.  Stability analyses can be performed from this second display, either by setting a starting ratio of predators and prey (with the arrow keys), or by pressing "S" for an automatic stability analysis which will start runs from one or many different predator/prey ratios.
	The -logistic model has three parameters.  r and K have meanings identical to the simple logistic model of population growth; r is the per capita population growth rate when the population is very small, and K is the carrying capacity, or the density at which the population achieves zero population growth.  The  parameter determines whether density dependence acts primarily at low population densities ( small) or high population densities ( large).
	The type 1 functional response is simply a straight line with a slope of C1.  The shapes of the type 2 and 3 functional responses, when they are given by the formulas used here, depend on 2 parameters.4t14+3
Competing Predators

	Hassell (1978) predicates his discussion of multi-parasitoid systems on a sequential model with one predator (P) acting first, and then a second (Q) acting on the surviving prey.  A system with simultaneous attack by both parasitoids would be equivalent if one is the superior competitor, winning in all cases of multiple parasitism.  This scenario is illustrated by equations of the general form

Nt+1 = Nt f1(Pt) f2(Qt)

Pt+1 = Nt [1 - f1(Pt)]

Qt+1 = Nt f1(Pt) [1 - f2(Qt)]

where f1(Pt) and f2(Qt) are prey survival probabilities after the searching of predators P and Q, and N gives the number of prey.  Since the most interesting outcomes of this model are cases that permit the stable coexistence of all three parties, Hassell includes parasitoid aggregation similar to that in the non-random searching model (above), giving the recursions9Nt+1 = Nt {1+(a1Pt/k1)-k1} {1+(a2Qt/k2)-k2}

Pt+1 = Nt [1-{1+(a1Pt/k1)-k1}]

Qt+1 = Nt {1+(a1Pt/k1)-k1} [1-{1+(a2Qt/k2)-k2}]

Here k1 and k2 are the negative binomial dispersion coefficients describing the clumping of P and Q in patches of high host density (May 1978, May and Hassell 1981).  When k1 = k2 =  this formulation reduces to a three-species Nicholson-Bailey model.

	The stability analysis of this model given by May and Hassell (1981) shows that three-species equilibria are most likely when both parasitoid-host links are stabilizing; i.e., when k < 1 for both parasitoids.  Stable coexistence is also more likely if the inferior competitor (Q) has the higher searching efficiency.11References

Hassell, M. P. 1978. The Dynamics of Arthropod Predator-Prey Systems. Monographs in Population Biology, Princeton University Press. Princeton, NJ. pp. 147-157.

May, R. M. 1978. Host-parasitoid systems in patchy environments: a phenomenological model.  J. Anim. Ecol. 47:833-44.

May, R. M. and M. P. Hassell.  1981.  The dynamics of multiparasitoid-host interactions. Am. Nat. 117:234-261.Options
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	This simulation illustrates a three-level interaction with a hyperparasite attacking previously-parasitized hosts.
 
N = initial prey population density, which can be set from 0 to 1E10.

P = initial parasitoid population density, which can be set from 0 to 1E10.

Q = initial hyperparasitoid population density, which can be set from 0 to 1E10.

lambda = the intrinsic growth rate of the prey (a finite growth rate that can take values from 0 to 5 per generation).

a1, a2 = the "areas of discovery" of the parasitoids and hyperparasitoids, respectively.  They measure searching efficiency, and are interpretable as the probability that a given parasitoid or hyperparasitoid will discover a particular prey individual during its lifetime.  Values can range from 0 to 1; high values indicate that the parasitoids are extremely efficient searchers.

k1, k2 = the clumping parameters of the negative binomial parasitoid and hyperparasitoid distributions.  If k < 1, the parasitoids are strongly aggregated in patches of high host density.  In contrast, large values of k (> 8) imply that the parasitoid distributions approach Poisson randomness.
	In this simulation, two parasitoid species compete for a single prey type.

N = initial prey population density, which can be set from 0 to 1E10.

P, Q = initial densities of the two parasitoid populations, which can be set from 0 to 1E10.

lambda = the intrinsic growth rate of the prey (a finite growth rate that can take values from 0 to 5 per generation).

a1, a2 = the "areas of discovery" of the parasitoids.  They measure searching efficiency, and are interpretable as the probability that a given parasitoid will discover a particular prey individual during its lifetime.  Values can range from 0 to 1; high values indicate that the parasitoids are extremely efficient searchers.

k1, k2 = the clumping parameters of the negative binomial parasitoid distributions.  If k < 1, the parasitoids are strongly aggregated in patches of high host density.  In contrast, large values of k (> 8) imply that the parasitoid distributions approach Poisson randomness.
	In this simulation the predator is assumed to consume two competing prey species, taking individuals of each species as they are randomly encountered, or switching to concentrate on the more abundant of the two prey types.

X, Y = the initial population densities of the two prey species, which can be set from 0 to 1E10.  (X and Y replace the single N used in other models in this series).

P = initial predator population density, which can be set from 0 to 1E10.

lambda, lambda' = the intrinsic growth rates of the two prey species (finite growth rates that can vary from 0 to 5 per generation).

g, g' = constants.

,  = the familiar competition coefficients that scale the density-dependent interspecific competitive feedback.

m = the interference constant, interpretable as the slope of the log-linear decline in search efficiency with predator density.

s = the switching constant, which can vary from 0 to 1 to express the degree of prey-switching shown by the predator (switching version only).
	This simulation illustrates a predator-prey interaction in which the predator much exceed some minimum prey consumption threshold to mature and begin reproducing.

N = initial prey population density, which can be set from 0 to 1E10.

P = initial parasitoid population density, which can be set from 0 to 1E10.

K = the prey carrying capacity (density-dependent version only) ranging from 0 to 1E10.

r = the intrinsic growth rate of the prey (an instantaneous growth rate that can take values from -5 to 5 per generation).

a = the "area of discovery" of the parasitoids.  This is a measure of searching efficiency, interpretable as the probability that a given parasitoid will discover a particular prey individual during its lifetime.  It can range from 0 to 1; high values indicate that the parasitoids are extremely efficient searchers.

c = the parasitoid's "numerical response."  This is a constant indicating the parasitoid's efficiency in converting prey into offspring.  Often for parasitoids c = 1, while for predators that must consume multiple small prey per offspring, the value of c will be smaller.

 = the minimum threshold prey consumption required before predators begin to reproduce; it can take values from 0 to 1E10.
	Here we assume that searching parasitoids encounter not only hosts, but other parasitoid individuals as well.  Behavioral interactions between the parasitoids then waste searching time and reduce the area of discovery.  Populus contains two interference models.  In the first there is a log-linear decline of searching efficiency, a, with parasitoid density; in the second case this reduction in efficiency is curvilinear.

N = initial prey population density, which can be set from 0 to 1E10.

P = initial parasitoid population density, which can be set from 0 to 1E10.

lamda = the intrinsic growth rate of the prey (a finite growth rate that can take values from 0 to 5 per generation).

a' = the instantaneous search rate of the parasitoid, during the available search time, T.  Note that if T is one full generation, then the product a'T is equal to a, the "area of discovery" searching parameter used in the other models from Hassell's book.  Values can range from 0 to 1.

b = the instantaneous encounter rate between parasitoids, analogous to a', their rate of encounter with hosts.  Values can range from 0 to 1.

c = the parasitoid's "numerical response."  This is a constant indicating the parasitoid's efficiency in converting prey into offspring.  Often for parasitoids c = 1, while for predators that must consume multiple small prey per offspring, the value of c will be smaller.

T = the total search time available to an individual, which may take values from 0 to 1 full generation.

Tw = the potential search time wasted at each encounter between parasitoids (curvilinear model only).

Q = a scaling constant used in the linear model to specify the decline of search efficiency with increasing parasitoid density.

m = the interference constant, interpretable as the slope of the log-linear decline in search efficiency with parasitoid density.
	In this simulation, parasitoids may aggregate in patches of high host density.  Because this behavior gives the hosts a refuge at low densities, it is a potentially important factor in stabilizing the host-parasitoid interaction.

N = initial prey population density, which can be set from 0 to 1E10.

P = initial parasitoid population density, which can be set from 0 to 1E10.

lambda = the intrinsic growth rate of the prey (a finite growth rate that can take values from 0 to 5 per generation).

a = the "area of discovery" of the parasitoids.  This is a measure of searching efficiency, interpretable as the probability that a given parasitoid will discover a particular prey individual during its lifetime.  It can range from 0 to 1; high values indicate that the parasitoids are extremely efficient searchers.

k = the clumping parameter of the negative binomial parasitoid distribution.  If k < 1, the parasitoids are strongly aggregated in patches of high host density.  In contrast large values of k (> 8) imply that the parasitoid distribution approaches Poisson randomness.

K = the prey carrying capacity (density-dependent version only) ranging from 0 to 1E10.
	This set includes three different functional response simulations.  In the Type I response, prey consumption rises linearly to a plateau as prey density increases.  Consumption of a Type II parasitoid rises asymptotically to saturation, and in the Type III case, prey consumption is a sigmoidally increasing function of prey density.

N = initial prey population density, which can be set from 0 to 1E10.

P = initial parasitoid population density, which can be set from 0 to 1E10.

lambda = the intrinsic growth rate of the prey (a finite growth rate that can take values from 0 to 5 per generation).

a' = the instantaneous search rate of the parasitoid, during the available search time, T.  Note that if T is one full generation, then the product a'T is equal to "a," the "area of discovery" used in the Nicholson-Bailey model.

c = the parasitoid's "numerical response."  This is a constant indicating the parasitoid's efficiency in converting prey into offspring.  Often for parasitoids c = 1, while for predators that must consume multiple small prey per offspring, the value of c will be smaller.

T = the total search time available to an individual, which may take values from 0 to 1 full generation.

Th = the parasitoid's handling time, the time required for it to oviposit on each host.  As host density rises, handling time consumes an increasing proportion of the parasitoid's time budget.

b = a constant used to scale the change of a' with prey population size in the Type III model.
	Populus contains both density-independent and density-dependent versions of the Nicholson-Bailey model.  In the DI version, prey population density is limited only by the interaction with parasitoids, but the DD version imposes an environmental carrying capacity above which prey populations would fall even in the absence of parasitism.  A time trajectory of prey and parasitoid populations (N,P vs. t) and a phase plane (P vs. N) can be viewed in either simulation.

N = initial prey population density, which can be set from 0 to 1E10.

P = initial parasitoid population density, which can be set from 0 to 1E10.

K = the prey carrying capacity (density-dependent version only) ranging from 0 to 1E10.

lamda = the intrinsic growth rate of the prey (a finite growth rate used in the DI model that can take values from 0 to 5 per generation).

r = the intrinsic growth rate of the prey (an instantaneous growth rate used in the DD version that can take values from -5 to 5 per generation).  Note that r = ln(lambda)

a = the "area of discovery" of the parasitoids.  This is a measure of searching efficiency, interpretable as the probability that a given parasitoid will discover a particular prey individual during its lifetime.  It can range from 0 to 1; high values indicate that the parasitoids are extremely efficient searchers.8t+314
Lotka-Volterra Competition

	Density-dependent growth models like the logistic equation simulate an intraspecific competitive process; resources become limiting as the population increases, and the per capita growth rate declines.  In this program, an additional term is added to the logistic to represent interspecific density dependent effects, and a pair of the resulting expressions comprise the "Lotka-Volterra competition equations," which provide a simple and historically important vehicle for thinking about competitive interactions.3
	In the Lotka-Volterra equations, densities of both species are subtracted from the carrying capacity to give a density-dependent feedback term, and the number of interspecific competitors is weighted by a term called the competition coefficient which varies with the species' similarity in resource requirements.  Thus

 ndN1n       K1n - N1n - N2  
     = r1               
nN1ndt            K1      

 ndN2n       K2n - N2n - N1  
     = r2               
nN2ndt            K2      

where N1 represents the density of species 1, K1 is the environmental carrying capacity of species 1, r1 is its intrinsic rate of increase, and  is the competition coefficient, a proportionality constant defining the amount of K1 used by every individual of species 2.  In the second expression,  is an analogous coefficient weighting the effect of each species 1 individual on K2.4
	Although we have no closed form solution for these equations, we can still gain interesting insights about their dynamics near the equilibrium when dN1/dt = dN2/dt = 0.  Trivial equilibria occur when r or N = 0; a more interesting case occurs when

N1 = K1 - N2 and N2 = K2 - N1

These are the equations for straight lines in N2 vs N1 coordinate space, the "Zero-Net-Growth Isoclines" which specify equilibrial density ratios.

References

MacArthur, R. H. 1972. Geographical Ecology. Harper & Row. New York. pp. 21-58.

Hutchinson, G. E. 1978. An Introduction to Population Ecology. Yale University Press.  New Haven.  pp. 117-151.

Krebs, C. J. 1985. Ecology: The Experimental Analysis of Distribution and  Abundance, Third Edition. Harper & Row. New York. pp. 235-272.t+314
Arthropod Predator-Prey Dynamics

	Empirical research with arthropod predator-prey systems has played a central role in the development of population biology because the animals are often amenable to laboratory experimentation and their simple life histories are compatible with idealized theoretical models.  This section of the Populus package contains a set of simulations taken from M. P. Hassell's Dynamics of Arthropod Predator-Prey Systems.  The models illustrate different components of predator-prey interaction and permit analyses of their dynamic consequences.  To conform with the discrete seasonality of most arthropods, the simulations are phrased as finite recursion equations of the basic form

Nt+1 = Nt f(Nt,Pt)

Pt+1 = cNt[1 - f(Nt,Pt)],

where Nt, Nt+1, and Pt, Pt+1 give the prey and predator population densities in successive generations, respectively,  is the geometric growth rate of the prey (which can remain constant or change as a function of prey density), and c is the number of predators produced for each prey individual attacked (the "numerical response" of the predator).  The function f(Nt,Pt) gives prey survival with respect to predator and prey densities and can be varied to reflect a variety of different predator foraging behaviors.5
	The biology of arthropod predators spans a wide range of complexity.  Among the simplest are hymenopteran and dipteran parasitoids that seek hosts and lay one or more eggs whose subsequent development kills and consumes the victim.  In contrast to predators where immatures and both adult sexes must locate and consume prey, only the adult female parasitoid searches; moreover, the number of progeny that she produces is likely to be a simple function of the number of hosts attacked.  The models in this group are framed without the complications of age structure or multiple foraging modes, consistent with the simple biology of parasitoids and their hosts.  Each model is preceded by a separate narrative introduction that describes its biology, mathematics, and dynamic behavior.  The models are ordered from simple to complex and their introductory narratives are sequential.


References

Hassell, M. P. 1978. The Dynamics of Arthropod Predator-Prey Systems. Monographs in Population Biology, Princeton University Press. Princeton, NJ.10t+314
Multiple Niche Polymorphism

	There are several means by which selection can protect and maintain variation at a polymorphic locus, including constant selection with a heterozygote advantage, selection favoring alternate alleles in males and females, and selection which varies with allelic frequency, favoring rare alleles.  This set of simulations is based on models by Levene (1953) and Dempster (1955); they assume that the environment is subdivided into patches of different type, and that alternate alleles at a polymorphic locus are favored in the different habitats.3	The two models invoke different density-dependent regulation processes.  Dempster assumes that a fixed proportion (ci) of the zygotes that result from panmictic mating settle in each patch type and then survive as dictated by their habitat-specific absolute fitnesses.  Survivors from all patch types then join to produce the next generation of zygotes.  The fitness coefficients are assumed to be constant and independent of local variations in allelic frequency.  As a result, polymorphism is only maintained if the weighted average of fitnesses in the respective habitats reveals a heterozygote advantage.  The Dempster recursion is

         ptw11 + qtw12     
pt+1 =               pt
                 w              

where

         T           
wij =  chwij,h
         h=1          

ch is the proportion of the total habitat of type h, and T is the total number of habitat types.8	In Levene's model, zygotes produced by population-wide random mating enter each patch type in proportion to their frequencies and the three genotypes survive with different relative fitnesses in the two habitats.  Each patch type contributes a proportion of the next-generation zygotes which is fixed by the composition of the local environment.  The recursion is

        T            ptw11,h + qtw12,h       
pt+1 =  chpt                              
         h=1     pt2w11,h + 2ptqtw12,h + qt2w22,h

The model is frequency-dependent, because each allele suffers reduced competition in the preferred patches when it is rare.  Relative fitnesses and the relative proportions of the patch-types determine whether this frequency-dependence suffices to maintain a segregating polymorphism.  In general the harmonic average fitness of heterozygotes must be higher than that of homozygotes to sustain a polymorphism (Cannings 1971).8References

Cannings, C. 1971. Natural selection at a multiallelic autosomal locus with multiple niches. J. Genetics 60:255-59.

Dempster, E. R. 1955. Maintenance of genetic heterozygosity.  Cold Spring Harbor Symposia in Quantitative Biology 20:25-32.

Levene, H. 1953. Genetic equilibrium when more than one ecological niche is available. American Naturalist 87:311-13.

Maynard Smith, J. 1966. Sympatric speciation. American Naturalist 100:637-650.

Roughgarden, J. 1979. Theory of Population Genetics and Evolutionary Ecology: An Introduction. Macmillan Publishing Co., Inc. New York. pp. 231-235.
	These simulations assume that the environment is subdivided into patches of different type, and that alternate alleles at a polymorphic locus are favored in the different habitats.

1. p sets the frequency of "A" alleles, and q = 1-p the frequency of "a" alleles.

2. c1, c2 ...cn are the relative frequencies of the two habitats in the environment; c1 + c2 + ...+ cn = 1.

3. The selection regime is specified by entering relative fitnesses for each of the three genotypes in each habitat.  The program will normalize these values to relative fitnesses ranging from 0 to 1 if you supply values outside that range.

4. The simulation can be set to run from 1 to 2000 generations before graphing results.

5. Runs are initiated by default from 6 different starting population-wide gene frequencies.  Alternatively, you may set 1 starting frequency of special interest.
Male Trait frequency, t2, and the frequency of females preferring the secondary male trait, p2, can take initial values from 0 to 1.

The a1 and a2 parameters set the strength of female preference for the respective male traits.  For example, p2 females will mate with t2 males a2 times more frequently than with t1 males.  a1 sets the preference of p1 females for t1 males; if a1 = 1, then p1 females mate randomly.  Permissible values range from 1 to 1E4.

D0 sets the initial linkage disequilibrium, and can take values between -0.25 and 0.25.

s sets the intensity of viability selection against the male type with the secondary sexual character.  It can take values from 0 to 1.

r sets the probability of recombination in gametes between the t and p loci and can range from 0 to 0.5.pkBGI Device Driver (EP24PIN) 2.00 - Feb 24 1992
Copyright (c) 1990 Ryle Design  (517) 773-0587
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V^Nutite uy
؃ y
ۃ   W33;rw;r+@[ tƋ t؃ _^]  t3Vt^                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      @                                                                                                                  @8px<iZ-KB$$B3333     "  "         HIJKLMNO"Epson LQ 180x180 dpi Landscape    "Epson LQ 360x360 dpi Landscape    "Epson LQ 180x360 dpi Landscape    "Epson LQ 180x360 dpi Landscape    "Epson LQ 180x180 dpi Landscape    "Epson LQ 360x360 dpi Landscape         '' N N''"Epson LQ 180x180 dpi Portrait     :''     3  @   *'  
               "Epson LQ 360x360 dpi Portrait     >=''?    +  @   *(  
   *(  
J"Epson LQ 360x180 dpi Portrait     >='?     3  @   *(  
               "Epson LQ 360x180 dpi Portrait     5'     3  @   *(  
               "Epson LQ 180x180 dpi Portrait     	5''	     3  @   *'  
               "Epson LQ 360x360 dpi Portrait     5''    +  @   *(  
   *(  
J @'' BGI.$$$ BGICAN.$$$ BGICAN.$$$ BGI.$$$ BGI.$$$ bgi.$$$ BGI.$$$ BGI.$$$ BGICAN.$$$ BGICAN.$$$ BGI.$$$ BGI.$$$ BGI.$$$ Insufficient memory - press a key >> File open error - press a key >> File write error (disk full?) press a key >> No paper - fix and press a key, <ESC> aborts >> Printer error - fix and press a key, <ESC> quits >> Printer offline - fix and press a key, <ESC> quits >>                                                        BGISTAT.$$$ Printer :  Plot    :   objects,  K buffer,   pass.  passes. Output  :  COM LPT 2 1 Pass    Pass   Pass  :  Rasterizing |          |            Printing (<ESC> exits) |          | * BGICNF.$$$ BGICNF.$$$ BGIPLT.$$$ BGIPLT.$$$ pkBGI Device Driver (EP9PIN) 2.00 - Feb 24 1992
Copyright (c) 1990 Ryle Design  (517) 773-0587
    |4                     |4 EP9PIN               U ]CB      p
5}
Q	K 	   	 j'                                    V~-7tw-Y...u3  -  5~-w-Y-㉗((-㋇((t-3u
~-G/;-w~-G7 P-3[--)-~-G7 ;-vz~-G/36-ȋG/36-tA*-  -H,-.-  6-  ^UVW-H;.-u.-.-~-W/+ЉV-FF
 3FF3tFF  F  3
t*<u#=(P#Y
t$-y /P#Y3~-G7 FF;Fr PYF  F~-7t w-(($F~ tbF$;(F<($-y %/P#Yд u}:(P#Yд ujv((!д uH$-y -/P"Yд t*F  j~-w-(($F (~-w-((#;VvVF~-G7 ;w˃~ u 3;Nrt.& FVEğ(ً-&t.يF& F F~-G7 Ћ~-G7 ;wA;Ns덊F$;(F<($-y V%/P!Yд u:(P!Yд uv6.6.O д u$-y V-/PT!Yд uF~-G7 ;Fv~-G7 ;~sh.-.-;-s),-@*-,--,-~-G/;,-wH,--6-~
t~	 P#YG
r _^]!8( u["-   P"-PY P>-PJ P@-P; PB-P, PD-P^-`- -˸ P~Y>8( t1PY~1P3PYY
t3PSY1PY1P1P3PYYд u
 PY/3PY2PSY8( uI( H( u; 8( u="-  P"-PH PN-P9 PF-P*^-`- U~-G3FG1F>N- u^N1]8( uI"-  P"-P P>-P P@-P>-&-@-(-^-`- U,@-FF8( uF"-  P"-Ph PN-PY>N- u PvvA^-`- ]ò U>9( tF@ F ^&0
2P PYYF u PY4`-  ^-  8( \-  Z-  -  ]cN-t= uj= ugj8( @( -  -  -   -  5( 6( ?( ( ( R- t5(R-  t@(R-@ t6(R- t?(R- t7(7( R-% $-y .~->5( tB~-G%1G#1G%1G#1G)1G'1$-㋇.1$-# -1~-G#1G%1G#1G%1G'1G)1G+1~-'z1 111~-G111v-  x-  1z-1|- PY1 '8( uL"-  P"-P  PN-P P@-P PB-P^-`- 8( uI"-  P"-P P>-P P@-P>-&-@-(-^-`- ,>- t78( u."-  P"-Pi P@-PZ^-`- 8( t >-;B-vB-B->-@-;D-vD-D-@-"-@  P"-P P>-P P@-P PB-P PD-P^-`- ø P6
Y>8( t
2Pn
Yi2P3P
YY
t3P
Y(2PK
Y32PC
Y;;2P3PT
YYд u	 PlV3PYC2P
Y8( u~8( u["- @ P"-P P>-P P@-P
 PB-P
 PD-P
^-`- i8( uL"-   P"-P
 P>-P
 P@-P
 PT-P
^-`- û  û  8( u["-  P"-PF
 P>-P7
 P@-P(
 PB-P
 PD-P

^-`- VWY->- u PY)  3 PR-3ɋ6~-D-3bRP\-Z- PKY PCY>( t 'P6:-YY u :- 3 <-  ?<- PY PY  PY><-u~-9 PY txG;>-stt7( u=(PfYt~-A PWY t>( t Pt
Y_^V( u<3((YYF--;w~-7t6.6.YYd-  b-  ^UVWlOF  3
 PR6`-6^-kVFF  F  36`-6^-3
 PRWtFV 2(3(4( &-  (-  V-X- 61613PP PFP
 u
F FFV FV;Fu;VuG PaYF  F  FF .;Ft.g PFP
 PFP
 PFP	 PFP	 PFP	FPvvC  PFP	 PFP	 PX-P	F ؃v}.V-mV-dV-x[V-RFV-I PFPE	~t	F4(+4( P- P2(P	 P3(P		 PFP PFP PFP PFPvvvv$ PFP PFP>5( t*&-;v-sF;v-sv&-;z-v6F;z-v-a(-;x-sF;x-sL(-;|-vF;|-v7vv6(-6&-F&-F(- PFP PFP PFP PFP>5( t*F;v-sF;v-sF;z-v6F;z-v-F;x-sF;x-sF;|-vF;|-v}vvvv PFPg PFPX PFPI PFP:>5( t*F;v-sF;v-sF;z-v6F;z-v- F;x-sF;x-s F;|-vF;|-v vvvv|9 PFP PFP PFP PFP>5( t,&-;v-sFF;v-rj&-;z-v8FF;z-v,S(-;x-sF+F;x-r>(-;|-vF+F;|-w)vvvvv  PFPF$1(~ ut
t	 PYF
|_^]<ENW`      @       @GiUVWv&-F~(-FHPHP@PF@P/WVWv vVWV vvvV Wvvv ~ u F FF+FPFPvVx +FPFPF+FPFPY ~
uJWV+FPFP? +FPFFP+FPFP  Wv+FPFFP	 _^]UVW~V^
;sϋыN^ً+ǉF+FF~ }!;F~F F@FF.F F;F~F @FFF F@FFػ F ~-G5FF   V-Ft2(1(PvWi~unFF;FunF  ~ uF N^KwB.YGv~3F+Fv~$G+vNv~GvG+v~N+v~ ~e_^]JJJJ,<UVWvF;v-sv-FF;z-vz-F;6x-s6x-F
;|-v|-F
>4(s6>4( u1(3( F~
FPVWcG;~vF;v
vK~-5uf\% FN F~;4( F؊- Ft3(1(PVWn~ uF G;~vF;v
v ~-G5F^3R3XvFF3R3Xv VF3vV3vV FFFF~K4( F؊- Ft3(1(PVWT FF;FuF  n~ uF G;~vFF;FuF  F~uF  F;v
wx_^]UVWVN;v-rg;z-wa;x-r[;|-wU>5( tNʋ~-W#+V+66-~ tğ(ڊ-&ğ(ڊ-& _^]U~u -F~ t+>4- t -  -F~ t
-  ^>!]U^^^VFشA!]UVWv~^~uF<4-   uF=0-  2-  A( F=3VF؊f!Fs 1uF -"uF -^    B!F-_^]UVWFVFV3;~s82-;0-u>A( t & F t^62-J(&2-G;~rȰ_^]Ã>- t-33Ҹ B!0-  2-  UVW~FVFV3;s8>4-@u  F~ u PY ^& 4-(4-F;rȰ_^]UV^FJ(-@ VFش?!Fr~@t 3F0-2-  >0- ut ^]U^F( -4-VFش@!Fr	F;4-t
4-   4-  ]U( uFFFI!]UVWv( u#ƱF tF^H!Fr;V8d-b-  w&r;8-s-F>->b-6b-d- V33_^]UFu F]ò
!
!ô Ê>C(  D(6E(UF >( t4 PFPF t  PzY'PY1F >?( u 6@( FfFt
 PYFFF u~ tT~ tZ F @(fF t
 PYFFt Ft FF u~ uF~ t]Ê>C( 6E(D( UVF F@BF3vvvv
F~ ~ u3Fu+F0FV!FPRVFY[f)FVF 3
 PRvv
VFFrF0FV!^]UF^F^&FV!F^&? u]UVWN'F33F~ ~ u#FuF0FV!F+F 
 3Gr0FV!_^]U>6( tF8( 
D(FE(F>H( u>I( tf-D(g-E(m
>D( tK^vv.qK2PY8H( u8(됸p2PYH( u8(q2PYH( u8(R22"3PYF~u+>H( u>I( tf-D(g-E(D( X3PNYFD(FE(]ð3P PYY
 P8(P P^-P PZ-P P-Pq PYUVWv>6( t$Kv."B(>C(FF%0 =0 u9(>H(t F  ^F(f-F~|
p-F(
r-G(i-m-G(k-G(o-G(q-G(s-G(u-G(g-㊇f-D(㊇g-E(I(ZI( L>D( v03PY'BRYH( t60H( t >D( vE(>E(v/E( E(  +F3;~}	G;~|G(E(G(G(m-k-G(o-G(q-G(s-G(u-G(g-l- j- n- p-
r-
I(㊇f-D(㊇g-E(_3PY6`-6^-aYY3PY6\-6Z-LYY3PY6-Y>-u3 3 ㊇f-D(㊇g-E(3PY>( t'i>?( t33PY>@(u3I3D㊇f-D(㊇g-E(><-
s3><-ds33P5Y6<-^Y4P%Y ㊇f-D(㊇g-E([4PYD(
&㊇f-D(㊇g-E(3*4PYD(t-E(u-)㊇f-D(㊇g-E(N4PYf-_^] <!!!!UVP4P3PYYд u F H؃v .V# PFPX3 PFPEF'FF ;wFƄ' (:-   PFP PFP P8-PFV--(d-  b-  @ PFPF ~-G5# PF(P PG(PH( PFP
t3PrY[4PY^]V"""#Uf4P3PYY F~ uz PFP0~ u( NF F:Fs( PFPF V؈'FF:FrFF Ƈ' (:- 3PYq4PY]UVW~-G-F-FG8FF  F;Fs0^㋇((FVVW~NF_^FF;Fr_^]UVW~>( t2WvvF t  P,Y'PkYF >?( t_3;r F B^& F@( FfFt
 PoYFFF u	~ us~ tF;rz3;rqF ]^& F F @(fF t
 PYFFt Ft FF u~ uF~ tF;s됰_^]UVvF~ t3FF F:Fs&F ؊ PWYF u 
FF:Frڰ^]ã>-@-B-D->-% N->-% F-@-% P-@-% H-B-% R-B-% J-D-% T-D-% L-UVWNv~..*-F..,-.HЀ>5( tG;NsFv-v-;vz->z-~-G#;F
sF
|-;6|-vx-  O6x-I~-9#sG#z->z-;z-vv-  v-;vsFx-6x-9V
v|-F
|-_^]UW~~ON Nu3F@_]YQ3YQ YQ YQ UVWF
V^Nutite uy
؃ y
ۃ   W33;rw;r+@[ tƋ t؃ _^]  t3Vt^                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     @                                                                                                                  @8px<iZ-KB$$B3333     "  "         -./01234"Eps/Pro 72x60 dpi Landscape       "Eps/Pro 144x120 dpi Landscape     "Eps/Pro 216x240 dpi Landscape     "Eps/Pro 72x60 dpi Landscape       "Eps/Pro 144x120 dpi Landscape     "Eps/Pro 216x240 dpi Landscape            g+  g+"Eps/Pro 60x72 dpi Portrait        /'.Z     3    2     K     
                                     "Eps/Pro 120x144 dpi Portrait      7'.     3    2     L     

    L     
J
                  "Eps/Pro 240x216 dpi Portrait      ~o;'(#    3    2     Z     
     Z     
     Z     
J
  "Eps/Pro 60x72 dpi Portrait        .5'./Z     3    2     K     
                                     "Eps/Pro 120x144 dpi Portrait      ^5'._     3    2     L     

    L     
J
                  "Eps/Pro 240x216 dpi Portrait      o5'(#    3    2     Z     
     Z     
     Z     
J
   @'' BGI.$$$ BGICAN.$$$ BGICAN.$$$ BGI.$$$ BGI.$$$ bgi.$$$ BGI.$$$ BGI.$$$ BGICAN.$$$ BGICAN.$$$ BGI.$$$ BGI.$$$ BGI.$$$ Insufficient memory - press a key >> File open error - press a key >> File write error (disk full?) press a key >> No paper - fix and press a key, <ESC> aborts >> Printer error - fix and press a key, <ESC> quits >> Printer offline - fix and press a key, <ESC> quits >>                                                        BGISTAT.$$$ Printer :  Plot    :   objects,  K buffer,   pass.  passes. Output  :  COM LPT 2 1 Pass    Pass   Pass  :  Rasterizing |          |            Printing (<ESC> exits) |          | * BGICNF.$$$ BGICNF.$$$ BGIPLT.$$$ BGIPLT.$$$ pkBGI Device Driver (HPLJ2) 2.00 - Feb 24 1992
Copyright (c) 1990 Ryle Design  (517) 773-0587
    R8                      R8 HPLJ2                U ]CB      Ç!Ly	Mb
    


 f
>			1                                   U Fd 3F~ tF0و0AFd )FF
 3F~ utF0و0AF
 )FF0و000P#Y F]UV/00 /  0G4FG2F1vY/㉗''/㋇''t/3u/;Fs>/r>/ u3C /꣺/0G436/ȋG436/tA/  /H//  /  ^]UVW/H;/u/./04+>/ǻ
 3F3tFF  F  3B
t%<uE'PF"Y
t0K P5"Y3oFF;Fr PcYF  F% = u㋇''0W2㋇''FV0w2vvB#F0G2;Fu40X P!Yд uo0w2vvh д unQ~ tR0Px!Yд u8voYд u&0PS!Yд uvvv д u0P&!Yд uF;s//;/s)/@////0G4;/wH///~
tv	 PYF
r _^]!@' u[/   P/PF P/P7 P/P( P/P P/P
/ 0 0˸ PkY>@' t5PY~5P3PYY
t3P@Y5PY5Pw5P3PYYд u
 PY/3PY5P@Y@' uQ' P' u;@' u=/  P/P5 P/P& P/P/ 0 U0G8FG6F>/ u^N5]y@' uI/  P/P P/P P/P///// 0 U/FF@' uF/  P/PU P/PF>/ u Pvv./ 0 ]ò U>A' tF@ F ^&05P PYYF u PY! 0  /  @' /  /  /  ]T/t= uO= uLO@' H' /  /  /  /  =' >' G' ' ' / t='/  tH'/@ t>'/ tG'/% /r 10>=' t;0G*5G(5G*5G(5G.5G,5G05/( 010G(5G*5G(5G*5G,5G.5G050&5 5550G6550  0  5050 PY5 &@' uL/  P/P( P/P P/P
 P/P/ 0 @' uI/  P/P P/P P/P///// 0 8/ t7@' u./  P/Pq P/Pb/ 0 @' t /;/v////;/v////@  P/P P/P P/P P/P P/P/ 0 ø P>
Y>@' t
5Pv
Yi5P3P
YY
t3P
Y5PS
Y	6PK
Y;6P3P\
YYд u	 PtV3PY6P
Y@' u@' u[/ @ P/P P/P P/P  P/P
 P/P
/ 0 u@' uL/   P/P
 P/P
 P/P
 P/P
/ 0 û  û  
@' u[/  P/PN
 P/P?
 P/P0
 P/P!
 P/P
/ 0 VW/>/ u PY1  3 PR/3ɋ60D23nRP// PSY PKY>' t &P6/YY u / 3 /  ?/ PY PY  PY>/u0> PY tG;>/stt?' uE'PrYt0K PcY t>' t P|
Y_^V' u#3''YYF/;w0  0  ^UVWpF  3
 PR6 06/VFF  F  36 06/3
 PR|tFV :';'<' /  /  // 65653PP4 PFP
 u
F FFV FV;Fu;VuG PYF  F  FF .;Ft.g PFP4
 PFP%
 PFP
 PFP
 PFP	FPvvC  PFP	 PFP	 P/P	F ؃v}./m/d/x[/RF/I PFPf	~t	F<'+<' P0 P:'P9	 P;'P*	 PFP	 PFP		 PFP PFPvvvv$ PFP PFP>=' t*/;0sF;0sv/;0v6F;0v-a/;0sF;0sL/;0vF;0v7vv6/6/F/F/ PFP1 PFP" PFP PFP>=' t*F;0sF;0sF;0v6F;0v-F;0sF;0sF;0vF;0v}vvvv PFP PFPy PFPj PFP[>=' t*F;0sF;0sF;0v6F;0v- F;0sF;0s F;0vF;0v vvvv|9 PFP PFP PFP PFP>=' t,/;0sFF;0rj/;0v8FF;0v,S/;0sF+F;0r>/;0vF+F;0w)vvvvv  PFP4F$9'~ ut
t	 P'YF
|_^](1:C      @       @~
~*yL

UVWv/F~/FHPHP@PF@P/WVWv vVWV vvvV Wvvv ~ u F FF+FPFPvVx +FPFPF+FPFPY ~
uJWV+FPFP? +FPFFP+FPFP  Wv+FPFFP	 _^]UVW~V^
;sϋыN^ً+ǉF+FF~ }!;F~F F@FF.F F;F~F @FFF F@FFػ F 0G:FF   /Ft:'9'PvWi~unFF;FunF  ~ uF N^KwB.<Gv~3F+Fv~$G+vNv~GvG+v~N+v~ ~e_^]----UVWvF;0s0FF;0v0F;60s60F
;0v0F
><'s6><' u9';' F~
FPVWcG;~vF;v
vK0:uf\% FN F~;<' F؊ 0 Ft;'9'PVWn~ uF G;~vF;v
v 0G:F^3R3XvFF3R3Xv VF3vV3vV FFFF~K<' F؊ 0 Ft;'9'PVWT FF;FuF  n~ uF G;~vFF;FuF  F~uF  F;v
wx_^]UVWVN;0s ;0v};0st;0wm>=' tNʋ0W(+V+/% = u0w2 3Ȁ~ tğ'/& ğ'/&  _^]U~u/F~ t+>/ t/  /F~ t
/  ^>!]U^^^VFشA!]UVWv~^~uF</   uF=/  /  I' F=3VF؊f!Fs 1uF/"uF/^    B!F/_^]UVWFVFV3;~s8/;/u>I' t & F t^6/R'&/G;~rȰ_^]Ã>/ t/33Ҹ B!/  /  UVW~FVFV3;s8>/@u  F~ u PY ^& /'/F;rȰ_^]UV^FR'/@ VFش?!Fr~@t 3F//  >/ ut ^]U^F'//VFش@!Fr	F;/t
/   /  ]U' uFFFI!]UVWv' u#ƱF tF^H!Fr;V800  w&r;/s/F>/>0600 V33_^]UFu F]ò
!
!ô Ê>K'  L'6M'UF >' t4 PFPF t  PzY&PY1F >G' u 6H' FfFt
 PYFFF u~ tT~ tZ F H'fF t
 PYFFt Ft FF u~ uF~ t]Ê>K' 6M'L' UVF F@BF3vvvv
F~ ~ u3Fu+F0FV!FPRVFY[j)FVF 3
 PRvv
VFFrF0FV!^]UF^F^&FV!F^&? u]UVWN'F33F~ ~ u#FuF0FV!F+F 
 3Gr0FV!_^]U>>' tF@' 
L'FM'F>P' u>Q' t0L'0M'm
>L' tK^vv.u!6PY8P' u@'됸F6PYP' u@'qg6PYP' u@'R666PYF~u+>P' u>Q' t0L'0M'L' .7PNYFL'FM']ôe7P PYY
 P@'P P/P P/P P/Pq PYUVWv>>' t$Kv.!J'>K'FF%0 =0 uA'>P't F  ^N'0F~|
0N'
0O'	0
0O'0O'0O'0O'0O'0O'0㊇0L'㊇0M'Q'ZQ' L>L' v0q7PY&BRYP' t60P' t >L' vM'>M'v/M' M'  +F3;~}	G;~|O'M'O'O'
00O'0O'0O'0O'0O'00 
0 0 0
0
Q'㊇0L'㊇0M'_|7PY6 06/aYY7PY6/6/LYY7PY6/Y>/u7 7 ㊇0L'㊇0M'7PY>' t&i>G' t77PY>H'u7I7D㊇0L'㊇0M'>/
s7>/ds77P5Y6/^Y7P%Y ㊇0L'㊇0M'[7PYL'
&㊇0L'㊇0M'3 8PYL'0M'0)㊇0L'㊇0M'$8PY0_^]@    UV&8P3PYYд u F H؃v .Z" PFPX3 PFPEF&FF ;wFƄ& '/   PFP PFP P/PFV//'0  0  @ PFPF 0G:# PN'P PO'PP' PFP
t3PrY18PY^]Z!!!"U<8P3PYY F~ uz PFP0~ u' NF F:Fs( PFPF V؈&FF:FrFF Ƈ& '/ 3PYG8PY]UVW0G2F/FG=FF  F;Fs0^㋇''FVVW~NF_^FF;Fr_^]UVW~>' t2WvvF t  P(Y&PgYF >G' t_3;r F B^& FH' FfFt
 PkYFFF u	~ us~ tF;rz3;rqF ]^& F F H'fF t
 PYFFt Ft FF u~ uF~ tF;s됰_^]UVvF~ t3FF F:Fs&F ؊ PSYF u 
FF:Frڰ^]ã/////% //% //% //% //% //% //% //% /UVWNv~1./F1./1HЀ>=' tG;NsF00;v0>00G(;F
sF
0;60v0  O60I09(sG(0>0;0v0  0;vsF0609V
v0F
0_^]UW~~ON Nu3F@_]YQ3YQ YQ YQ UVWF
V^Nutite uy
؃ y
ۃ   W33;rw;r+@[ tƋ t؃ _^]  t3Vt^                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             @                                                                                                                  @8px<iZ-KB$$B3333     "  "         DEFGHIJK*p0X*p+1Y43214321*bW    %HP LJ/DJ 75 dpi Letter Landscape       %HP LJ/DJ 100 dpi Letter Landscape      %HP LJ/DJ 150 dpi Letter Landscape      %HP LJ/DJ 300 dpi Letter Landscape      %HP LJ/DJ 75 dpi Legal Landscape        %HP LJ/DJ 100 dpi Legal Landscape       %HP LJ/DJ 150 dpi Legal Landscape       %HP LJ/DJ 300 dpi Legal Landscape        %HP LJ/DJ 75 dpi Letter Portrait        W@''K     *t75R*r0a 	*rB&l0H   *b75W                   %HP LJ/DJ 100 dpi Letter Portrait       @''d     *t100R*r0a	*rB&l0H   *b100W                  %HP LJ/DJ 150 dpi Letter Portrait       @''     *t150R*r0a	*rB&l0H   *b150W                  %HP LJ/DJ 300 dpi Letter Portrait       _	@'',    *t300R*r0a	*rB&l0H   *b300W                  %HP LJ/DJ 75 dpi Legal Portrait         W@2'K     *t75R*r0a 	*rB&l0H   *b75W                   %HP LJ/DJ 100 dpi Legal Portrait        @2'd     *t100R*r0a	*rB&l0H   *b100W                  %HP LJ/DJ 150 dpi Legal Portrait        @2'     *t150R*r0a	*rB&l0H   *b150W                  %HP LJ/DJ 300 dpi Legal Portrait        _	;>e',<    *t300R*r0a	*rB&l0H   *b300W                   WW@'' BGI.$$$ BGICAN.$$$ BGICAN.$$$ BGI.$$$ BGI.$$$ bgi.$$$ BGI.$$$ BGI.$$$ BGICAN.$$$ BGICAN.$$$ BGI.$$$ BGI.$$$ BGI.$$$ Insufficient memory - press a key >> File open error - press a key >> File write error (disk full?) press a key >> No paper - fix and press a key, <ESC> aborts >> Printer error - fix and press a key, <ESC> quits >> Printer offline - fix and press a key, <ESC> quits >>                                                        BGISTAT.$$$ Printer :  Plot    :   objects,  K buffer,   pass.  passes. Output  :  COM LPT 2 1 Pass    Pass   Pass  :  Rasterizing |          |            Printing (<ESC> exits) |          | * BGICNF.$$$ BGICNF.$$$ BGIPLT.$$$ BGIPLT.$$$ pkBGI Device Driver (HPPJ) 2.00 - Feb 24 1992
Copyright (c) 1990 Ryle Design  (517) 773-0587
    I:                       I: HPPJ                 U ]CB      d%
 =   *K


 `a	T	S	U	t                                   V 3  53w-Y3㉗''3㋇''t33u3G/;3v>3r>3 u383G/363ȋG/363tA3  3H33  3  ^UVW~3H;3u3.33/+>3ǻ
 3F3tFF  F  31
t*?<u##'P"Y
t3 65P"Y3YFF;Fr PYF  F37u@3 H5Px"Yд t3w-'',!д t 떋㋇''FV3G7 PG-3[FF  \3G7 H;Fv
3 H53 l5P!YЊ´ u-vvv д uFFF3G7 ;FwF;s33;~3s)3@33333G/;3wH333~
tv	 PYF
r _^]!' u[3   P3P P3P P3P P3P P3P33 N4˸ PY>' t7P:Y~7P3PMYY
t3PY7PY7Pf7P3P YYд u
 P8Yf/3PY7PY' uF/' .' u; ' u=3  P3P P3P P3P33 U7 3G3FG1F>3 u^N7] ' uI3  P3PQ P3PB P3P3333333 U3FF' uF3  P3P P3P>3 u Pvv33 ]ò U>' tF@ F ^&07P PsYYF u PY3  3  ' 3  3  ~3  ]3t= uc= u`c' &' ~3  3  3  3  ' ' %' ' ' 3 t'3  t&'3@ t'3 t%'3 t'' 3% 3 43>' t;3G%7G#7G%7G#7G)7G'7G+73# ^413G#7G%7G#7G%7G'7G)7G+73&7 7773G1773  3  7373 PxY7 &f' uL3  P3P P3P P3P P3P~33 
' uI3  P3PR P3PC P3P4333333 3 t7' u.3  P3P P3P33 t' t 3;3v3333;3v3333@  P3P P3P P3Pp P3Pa P3PR33 ø P
Y>' t
7P
Yi7P3P
YY
t3P
Y7P
Y 8P
Y;%8P3P
YYд u	 PV3P[
Y8P
Y' u	\' u[3 @ P3P P3P P3P P3Pt P3Pe33 ' uL3   P3P9 P3P* P3P P3P33 û  û  ' u[3  P3P
 P3P
 P3P
 P3P
 P3P
33 VW~3>~3 u PY  3 PR33ɋ63D-3RPB33 PY PY>' t &P63YY u 3 3 3  ?3 PY PY  P{Y>3u39 PY tG;>~3stt' u#'PYt3K PY t>' t P
Y_^V' u$3;63s(''Q
YYF;63r3  3  ^UVWF  3
 PR6363VFF  F  363633
 PRtFV ''' 3  3  33 67673PP PFP" u
F FFV FV;Fu;VuG PYF  F  FF .;Ft.g PFP
 PFP
 PFP
 PFP
 PFPw
FPvvC  PFPX
 PFPI
 P3P:
F ؃v}.3m3d3x[3RF3I PFP	~t	F'+' PF4 P'P	 P'P	 PFP	 PFP	 PFP|	 PFPm	vvvv$ PFPO	 PFP@	>' t*3;3sF;3sv3;3v6F;3v-a3;3sF;3sL3;3vF;3v7vv6363F3F3 PFP PFP PFP PFP>' t*F;3sF;3sF;3v6F;3v-F;3sF;3sF;3vF;3v}vvvv PFP
 PFP PFP PFP>' t*F;3sF;3sF;3v6F;3v- F;3sF;3s F;3vF;3v vvvv|9 PFPa PFPR PFPC PFP4>' t,3;3sFF;3rj3;3v8FF;3v,S3;3sF+F;3r>3;3vF+F;3w)vvvvv  PFPF$'~ ut
t	 PYF
|_^]w




      @       @=+

4

UVWv3F~3FHPHP@PF@P/WVWv vVWV vvvV Wvvv ~ u F FF+FPFPvVx +FPFPF+FPFPY ~
uJWV+FPFP? +FPFFP+FPFP  Wv+FPFFP	 _^]UVW~V^
;sϋыN^ً+ǉF+FF~ }!;F~F F@FF.F F;F~F @FFF F@FFػ F 3G5FF   3Ft''PvWi~unFF;FunF  ~ uF N^KwB.Gv~3F+Fv~$G+vNv~GvG+v~N+v~ ~e_^]JWgwUVWvF;3s3FF;3v3F;63s63F
;3v3F
>'s6>' u'' F~
FPVWcG;~vF;v
vK35uf\% FN F~;' F؊3 Ft''PVWn~ uF G;~vF;v
v 3G5F^3R3XvFF3R3Xv VF3vV3vV FFFF~K' F؊3 Ft''PVWT FF;FuF  n~ uF G;~vFF;FuF  F~uF  F;v
wx_^]U
VWVv;3s;3v ;63s ;63v >' t
3W#+׋F% F+6337u@~ t*Nğ'^& *Nҋğ'^& x3G7 PG-3[㋇''VFV*NF*NЈF3Vt^F&	^F& ~A3G7 ;w_^]U~u3F~ t+>3 t3  3F~ t
3  ^>!]U^^^VFشA!]UVWv~^~uF<3   uF=3  3  '' F=3VF؊f!Fs 1uF3"uF3^    B!F3_^]UVWFVFV3;~s83;3u>'' t & F t^630'&3G;~rȰ_^]Ã>3 t333Ҹ B!3  3  UVW~FVFV3;s8>3@u  F~ u PY ^& 3p'3F;rȰ_^]UV^F0'3@ VFش?!Fr~@t 3F33  >3 ut ^]U^Fp'33VFش@!Fr	F;3t
3   3  ]U' uFFFI!]UVWv' u#ƱF tF^H!Fr;V833  w&r;3sl3F>j3>3633 V33_^]UFu F]ò
!
!ô Ê>)'  *'6+'UF >' t4 PFPF t  PzY&PY1F >%' u 6&' FfFt
 PYFFF u~ tT~ tZ F &'fF t
 PYFFt Ft FF u~ uF~ t]Ê>)' 6+'*' UVF F@BF3vvvv
F~ ~ u3Fu+F0FV!FPRVFY[n)FVF 3
 PRvv
VFFrF0FV!^]UF^F^&FV!F^&? u]UVWN'F33F~ ~ u#FuF0FV!F+F 
 3Gr0FV!_^]U>' tF' 
*'F+'F>.' u>/' t3*'3+'m
>*' tK^vv.O8PY8.' u'됸=8PY.' u'q^8PY.' u'R888PYF~u+>.' u>/' t3*'3+'*' %9PNYF*'F+']Î\9P PYY
 P'P P3P P3P P~3Pq PYUVWv>' t$Kv. ('>)'FF%0 =0 u'>.'t F  ^,'3F~|
3,'
3-'33-'3-'3-'3-'3-'3-'3㊇3*'㊇3+'/'Z/' L>*' v0h9PY&BRY.' t60.' t >*' v+'>+'v/+' +'  +F3;~}	G;~|-'+'-'-'33-'3-'3-'3-'3-'33 3 3 3
3
/'㊇3*'㊇3+'_s9PY6363aYY~9PY6363LYY9PY6~3Y>~3u9 9 ㊇3*'㊇3+'9PY>' t&i>%' t99PY>&'u9I9D㊇3*'㊇3+'>3
s9>3ds99P5Y63^Y9P%Y ㊇3*'㊇3+'[9PY*'
&㊇3*'㊇3+'39PY*'3+'3)㊇3*'㊇3+':PY3_^] f   UV:P3PYYд u F H؃v .4" PFPX3 PFPEF&FF ;wFƄ& '3   PFP PFP P3PFVl3j3'3  3  @ PFPF 3G5# P,'P P-'P.' PFP
t3PrY(:PY^]4!{!!!U3:P3PYY F~ uz PFP0~ u' NF F:Fs( PFPF V؈&FF:FrFF Ƈ& '3 3PY>:PY]UVW3G-F3FG8FF  F;Fs0^㋇''FVVW~NF_^FF;Fr_^]UVW~>' t2WvvF t  P,Y&PkYF >%' t_3;r F B^& F&' FfFt
 PoYFFF u	~ us~ tF;rz3;rqF ]^& F F &'fF t
 PYFFt Ft FF u~ uF~ tF;s됰_^]UVvF~ t3FF F:Fs&F ؊ PWYF u 
FF:Frڰ^]ã33333% 33% 裬33% 33% 裮33% 33% 裰33% 33% 裲3UVWNv~4 .3F4 .34 HЀ>' tG;NsF33;v3>33G#;F
sF
3;63v3  O63I39#sG#3>3;3v3  3;vsF3639V
v3F
3_^]UW~~ON Nu3F@_]YQ3YQ YQ YQ UVWF
V^Nutite uy
؃ y
ۃ   W33;rw;r+@[ tƋ t؃ _^]  t3Vt^                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     @                                                                                                                  @8px<iZ-KB$$B3333     "  "         "HP PJ 90 dpi 2 Color Landscape    "HP PJ 90 dpi 16 Color Landscape   "HP PJ 180 dpi 2 Color Landscape   "HP PJ 180 dpi 8 Color Landscape   "HP PJ 90 dpi 2 Color Portrait     @''Z     *t90R*r0a*r1U *r0B            *b90W                                                                 "HP PJ 90 dpi 16 Color Portrait    @''h   *t90R*r0a*r4U *r0B            *b90V                             *b90W                             "HP PJ 180 dpi 2 Color Portrait    @''     *t180R*r0a*r1U*r0B            *b180W                                                                "HP PJ 180 dpi 8 Color Portrait    @''   *t180R*r0a*r3U*r0B            *b180V                            *b180W                             @'' 	

BGI.$$$ BGICAN.$$$ BGICAN.$$$ BGI.$$$ BGI.$$$ bgi.$$$ BGI.$$$ BGI.$$$ BGICAN.$$$ BGICAN.$$$ BGI.$$$ BGI.$$$ BGI.$$$ Insufficient memory - press a key >> File open error - press a key >> File write error (disk full?) press a key >> No paper - fix and press a key, <ESC> aborts >> Printer error - fix and press a key, <ESC> quits >> Printer offline - fix and press a key, <ESC> quits >>                                                        BGISTAT.$$$ Printer :  Plot    :   objects,  K buffer,   pass.  passes. Output  :  COM LPT 2 1 Pass    Pass   Pass  :  Rasterizing |          |            Printing (<ESC> exits) |          | * BGICNF.$$$ BGICNF.$$$ BGIPLT.$$$ BGIPLT.$$$ pkBGI Device Driver (PSBGI) 2.00 - Feb 24 1992
Copyright (c) 1990 Ryle Design  (517) 773-0587
                           PSBGI                U ]CB      ËM      s1 IJ@;:<                                   U
VW t>46F8F:F(Fv*FHPHP@PF@PVWVv]vWVWQvvvWAVvvv1~ u F FF+FPFPvW+FPFPF+FPFP~uJVW+FPFP+FPFFP+FPFPVv+FPFFPPR _^]<U uqPY> t>! t P>YF! ^
[
~ t! uIP PyYY!  u
(  *  R  P   ]= uGZ4uDt= t%zD PYPY9PR UZG4FG2F>D u^N`] u,66646*6(94(6*PR >DsDò U> tF@ F ^&0R  P   PKYn]%Dt= u1= u.1  "  $     ! H tH  tH@ tH tH% &8 \Z> t;ZG*NG(PG*RG(TG.VG,XG0Z&( 1ZG(NG*PG(RG*TG,VG.XG0ZZGL M\]ZG2^_ P)Y P!Y'L GUVW
 t8;NtNPYPYL;Du
>DtaDLDvO.PY=Zw6渪PzYVYPmYVYЋZG6 PMYVaYP@YZw6VKY뚋ZG6 PYV+YP
YZG6 PY^PY33 F F  VF6t%uZG6PYPY3F %F"tZG6PYPY3G3GnF~rtZG6PYZG6PmYqPR _^]9Dl u4(6*PR  uH4;8v8846;:v::66:686664 PR à uPY   PY?tUj u[> t6F46ZG(+F4ZG*+6664`YP?Y66PYP/YR PR ]û  û  
 u 6:686664qPR ðPYð
P
Y
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> t֋vZG(+F׋~G(+FZG*+ƋG*+ǋPYvYPoYVYPbYvqYPPYVdYPCYfvRYP1YWEYP$YGv3YPYW&YPY(  P Y$P Y_^]U;P3PYY u$ P7Y 8~tFPY t PFP u3PY]UVWVv~
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8t+314
Inbreeding

	This module simulates inbreeding in a finite population, and shows both expected and realized values of the inbreeding coefficient F.  No selection operates; changes in allele frequencies and F are due entirely to chance effects.

	"Identity by descent" means that two alleles are copies of the same ancestral allele.  The inbreeding coefficient F is defined as the probability that an individual's alleles at a particular locus are indentical by descent.  F = zero means no inbreeding, while F = 1 means complete inbreeding.  In the latter case all individuals in the population are genetically identical.9	In the absence of new genetic variation by mutation or immigration, F increases in finite populations over time.  Simply by chance some alleles will be lost and others will increase in frequency, the result being that individuals in later generations have greater and greater probabilities of carrying copies of the same ancestral allele.

	The exact rate at which the inbreeding coefficient is expected to
increase is given by the following equation:

Ft = 1 - (1 - 1/2N)t

where Ft  is the inbreeding coefficient in generation t, N is the population size.  It is assumed that there is random mating and F in generation 0 is zero.  The equation (and intuition) tell us that the inbreeding coefficient is expected to increase more slowly in larger populations and more rapidly in smaller populations.7
	Not every population of a given size will have exactly the same inbreeding coefficient, even if they have been random mating for the same number of generations.  The equation gives an expected F, but there is variation around that expectation.  Just by chance a finite population can evolve to F = 1 more quickly than expected, or even move temporarily towards F = 0.  The equation tells us what to expect on average for a large cohort of finite populations.

	Unless there are other forces operating, drift leads inevitably to fixation of one allele, at which point F = 1.  All finite populations are headed to the same end, but they vary in the rate at which they approach that end.


Further reading:

Hartl, D.  Principles of Population Genetics, Chapter 3.
1. Like all of the Populus selection models, this simulation produces four different output graphs: allelic frequencies vs time, genotypic frequencies vs time, change in allelic frequency per generation vs allelic frequency, and the adaptive topography, population mean fitness vs allelic frequency.  You may choose among these outputs on the parameter screen, or toggle between them with the space bar after the first is plotted.

2. "s" is the selection coefficient against "aa" genotypes.  It can take values from 0 to 1.

3. "h" sets the dominance of the "a" allele relative to "A" alleles.  If h = 0, then "a" is completely recessive; if h = 1, it is fully dominant; if h = 1/2 the heterozygote phenotype is exactly intermediate between the two homozygote phenotypes.

4.  sets the mutation rate (per generation) from A to a alleles.  It can take values from a maximum of 1 to a minimum of 1E-10.

5. The last lines on the screen allow you to set the number of generations that the simulation will run, and to choose between starting from a single initial frequency (that you can set) or from six different initial frequencies for a stability analysis.
	This simulation produces four different output graphs: allelic frequencies vs time, genotypic frequencies vs time, change in allelic frequency per generation vs allelic frequency, and the adaptive topography, population mean fitness vs allelic frequency.  You may choose among these outputs on the parameter screen, or toggle between them with the space bar after the first is plotted.

	With positive mutation rates this simulation reaches equilibrial frequencies with the deleterious allele still present.  To see the polymorphic equilibrium more clearly use the zoom function (Alt-Z), expanding the lowest frequency range to fill the full screen.  The grid function (Alt-G) will help to read equilibrial frequencies from the ordinal axis.
	Populus produces four different output screens for Kirkpatrick's model; you can choose one on the input screen, or see them all in sequence by pressing the space bar between displays.  The output screens give 1) t2 vs. p2, the frequency of males with the secondary sexual characteristic vs the frequency of females that prefer the secondary males; 2) D vs. time, the linkage disequilibrium correlating the male trait and female choice alleles as selection proceeds; 3) male viability vs. time, illustrating declines in male viability if the secondary trait increases in frequency; 4) D vs. t2 and p2, in three dimensions.
	The bold green line in the t2 vs. p2 graph gives the line of equilibria where the viability penalty experienced by males bearing the T2 trait is exactly balanced by their enhanced attractiveness to females.10t14+3
Non-Random Searching

	The Nicholson-Bailey assumption that parasitoids forage randomly is obviously oversimplified.  Real hosts are likely to be distributed in a patchwork of high and low densities, and parasitoids can be expected to respond by orienting toward high-density patches.  Because this parasitoid aggregation has the effect of giving hosts a refuge at low densities, it is a potentially important stabilizing factor in the dynamic interaction (Murdoch and Oaten 1975, Hassell 1978).5	This simulation reproduces a model by May (1978), who assumed that parasitoid attacks show a negative binomial pattern.  A negative binomial distribution is specified by its mean, and a clumping parameter, k.  Distributions with a small k value (< 1) are strongly clumped, while thoses with large k (> 8) approach Poisson randomness.  By specifying the statistical distribution of parasitoid attacks, May was able to model the dynamic effects of aggregation without explicitly including details of orientation and foraging behavior.  His recursions are

Nt+1 = Nt [1+(aPt/k)]-k

Pt+1 = Nt[1-(1+{aPt/k})-k]

where all parameters are as given for the basic Nicholson-Bailey version, save for k, the negative binomial dispersion parameter, which can be interpreted as a coefficient of variance of parasitoid density among patches.  Note that this recursion approaches Nicholson-Bailey as k approaches infinity.  The dynamics of this model show diverging oscillations if k > 1, but the effect of parasitoid aggregation produces damped oscillations or a monotonic approach to equilibrial parasitoid and host densities if k < 1.12	We have also included a version of this model with density-dependent prey growth.  In this case, the prey recursion is

Nt+1 = Nt [exp(-{r/K}Nt)] [1+(aPt/k)]-k

where this upper case K is the environmental carrying capacity and again, r = ln .  Interested students should see Hassell (1978) or Bedington, Free, and Lawton (1975, 1976) for the complex stability analysis of this model; but in general, stability varies inversely with host growth rate and directly with density dependence and the ratio r/K.4References

Beddington, J. R., C. A. Free, and J. H. Lawton. 1975. Dynamic complexity in predator-prey models framed in difference equations. Nature 225:58-60.

Beddington, J. R., C. A. Free, and J. H. Lawton. 1976. Concepts of stability and resilience in predator-prey models. J. Anim. Ecol. 45:791-816.

Chesson, P. and W. W. Murdoch. 1986. Aggregation of risk:  relationships among host-parasitoid models. Am. Nat. 127:696-715.

Hassell, M. P. 1978. The Dynamics of Arthropod Predator-Prey Systems. Monographs in Population Biology, Princeton University Press. Princeton, NJ. pp. 50-79.

May, R. M. 1978. Host-parasitoid systems in patchy environments: a phenomenological model. J. Anim. Ecol 47:833-44.

Murdoch, W. W. and A. Oaten. 1989. Aggregation by parasitoids and predators:  effects on equilibrium and stability. Am. Nat. 134:288-310.

Murdoch, W. W. and A. Oaten. 1975. Predation and population stability. Adv. Ecol. Res. 9:2-131.
	Populus produces two different outputs from the Nicholson-Bailey model, a time trajectory of the prey and parasitoid population sizes (N,P vs. t) and a phase plane (P vs. N).  The model's unstable oscillations usually increase until one or both parties are driven to extinction.  Comparison of the density-dependent and density-independent versions will show that density-dependent host growth is a strongly stablizing factor, prolonging the oscillations or damping them to a steady-state equilibrium.  Sometimes when the parasitoid goes extinct host populations can reach extreme density and scaling of the graph obscures the oscillations.  A semi-log plot (press <Alt-L>) helps in interpreting such cases.

	These N,P vs t graphs are artificial in allowing population densities to fall to a fraction of an individual; when this happens to predators, prey begin a log growth phase that ends only when predators recover enough population density to respond.  For practical purposes, one might consider the simulation run ended when density falls below 1 (or 2!).
	As in all of the Populus Arthropod Predator-Prey simulations, there are two different outputs from the Functional Response models, a time trajectory of the prey and parasitoid population sizes (N,P vs. t) and a phase plane (P vs. N).  The Type I model is identical to Nicholson-Bailey.  Type II simulations are less stable, and have increased oscillatory tendencies.  Depending on the parameter values that are supplied, Type III runs can be more stable than either Type I or Type II, but the functional response is not sufficient to neutralize or damp the oscillation; cycling should still increase until one or both parties are driven to extinction.
	As in all of the Populus Arthropod Predator-Prey simulations, there are two different outputs from the Nonrandom Search models, a time trajectory of the prey and parasitoid population sizes (N,P vs. t) and a phase plane (P vs. N).  The tendency of parasitoids to aggregate in patches of high host density is a powerful stabilizing force in the interaction.  This aggregation is most extreme at low values of k (that's the lower case k).  The model approaches Nicholson-Bailey as k approaches infinity.  As in the Nicholson-Bailey model, density dependent host growth is also a stabilizing factor.
	As in all of the Populus Arthropod Predator-Prey simulations, there are two different outputs from the Interference models, a time trajectory of the prey and parasitoid population sizes (N,P vs. t) and a phase plane (P vs. N).  In the linear version of this model, the foraging efficiency of parasitoids decreases with density in correlation with the interference constant, m.  Increasing host growth also tends to prolong oscillation.  In the curvilinear version, the density-dependent decline in foraging efficiency is set by specifying Tw, the time wasted as a result of the parasitoids' interaction, as in the functional response models.5t+314
Polyphagous Predators

	The preceding models in this set have all dealt with single pairs of predator and prey species.  Natural interactions are rarely so isolated, and this simulation allows exploration of a more complex system with one predator and two competing prey species.  It uses a discrete analog of the Lotka-Volterra competition model to describe the interaction between competing prey as

Xt+1 = Xt exp[-g(Xt + Yt)]

Yt+1 = 'Yt exp[-g'(Yt + Xt)]

where X and Y denote the two competing prey species, g and g' are constants, and  and  are competition coefficients like those of the Lotka-Volterra model.  This competition model retains the linear Lotka-Volterra isoclines, but its dynamics are complicated by the lags inherent in the discrete formulation (May 1974).8	Adding a predator to this system gives the recursions
 
Xt+1 = Xt exp[-g(Xt + Yt) -aXPt1-m]

Yt+1 = 'Yt exp[-g'(Yt + Xt) -aYPt1-m]

Pt+1 = Xt[1 - exp(-aXPt1-m)] + Yt[1 - exp(-aYPt1-m)]

where, "X" and "Y" are the two competing prey, "P" is the predator, "a" is the area of discovery of the searching predator, and "m" is an interference constant, interpretable as the slope of the decline in search efficiency with predator density.  When m=0, this simulation is analogous to the basic Nicholson-Bailey model and again, increasing interference lends additional stability.  With complete niche overlap (  1) competitive coexistence is impossible, but with  < 1 a predator can stabilize an otherwise unstable competitive interaction if it prefers the superior competitor (i.e., if aX/aY  1).6	If this random predator is replaced by one that switches, changing its preference between prey types to focus on the more common one (Murdoch 1969), it can be a stronger stabilizing influence.  Hassell models two competing prey and a switching predator with the following system:

Xt+1 = Xt exp[-g(Xt + Yt) -(1+E)aXPt1-m]

Yt+1 = 'Yt exp[-g'(Yt + Xt) -(1-E)aYPt1-m]

Pt+1 = Xt[1 - exp{-(1+E)aXPt1-m}] + Yt[1 - exp{-(1-E)aYPt1-m}]

where
E = s(Xt - Yt)/(Xt + Yt)

and s varies from 0 to 1 to express the degree of switching.  When density-dependent competition between prey species is strong ( near 1.0) this switching predator exerts a powerful stabilizing influence.13References

Hassell, M. P. 1978. The Dynamics of Arthropod Predator-Prey Systems. Monographs in Population Biology, Princeton University Press. Princeton, NJ. pp. 122-146.

May, R. M. 1974. Biological populations with non-overlapping generations: stable points, stable cycles, and chaos.  Science 186:645-647.

Murdoch, W. W. 1969. Switching in general predators: experiments on predator specificity and stability of prey populations. Ecological Monographs 39:335-354.
	As in all of the Populus Arthropod Predator-Prey simulations, there are two different outputs from the Polyphagous Predators model, a time trajectory of the prey and predator population sizes (X,Y,P vs t) and a phase space (X vs Y vs P).  This model represents the trajectories of three species (two competing prey and a predator, note color coding on the axis) and the phase plot is consequently three-dimensional.  In the 3-D plot each point is shadowed on the back planes (which may be gridded if you like) to aid determination of the numerical values.  The polyphagous predators model maintains a stable 3-species equilibrium when the predator is more effective in searching for the superior competitor, or under many conditions when the predator switches its preference to focus on the more abundant prey type.
	As in all of the Populus Arthropod Predator-Prey simulations, there are two different outputs from the Competing Predators model, a time trajectory of the prey and predator population sizes (N,P,Q vs t) and a phase space (N vs P vs Q).  This model represents the trajectories of three species (one prey and two competing predators, note color coding on the axes) and the phase plot is consequently three-dimensional.  In the 3-D plot each point is shadowed on the back planes (which may be gridded if you like) to aid determination of the numerical values.  The competing predators model is most likely to maintain a stable 3-species equilibrium if each of the predator-prey links is stabilizing, i.e., if the k values are small and both predators aggregate in patches of high host density.  Stability is further favored if the inferior competitor (Q) has the higher searching efficiency.
	As in all of the Populus Arthropod Predator-Prey simulations, there are two different outputs from the Host, Parasite, Hyperparasite model, a time trajectory of the prey, parasitoid and hyperparasitoid population sizes (N,P,Q vs t) and a phase space (N vs P vs Q).  This model represents the trajectories of three species (one prey, one parasite, and one hyperparasite, note color coding) and the phase plot is consequently three-dimensional.  In the 3-D plot each point is shadowed on the back planes (which may be gridded if you like) to aid determination of the numerical values.  A stable three-party equilibrium is most likely when both the parasitoid and hyperparasitoid aggregate in patches where the density of their respective prey is high (k values are small), and is further enhanced when the hyperparasitoid has a higher searching efficiency than the parasitoid.
	As in all of the Populus Arthropod Predator-Prey simulations, there are two different outputs from the Threshold Predator Reproduction model, a time trajectory of the prey and predator population sizes (N,P vs. t) and a phase plane (P vs. N).  One of the most interesting features of this model is the absence of global equilibria when c > 0.  This implies that there are some predator/prey abundance ratios that lead to stable coexistence and others that lead to extinction from the same model parameter values; if a predator of this sort were used as a biological control agent, initial densities produced by the release would be critical to the dynamic outcome.  The best way to observe these local equilibria is to run extensive stability analyses on the P vs. N graph.
	In this mode you can move a cursor to establish a new initial starting point.  Press <Enter> or <f> to draw a trajectory from that point.  If you press <m>, trajectories are initiated from multiple points, either around the perimeter of the plot area, or in a gridded pattern (set from the options menu).

	Note that on some plots (such as those with 4 or more equations) not all of the dependent variables are plotted.  In those cases, the initial value (i.e. the values set during data entry) of the unseen variables are used.

	Some models give you the option of changing variables other than those being plotted.  If this is the case, <space> will toggle through the different variables that you are allowed to change.

To summarize, here are the commands:

  Alt-S      - Activate/deactivate
  F, Enter   - Draw trajectory forward in time
  B          - Draw trajectory backward in time
  M          - Draw multiple trajectories (options set in options menu)
  C          - Continue trajectory from where it last left off
  E          - Erase all trajectories
  Alt-E      - Erase the last trajectory
  Alt-D      - Erase all trajectories from memory only.  This means that
               the trajectories cannot be redrawn if you leave the plot
               and come back.
  Alt-C      - Close (hide and remove from memory)
  Esc        - Deactivate and return to data entry

  Arrows     - Position the cross hairs cursor in the x and y directions
  Ctrl-left & right arrows - Position the cross hairs in the z direction

and sometimes,

  Space Bar  - Change to a different set of variables7t14+3
Drift and Selection

	This module simulates the operation of natural selection in a finite population.  Both genetic drift and natural selection affect allele frequencies.  By adjusting population size and relative fitnesses the user can study the interaction of these two evolutionary forces.

	Drift tends to eliminate genetic variation. In any finite population, one allele will eventually increase to fixation, at a rate that depends on the population size.

	Selection can either maintain or eliminate genetic variation. Selection in favor of the heterozygote creates a stable polymorphism, but selection in favor of one of the homozygotes eliminates variation in deterministic models.3	When both selection and drift operate, there is an opposition of evolutionary forces if the heterozygote is most fit, with selection "trying" to maintain variation and drift "trying" to eliminate it.  Which force wins the battle depends on the relative strengths of drift and selection.  Drift is very strong if population size is small, and is weak if population size is large.  Selection is strong if the "w" parameters differ greatly, and is weak is the w's are similar.

	Kimura has provided a rule of thumb for comparing the strengths of selection and drift, as follows.  Define s, the selection coefficient such that homozygotes have fitnesses 1-s relative to the heterozygote fitness of 1.  N is the population size.  Selection predominates when 4Ns >> 1, and drift predominates when 4Ns << 1.  By predominate we mean selection (or drift) usually wins the battle to maintain (or lose) variation.  When 4Ns is close to 1 then we cannot predict the evolutionary outcome with any certainty.


Reference

Kimura, M. 1983. The Neutral Theory of Molecular Evolution.  In:  Evolution of Genes and Proteins, M. Nei and R. K. Koehn, eds.  Sinauer Associates, Sunderland, MA pp208-233.
	The user must supply four payoffs that describe the interaction between pure strategies A and B.

	There are also two parameters to be entered for each mixed strategy, C and D.  The first parameter on the C line indicates how often C behaves like A, while the second indicates how often C behaves like B.  These two values must sum to 1.0.  Likewise, parameters on the D line specify how often D behaves like A and B, respectively.

	Finally, the user must indicate which invader and invaded population will be tested.  Choose one invader and one resident strategy from the list.  The invader and the resident must be different.

	The default values correspond to the Hawk-Dove game of behavioral interactions.11t+314
Selection on an X-Linked Locus

	This program shows how allelic frequencies change when the locus under selection is on a sex-determining chromosome.  There are five possible genotypes:  three in the homogametic sex (usually females) denoted XX, Xx, and xx, and two in the heterogametic sex (usually males), denoted XY and xY.  There are two allelic frequencies to keep track of, p in females and p in males, where p is the frequency of the "X" allele.4	Selection on a sex-linked locus differs from selection on an autosomal locus in several respects:

	1) Oscillations of allelic frequency - In the one-sex autosomal model there are never oscillations in allelic frequencies over time.  However, in the sex-linked model there can be oscillations if allelic frequencies are initially different in males and females.  The default parameter values illustrate oscillations.  Notice that frequencies eventually become equal in the two sexes.

	2) Genetic polymorphism - In the two-allele autosomal model there is only one way to maintain genetic polymorphism:  the heterozygote must have the highest fitness.  For the sex-linked case there are two ways:  either by heterozygote advantage, or by differential selection in the two sexes (i.e., one allele is favored in one sex and the other allele is favored in the other sex).  See if you can find examples of both types of polymorphism.


For further reading, see chapter 4 of:

D. L. Hartl and A. G. Clark (1989) Principles of Population Genetics, Second edition.  Sinauer, Sunderland, Mass.
1.  wAA, wAa, and waa are the relative fitnesses of the three genotypes. Specify a real number ranging from 0 to 1.  By convention, the most fit of the three genotypes is given a relative fitness of 1.  A genotype with wxx = 0.25 will contribute  as many progeny per capita as the best-fit genotype to the next generation.  If wxx = 0, the genotype is lethal; all such individuals fail to survive or reproduce.

2.  How many generations should you simulate?  The answer will depend on the strength of the selection process specified by your choice of relative fitnesses.  Reasonable values might range from 10 to 1000 generations.

3.  By default, the program automatically runs the simulation from six different initial allelic frequencies.  Sometimes in cases of heterozygote advantage (wAA<wAa>waa) or disadvantage (wAA>wAa<waa) it is interesting to specify a precise starting frequency.  If so, set this box to "y" and

4.  enter some starting frequency between 0 and 1.1114t+3
Adaptive Landscapes

	For the one-locus selection model with constant fitnesses, a simple principle applies for all possible parameter values:  Natural selection changes allelic frequencies each generation in such a way that the average fitness of the population (W) is always increasing over time.  Populations climb "adaptive peaks", but never go down into "adaptive valleys".9	The average fitness of a population (W) is a function of allelic frequencies.  In particular, if zygotes are present in Hardy Weinberg proportions prior to the operation of natural selection then the average fitness in the population will be

W = p2wAA + 2pqwAa + q2waa.

If fitnesses are scaled such that the largest is equal to 1.00, then W can be thought of as the percentage of the population of zygotes that will survive viability selection.

	Since q = 1-p, W is a quadratic function of p (or q).  That means that a graph of W versus p will be concave upwards or downwards, with at most one maximum or minimum.  This picture of W versus p is called an "adaptive topography" or "evolutionary landscape".  It shows the average fitness of the population for all possible genetic configurations.10	Wright showed that the following general equation applies to the one-locus selection model:

       npq dW
p =      
       nW  dp

where p is the change in p from one generation to the next.  The first term of the equation is always positive, so the sign of p depends only on the second term.  If the derivative is positive then p will be positive, which means p increases and W increases.  On the other hand, W might be decreasing with p, in which case p will be negative, p will decrease, and W will increase.  In either case W increases from generation to generation.9	Another consequence of Wright's equation is that equilibria are always at critical points (maxima or minima) of W.  That is, p is zero (i.e., there is no change in frequency from generation to generation) only when

	i)   p = 0     There is no genetic variation, or
	ii)  q = 0     There is no genetic variation, or
	iii) dW/dp = 0 The population is at a maximum or minimum of W.

	While both maxima and minima of W correspond to equilibria only the maximum is a stable equilibrium, that is, one that "attracts" the population.  A case in which the heterozygote has the highest fitness will exhibit such a stable equilibrium.  A minimum of W corresponds to an unstable equilibrium; populations will move away from the unstable point.  A case in which the heterozygote has the lowest fitness will exhibit an unstable equilibrium.12References

For a clear explication of the ideas of adaptive topography, see Chapter 4 of:

Roughgarden, J.  (1979) Theory of Population Genetics and Evolutionary Ecology:  An Introduction.  Macmillan, New York.

For a discussion of circumstances under which the adaptive topography rule fails, see:

Curtsinger, J. W.  (1984) Evolutionary landscapes for complex selection.  Evolution 38:359-367.
1.  The w's are fitnesses of the five possible genotypes (three kinds of females and two kinds of males).  Enter any positive numbers that reflect relative fitnesses.  The default values, (all w's = 1) correspond to the situation where there is no fitness difference between the genotypes.

2.  p(females) and p(males) are the initial frequencies of the "X" allele in females and males.  Enter positive real numbers in the range (0,1).  If you want to see oscillations, make the initial allele frequencies different in males and females.

3.  The choice of a number of generations depends on what part of the model you are interested in.  If you want to see oscillations in gene frequencies, enter less than 20 generations, since the oscillations die down quickly.  If you want to see equilibrium gene frequencies, enter a larger number.  How large will depend on the strength of selection.914t+3
Genetic Drift: A Markov Model

	This simulation uses a "Markov model" to predict the distribution of allelic frequencies in an infinite series of small populations with one randomly-drifting diallelic locus.  It assumes that all populations begin drifting from the same starting point, and calculates the proportion which are expected to show each possible composition in subsequent generations.  The model is completely deterministic.  To run it, you must specify a population size, and the number of "A" or "a" alleles in each population at the outset.  Note that if population size = N, then the number of "A" alleles could range from 0 to 2N.11	Suppose that a small population consists of N = 2 diploid individuals.  At a single gene locus with 2 alleles ("A" and "a"), the population could be in any one of 5 possible states; that is, it could contain 0&4, 1&3, 2&2, 3&1, or 4&0 "A" & "a" alleles, respectively.  In general, a population of size N could have 2N+1 such compositions, with 0, 1, 2, ...2N "A" alleles.  In two of these states (0 and 2N "A" alleles) only one allelic type remains and we say that the population is "fixed" for "a" or "A," respectively.  Neither sampling error nor genetic drift occurs when gametes are chosen from a fixed population.  In contrast, a population which is not fixed still contains both allelic types; it can jump to any other state in a single generation of sampling, and the likelihood of large or small jumps can be calculated using the present allelic frequencies and simple probability arguments.9	A Markov Model of genetic drift assumes an infinite number of small populations, all of the same size, N.  Each population is in 1 of the 2N+1 composition states, with 0 ... 2N "A" alleles, and the status of the entire collection is represented by a vector of 2N+1 elements, specifying the proportion of populations in each of the 2N+1 possible states.  The model uses a transformation matrix with 2N+1 rows and columns to simulate gene-frequency changes in the entire population ensemble.  The matrix incorporates transition probabilities for every possible pair of states.  Multiplication of the state vector and the transformation matrix yields a new state vector, updated to reflect the fraction of populations in each state after 1 additional generation of random sampling drift.  A detailed discussion of this model is given by

Roughgarden, J. 1979. Theory of Population Genetics and Evolutionary Ecology: An Introduction. MacMillan Publishing Co., Inc. New York. pp 57-80.
1.  The simulation produces output graphs showing either the population-wide frequency of altruistic alleles, or the interdemic variance in allelic frequency.

2.  Deme size, "N", is the number of individuals in each of the 10 randomly breeding subpopulations.

3.  The demic survival coefficients determine the functional relation between demic extinction rates and the local frequency of altruists.  Values ranging from 0 to 10 are permitted.

4.  Individual selection coefficients set the relative fitness of the three genotypes.  Values should range from 0 to 1.

5.  The migration rate, "m", specifies the fraction of individuals per generation which are replaced by migrants from the population-wide pool.

6.  Runs may be initiated with a single altruistic mutant, or by specifying a starting frequency of altruistic alleles.  The graphical output averages results from several runs (specify a number from 1 to 25).
	The A vs. T graph plots frequencies of the altruistic allele in the population at large.  Results of this Monte Carlo simulation will vary from run to run, and general tendencies can be clarified by averaging a larger number of runs (at the expense of increased computation time).

	The  vs. T. graph plots interdemic variance in gene frequency over the same run interval.  Since the interdemic variance is a critical determinant of evolutionary change under group selection, this plot can be of considerable value in interpreting the A vs. T results.3t+314
Kirkpatrick's Haploid, Arbitrary
Model of Sexual Selection

	Elaborate sex-limited ornaments like the tails of peacocks and birds of paradise appear to increase their bearers' attractiveness to potential mates at a cost to their viability.  These observations have posed an interesting challenge to evolutionists because, while it is plausible that some such traits might function in the competition between males for matings (Darwin 1871), an explanation for female preference is much more difficult; why should females evolve a predilection to choose traits that reduce male viability?

	R. A. Fisher suggested that the evolution of female choice would initially require a reproductive advantage.  For example, a female who chose mates possessing some attribute that conferred high viability might have highly viable progeny.  If females with the strongest preferences choose males with the most pronounced traits, the genetic correlation between female choice and male attribute could produce a "runaway process" in which the male character evolves beyond the point where it shifts from viability asset to liability under the impetus of enhanced mating success.10	This simulation reproduces a two-locus haploid model by Mark Kirkpatrick (1982).  It assumes that there is a sex-limited diallelic locus in males (T), one allele (T1) conferring a "normal" or "cryptic" phenotype with high viability, and one (T2) conferring some arbitrary trait with reduced viability (1 - s, where s > 0).  In females, there is a corresponding diallelic polymorphism (P) such that some females (P1) mate randomly or prefer normal males, while the others (P2) prefer males with the reduced-viability trait.  The intensity of female preferences is set by the parameter values a1 and a2; P2 females prefer to mate with T2 males by the factor a2 and a1 indicates the preference of P1 females for T1 males.   There is no cost to the females of choosing mates.  The recursion equations that Kirkpatrick developed from these assumptions are moderately complex and will not be given here; interested students are directed to the source.
	Populus produces four different output screens for Kirkpatrick's model; you can choose one on the input screen, or see them all in sequence by pressing the space bar between displays.  The output screens give 1) t2 vs. p2, the frequency of males with the secondary sexual characteristic vs the frequency of females that prefer the secondary males; 2) D vs. time, the linkage disequilibrium correlating the male trait and female choice alleles as selection proceeds; 3) male viability vs. time, illustrating declines in male viability if the secondary trait increases in frequency; 4) D vs. t2 and p2, in three dimensions.

	These simulations produce lines of equilibria where the viability penalty experienced by males bearing the T2 trait is exactly balanced by their enhanced attractiveness to females (given as a heavy green line on the t2 vs. p2 graph).  At low frequencies of the p2 female choice allele, this mating advantage is often insufficient to maintain T2 in the population, but at higher P2 frequencies polymorphic equilibria or even fixation of the T2 allele are possible.  There is no direct selection on females; p frequencies change only as a correlated response to changes in male trait frequency, so linkage disequilibrium between the male and female loci is critical to the evolution of female choice.  At polymorphic equilibria where P1, P2, T1, and T2 are all maintained in the population, this linkage disequilibrium (caused by non-random mating) is a permanent feature even with very high recombination rates.  Finally, note that when the slope of the line of polymorphic equilibria is steep, small changes in the frequency of female choice alleles can effect large shifts in the composition of the male population.11References

Darwin, C. 1871. The Descent of Man and Selection in Relation to Sex.  John Murray, London.

Fisher, R. A. 1958. The Genetical Theory of Natural Selection. 2nd ed., Dover, N.Y.

Kirkpatrick, M. 1982. Sexual selection and the evolution of female choice. Evolution 36:1-12.

Maynard Smith, J. 1991. Theories of sexual selection. TREE 6:146-51.
1.  The model assumes that population size, N, remains constant, and the gene pool contains 2N alleles for each locus.  The program will accept population sizes from 1 to 200.

2.  Independent drift of from 1 to 6 gene loci can be simulated, and 

3.  the initial frequency of "A" and "a" alleles at each locus can be set independently (by specifying values ranging from 0 to 1.0).

4.  By default, the simulation will run for 3N generations.  Since the average time to fixation or loss of an allele beginning at 0.5 frequency is near 2.8N generations, this run length will fix most, but not all loci. Alternate values ranging from 1 to 999 may be entered in this box.
	Histograms show the proportion of an infinite series of small populations expected to have each of 2n+1 possible allelic frequencies.  Pressing any key will advance the view to represent the effect of 1 or more additional generations of genetic drift.  If the populations are polymorphic at the outset (0 < p < 1) the initial distribution broadens and becomes U-shaped as more and more of the populations become fixed for either "A" or "a" alleles.
1.  The computations involved in a markov model of genetic drift are feasible on a personal computer only with relatively small populations. This program permits population sizes ranging from 1 to 16.

2.  The drifting gene locus is assumed to have both "A" and "a" alleles, and their initial frequency is set by specifying the number of "A" alleles present in each population.  Values can vary from 0 to 2N.  Note that the two extreme values, 0 and 2N, imply that the populations are "fixed" for "a" and "A" alleles, respectively, and no genetic drift will occur.

3.  The initial output is a histogram showing the gene frequency set by the "A" allele specification, above.  After pressing enter, a new histogram will update the distribution of allelic frequencies to reflect 1 or more generations of genetic drift.  This box sets the number of generations which are run between successive output views, and may take values from 1 to 10.
1.  Demesize sets the number of individuals in each of the six subpopulations.  If this number is small, then genetic drift will cause substantial differentiation in allelic frequency among demes.

2.  Starting Frequency sets the initial allelic frequency.  All six demes will begin at this value.

3.  Migration Rate sets the proportion of deme members which are exchanged by migration in each generation.  For each individual in each deme, a random number is tested against the migration rate.  If the draw is less than the migration rate, that individual is replaced by a randomly chosen member of the migrant pool, which has the population-wide allelic frequency.

4.  The simulation will pause after some specified number of generations, so that both output graphs can be studied at intervals through a run.7t+314
Selection, Gene Flow and Clines.

	Gene flow is often viewed as the "glue" which holds biological species together; it homogenizes differences caused by genetic drift or geographic variation in the strength and direction of natural selection.  Since gene flow and locally varying selection are potentially counteracting processes which affect the geographic variation of populations, it is interesting to model their combined action.

	This simulation duplicates a model of gene flow and selection published by John Endler (1973).  It is a "stepping stone model," assuming that the population is arrayed in a linear string of semi-independent demes across some environmental gradient.  Each deme experiences selection pressures which vary with its position on the gradient, and exchanges migrants once each generation with the two neighboring demes.5
	Our simulation represents a series of 50 demes, and allows the selection regime to vary from deme-to-deme in four different ways, as follows:
(a) In the "gradient mode," the fitness of AA genotypes decreases linearly while that of aa genotypes increases, from deme 1 to deme 50.  Heterozygote fitness remains constant at a value halfway between the homozygote maxima and minima.
(b) In the "heterozygous advantage mode," homozygote fitnesses remain as they were in the gradient mode, but the heterozygotes have a spatially constant fitness which is always greater than either homozygote by a minimum amount, h1.
(c) In the "local heterozygous advantage mode," homozygote fitnesses are again as before, but heterozygote fitness also varies spatially, remaining a fixed amount, h2, greater than the most fit homozygote.
(d) In the "frequency-dependent mode," it is assumed that there is a locally optimal frequency of "A" alleles which decreases from deme 1 to deme 50.  The local fitness of each genotype is decremented by an amount which varies with the deviation of that genotype's frequency from the local optimum.
	A more detailed discussion of these selection regimes can be found in Endler's paper.5
	In addition to these different patterns of selection, our program allows its overall intensity to be altered as well.  "s" represents the maximum genotype-specific change in relative fitness across the environmental gradient; if s = 1.0, then the relative fitness of AA genotypes declines from 1 in deme 1 to (1-1/49) in deme 2, (1-2/49) in deme 3, and so forth.  Gene flow migration is adjusted by varying the parameter "g," the proportion of individuals in each deme coming from the adjacent demes in each generation.  Half of the migrants come from the deme above, and half from the deme below on the environmental gradient.  Demes 1 and 50 receive the full "g" proportion of migrants from their only neighboring deme.

	Those who refer to the original Endler paper should note a trivial error that demonstrates John's professorial aptitude.  Figs 4 and 6, which purport to graph the equilibrial p values actually graph equilibrial q's.  Populus output presents the p values, hence our graphs have similar shapes, but reversed slopes. 

References

Endler, J. A. 1973. Gene flow and population differentiation. Science 179:243-50.
	Graphical outputs of this model present the frequency of A alleles in each of the 50 stepping-stone demes arrayed along an environmental gradient.  The simulation terminates when frequencies in each of the change by no more than 1% per generation.  The model shows that clines can develop even when local gene flow is quite heavy; and the shape of these clines depends on the mode of selection.
1.  "h1" and "p1" are the initial frequencies of "1" alleles in the host and parasite populations, respectively.  Entries can range from 0 to 1.0.

2.  "s" and "t" are the fitness penalties for parasitized hosts and thwarted parasites, respectively.  They require entries ranging from 0 to 1.0; a low value implies that the attack or rebuff has little effect on the victim, while a high value (1.0) denotes a lethal encounter.

3.  The model incorporates two-way mutation or migration in both the host and parasite populations.  Thus h1 and p1 alleles are converted to h2 and p2 alleles (and vice versa) at a rate "m," ranging from 0 to 1.0.

4.  The simulation can be run for intervals ranging from 1 to 10,000 host generations, and in many cases a substantial run-time is necessary to make the final outcome completely clear.

5.  Many parasites have shorter life histories than their hosts, and the model allows up to 100 parasite generations for each host generation. Enter a value from 1 to 100.
	Allelic frequencies in the host population are graphed against those of the parasites.  The output cycles because type1 hosts do well when type1 parasites are rare, etc.  For many combinations of the input parameters these oscillations will reach a stable "limit cycle," traversing the same range of frequencies again and again.  The model is unrealistic in being completely deterministic; for example, when a limit cycle passes extremely close to the axis, the rare gene is likely to drift to extinction.
	The graphs show frequencies of all three alleles in both host and parasite populations as the coevolutionary interaction proceeds.  The model shows cyclic behavior similar to that of the two-allele case, but is somewhat more stable.  Even when oscillations have progressed to the point where both host and parasite tragectories are near edges of the allelic-frequency space, both populations continue to make occasional passes back through intermediate frequencies.614t+3
Selection on a
Single Autosomal Locus


	This model runs a simple deterministic simulation of natural selection.  It assumes that the population is infinitely large and that the relative fitness of individuals is determined by two alleles ("A" and "a") segregating at a single autosomal locus.  Diploid individuals carry two gene copies and hence may be homozygous (AA or aa) or heterozygous (Aa).

	The model requires specification of relative fitnesses for each of the three genotypes "wAA," "wAa," and "waa" (ranging from 0.0 to 1.0, and assumed to remain constant) and the number of generations you would like to simulate.  It produces a graph predicting gene frequencies in future generations.4	Assume that a constant proportion, l, of zygotes with the AA genotype survives from birth to reproduction, and that having done so, each survivor produces 2m gametes which fuse randomly to form the next generation of zygotes.  The progeny per capita, or fitness of AA's is WAA = lAAmAA.  If the phenotypes produced by Aa and aa have different survival and reproductive abilities, then WAa and Waa may be larger or smaller.  Since it is the relative value of these genotypic fitnesses which determines selective change in gene frequencies, they are customarily normalized (dividing each by the largest value) so that the highest relative fitness (small "w") is 1.0, and the others are fractions ranging from 0 to 1.0.  For example, if wAA = wAa = 1.0 and waa = 0, then the "a" allele is a recessive lethal.

	Different patterns of gene expression can be modeled by choosing appropriate relative fitnesses.  For example, if wAA = wAa = 1.0 and waa = 0.5, the "A" allele is dominant (AA and Aa individuals have the same relative fitness); since aa homozygotes survive and reproduce only half as well, these values specify selection against a recessive.  Values of wAA = wAa = 0.5 and waa = 1.0 would model selection against a dominant.8	If we know (a) relative rates of survival and reproduction for the three genotypes, and (b) frequencies of the two alleles in generation t, we can calculate the allelic frequencies expected at this locus in generation t+1, using the equation:

        (ptwAA + qtwAa)pt
pt+1 =                                
        (pt)2wAA + 2ptqtwAa + (qt)2waa

where pt gives the frequency of "A" alleles in generation t, qt = (1 - pt) gives the corresponding frequency of "a" alleles, and wAA, wAa, and waa are the relative fitnesses of the three genotypes.  The numerator is the weighted average of fitnesses for the two genotypes which have "A" alleles, while the denominator is the weighted average of all three genotypic fitnesses, or the mean fitness of the population.11	This simple model of natural selection is discussed in many texts on ecology, evolution, and population genetics.  For examples, see

Ehrlich, P. R., and J. Roughgarden. 1987. The Science of Ecology. MacMillan Publishing Co. New York. pp. 113-117.

Futuyma, D. J. 1986. Evolutionary Biology. Sinauer Associates, Inc. Sunderland, MA. pp. 155-159.

Hartl, D. L. 1988. A Primer of Population Genetics. Sinauer Associates, Inc. underland, MA. pp. 116-128.414t+3
Interdemic Group Selection

	"Altruistic" traits which appear to benefit others at some cost to their bearer pose an interesting problem for evolutionary biologists, because natural selection is usually conceived as a competitive interaction among individuals.  Darwin suggested an explanation which has come to be known as group selection; he speculated that the evolution of hymenopteran sterile castes must involve differential success of colony groups which produce such workers to varying degree.

	This simulation duplicates a group selection model presented by Levin and Kilmer (1974).  It assumes that a population is subdivided into separate, randomly interbreeding groups called demes, and that frequencies of altruistic and egoistic or selfish alleles at a single polymorphic locus are affected by (a) selection on individuals within each deme, (b) genetic drift, (c) the exchange of migrants between demes, and (d) demic survival rates which vary with the local frequency of altruistic and egoistic individuals.  Extinguished demes are reformulated by colonists drawn randomly from the population at large.3	Simulation runs can be initiated by introducing a single altruistic mutation, or by a binomial sampling procedure which establishes the altruist frequencies in every deme near some arbitrary starting value.  In the later case, 2N random numbers are drawn for each deme, and one altruistic allele is tallied for every draw which is smaller than the specified starting frequency.

	Each generation incorporates four processes, beginning deme-by-deme with (a) natural selection on individuals.  Our algorithm is identical to the one described for simulations of selection on a single autosomal locus, except that it requests selection coefficients (sAA, sAE, sEE) rather than relative fitnesses (wAA, wAE, wEE).  Note that sAA = 1 - wAA.

	(b) A binomial sample based on the altruist frequency resulting from individual selection is used to simulate genetic drift in each deme.

	(c) A portion of each deme is replaced by migrants from the population at large.  N random numbers are drawn for each deme, and the number of draws which are smaller than the specified migration rate sets the number of individuals to be replaced.  The appropriate alleles are eliminated from each deme by binomial sampling, and replaced from the population at large.  Sampling from the population-wide gene pool occurs without replacement.3	(d) Finally, some demes are extinguished and replaced by colonists drawn from the population at large.  Probabilities of survival (PS) are calculated for each deme as

PSi = a + bqic

where qi is the frequency of altruistic alleles in the ith deme and coefficients a, b, and c define the functional dependence of survival on the ratio of altruistic and selfish genes.

	For a detailed discussion of this model and some of the results it produces, students can consult the original paper by Levin and Kilmer (1974).  Note also that George Williams' influential Adaptation and Natural Selection (1966) has become a basic starting point for informed discussions of group selection.


References

Williams, G. C. 1966. Adaptation and Natural Selection. Princeton University Press. 307 pp.

Levin, B. R., and W. L. Kilmer. 1974. Interdemic selection and the evolution of altruism: a computer simulation study. Evolution 28:527-545.914t+3
Intrademic Group Selection

	Although a deme or randomly interbreeding subpopulation is the unit many evolutionists envision while thinking about group selection, D. S. Wilson has suggested alternative models focusing on the evolutionary consequence of various demic substructures.  For example, while many life histories have a dispersing phase which makes demes very large, ecological interactions which affect fitness often take place in much more localized units, which Wilson calls "trait groups."  After natural selection operates within these trait groups, demic frequency estimates require a weighted average among all trait groups in the deme.5	This simulation is an amalgam of several Wilson models, based on the following scenario:

(1) A large, randomly interbreeding deme with altruistic and egoistic or selfish alleles segregating at any desired starting frequency is randomly subdivided into 100 trait groups, each of size N.

(2) Natural selection operates within each trait group for one or more generations.  Our algorithm is the same as that described for selection on a single autosomal locus, except that fitnesses of the genotypes are

WAA = (1 + bqi)(1 - 2s)
WAE = (1 + bqi)(1 - s) 
WEE = (1 + bqi)        

where A and E represent the altruistic and egoistic alleles, b is the benefit realized by recipients of the altruism, s is the cost sustained by altruists, and qi is the local frequency of altruistic alleles in the ith trait group.  This process affects both the allelic frequencies and sizes of the trait groups.6(3) After selection within the trait groups, a new weighted average allele frequency is calculated for the deme as a whole, and 100 new trait groups (again of size N) are drawn, using a binomial sampling procedure based on this updated frequency.  The program draws 2N random numbers for each trait group and tallies one altruistic allele for each draw which is smaller than the overall demic frequency.
8References

Wilson, D. S. 1975. A theory of group selection. Proc. Nat. Acad. Sci. USA 72:143-146.

Wilson, D. S. 1983. The group selection controversy: history and current status. Ann. Rev. Ecol. Syst. 14:159-87.1014t+3
Genetic Drift: A Monte Carlo Model

	This simulation uses a random number generator to sample genes from a small parental population and pass them on to offspring.  Population size is assumed to remain constant from generation to generation, and gene-frequency changes result only from the random-sampling process.  Drift can be simulated for 1 to 6 diallelic loci simultaneously.  To run the model, you must specify a population size, "N," and initial gene frequencies for each locus.6	Suppose that a population consists of one male and one female, and that both are heterozygous at a locus that has two alleles.  There are four alleles in the total gene pool, 2 "A" alleles and 2 "a" alleles, so p = q = 0.5.  The female will produce "A" and "a" eggs in equal frequency, and the male will produce half "A" and half "a" sperm.  The probability of two independent events occurring together is the product of their individual probabilities, so if gametes are chosen at random and fused to form a filial population of two individuals, the probability that the first individual will be an "AA" is (0.5*0.5) = 0.25, and the probability that both progeny will be "AA's" is (0.5*0.5)*(0.5*0.5) = 0.0625.  Thus if N = 2, there is 1 chance in 16 that allelic frequency will change from p = 0.5 to p = 1.0 in a single generation simply through the random sampling of gametes.

	In addition to the drift from p = 0.5 to p = 1.0, there are other possible outcomes; "p" could change to 0, 0.25, or 0.75, and the likelihood of these events is calculated similarly (you should be able to do it).  A computer model which uses random numbers to mimic this stochastic sampling process is called a "Monte Carlo Simulation."11	The process of genetic drift and its implications are discussed in most treatments of population genetics or evolutionary biology.  For examples, see

Hartl, D. L. 1988. A Primer of Population Genetics, 2nd Edition. Sinauer Associates, Inc. Sunderland, MA. pp 69-77.

Futuyma, D. J. 1986. Evolutionary Biology. Sinauer Associates, Inc. Sunderland, MA. pp 129-131.

Roughgarden, J. 1979. Theory of Population Genetics and Evolutionary Ecology: An Introduction. MacMillan Publishing Co., Inc. New York. pp 57-80.4t+314
Host-Parasite Coevolution+214
a diallelic haploid model

	This model simulates allelic frequency tragectories in a simple host-parasite interaction.  Both host and parasite are assumed to be haploid, with a single diallelic locus.  Allelic frequencies are h1 and h2 = 1 - h1 in the host, and p1 and p2 = 1 - p1 in the parasite.  Parasite1 is most successful on host1, and suffers a fitness penalty when it encounters host2.  Host1 suffers a fitness penalty when it is attacked by parasite1, but not from encountering parasite2.  Encounters are assumed to occur in proportion to the genotypic frequencies.

	To run the model, specify initial allelic frequencies for the host and parasite, the fitness penalty for parasitized hosts and rebuffed parasites, a mutation or migration rate for the parasites, the number of host generations to be simulated, and the number of parasite generations per host generation.12	This model incorporates variations in parasite specificity and host resistence.  Because type1 parasites prosper on type1 but not on type2 hosts, the success of type1 parasites will depend on the relative frequencies of type1 and type2 hosts.  Similarly, the success of type1 hosts will vary with the relative frequency of the parasite types.  This frequency-dependent coevolutionary interaction implies that rare types will realize a fitness advantage relative to the more common ones, giving the model an inherrent tendency toward oscillation.  Depending on the fitness coefficients and mutation-migration rates, these oscillations can damp to equilibrium or form stable limit cycles.5	The transformation equations which are iterated for each new generation of the simulation are as follows:

       h1(1 - sp1)
H1' =                                 
       (sh1) + 1 + (sp1) - s - (2sp1h1)


       p1(h1 + (1 - t)h2)
P1' =                                      
       p1(h1 + (1 - t)h2) + p2(h2+(1 - t)h1)

s = penalty to host of being parasitized,
t = penalty to parasite of being rebuffed.

	For a detailed discussion of this host-parasite coevolution model and its implications, see

Seger, J., and W. D. Hamilton. 1988. Parasites and Sex. pp 176-193 in The Evolution of Sex, R. E. Michod and B. R. Levin, eds. Sinauer Associates, Inc. Sunderland, MA.  see especially Figures 1 & 2 and related text.
1.  "h1" and "p1" are the initial frequencies of "1" alleles in the host and parasite populations, respectively.  Entries can range from 0 to 1.0.

2.  "s" and "t" are the fitness penalties for parasitized hosts and thwarted parasites, respectively.  They require entries ranging from 0 to 1.0; a low value implies that the attack or rebuff has little effect on the victim, while a high value (1.0) denotes a lethal encounter.

3.  The model incorporates two-way mutation or migration in both the host and parasite populations.  Thus h1 and p1 alleles are converted to h2 and p2 alleles (and vice versa) at a rate "m," ranging from 0 to 1.0.

4.  The simulation can be run for intervals ranging from 1 to 5,000 host generations, and in many cases a substantial run-time is necessary to make the final outcome completely clear.

5.  Many parasites have shorter life histories than their hosts, and the model allows up to 100 parasite generations for each host generation. Enter a value from 1 to 100.8t+314
Population and
Quantitative Genetics

	This module illustrates connections between simple population and quantitative genetic models.  For a given set of fitnesses, the program will show how mean fitness, heritability, and allelic frequencies change over time, and also the adaptive topography.
	The graphs show four kinds of output, illustrating a one-locus deterministic selection model with random mating and discrete generations.8
1.  For any given set of relative fitnesses, there exists a curve called the adaptive topography that shows how the average fitness in the population changes as a function of allelic frequency.  Sewall Wright proved that allelic frequencies always change in such a way that the mean fitness increases over time, i.e., populations climb "adaptive peaks" and do not go down into "adaptive valleys".  To see this, try several different types of fitnesses.  If the heterozygote is the most fit genotype, then the adaptive topography will have a peak at intermediate allelic frequencies; the population will "climb" this peak.  If the heterozygote is the least fit, then there will be a valley at intermediate frequencies (verify that the population does not go down into the valley).  If one of the homozygotes is the most fit, then the function will have no peak, and the population will move to fixation for the fittest genotype.6
2.  Heritability is defined as

Additive genetic variance for fitness
                                     
Total variance for fitness

The additive variance is that part of the variance that is attributable to the effects of alleles.  It does not include the variance that is due to interaction between alleles at a locus (that's dominance variance), to interaction between alleles at different loci (that's epistatic variance), or to environmental variation (that's environmental variance, ve).
	Notice that the heritability might rise or fall at first in any particular simulation, but eventually it goes to zero if you have specified enough generations to reach equilibrium.  This is because the additive variance (which is the numerator of the heritability) inevitably goes to zero for simple deterministic models.8
3.  Mean fitness in a population changes as a function of time.  It differs from the adaptive topography, which shows how mean fitness changes as a function of allelic frequency.  Notice that for any fitnesses, the mean fitness will always increase over time.

4.  Allelic frequency trajectories, i.e., allelic frequency as a function of time.  Allelic frequencies can increase to one, decrease to zero, or reach a stable intermediate frequency, depending on the fitnesses that you put in.  If one of the homozygotes is the most fit genotype, then the frequency will go to zero or one.  If the heterozygote is the most fit then there will be a stable intermediate equilibrium.  If the heterozygote is the least fit then there will be historical effects; i.e., the ultimate state reached by the population will depend on the initial allelic frequency.7t+314
Selection on Two Loci

	This module shows how selection operates when two loci influence fitness.  There are more variables to keep track of than in the one-locus case.  With alleles "A" and "a" at one locus, and "B" and "b" at the other locus, there are four types of gametes:  AB, Ab, aB, ab.  A new parameter "D" measures the statistical association between alleles at the two loci.

	Under a special circumstance selection at two loci behaves just like selection at one locus.  The special condition is additivity of fitness effects, i.e., the fitness of AABB individuals is just the sum of fitness effects of AA plus effects BB.  Under additivity, equilibria for each locus appear just as they would if the other locus were not there.6	Another property of additive models is that D goes to zero, i.e., alleles at the two loci become randomly associated, even if they were initially non-randomly associated.  This also occurs when there is no selection, as shown by the default parameters.  The rate at which D goes to zero in these cases depends on R, the recombination fraction.

	If fitness effects are non-additive ("epistasis"), then the situation can become very complicated.  D will not necessarily go to zero.  There can be many polymorphic equilibria, some stable and some unstable.  The principle of maximized mean fitness no longer applies.  The existence of certain equilibria is sensitive to R.  Usually, strong non-additive selection and small R causes equilibria with non-zero D, i.e., selection builds up combinations of alleles that work well together, even though recombination tends to tear them apart.



Further reading:

Hartl and Clark, Principles of Population Genetics, Chapter 4.
Alt-Z - Activate/deactivate

When it is active,

  Enter     - Zoom in
  Z         - Zoom out by a factor of 2
  R         - Restore ranges to those set in the model
  Tab       - Go to next graph (if more than one is on screen)
  Shift-Tab - Go to previous graph (if more than one)
  Alt-C     - Close (hide and remove from memory)
  Esc       - Deactivate and return to data entry

  Movement:
    N           - Set cursor to opposite corner of zoom box
    Arrows      - Up, down, right, left
    Ctrl-left & right arrows - In, out (for 3D plots)
    PgUp, PgDn  - All the way up or down
    Home, End   - All the way left or right

t+314
Frequency-Dependent Selection:
ESS Model

	One way to deal with the complexities of frequency dependent selection is to simplify the genetics, while retaining the frequency dependent nature of the fitnesses.  The theory of Evolutionary Stable Strategies (ESS) does this by modeling frequency dependent evolution for a hypothetical asexual species.

	This module simulates the evolution of discrete phenotypes in a large asexual population.  In ESS terminology, the phenotypes are called "strategies", here denoted A, B, C, and D.  Individuals interact in randomly formed pairs and affect each others fitness.  A and B are "pure strategies", i.e., these two types of individuals always behave the same way in their interactions with other individuals. C and D are "mixed strategies"; they sometimes behave like A, and sometimes like B.

	An evolutionarily stable strategy is defined as a strategy that cannot be invaded by any other strategy when almost everyone in the population adopts it.  This module will allow you to specify two pure and two mixed strategies, and to determine which strategies can invade and which can be invaded.5	The set up of the model is as follows.  There is a "payoff matrix" for the two pure strategies, specifying the increment to fitness obtained by A or B when interacting with A or B:

							   Interacting with

								A 		B

							A	E1		E2
				Payoff to
							B	E3		E4

The fitness of strategies A or B is assumed to be a constant, plus terms that reflect payoffs in random encounters with other A's and B's:

Fitness of A = Constant + P x E1 + Q x E2

Fitness of B = Constant + P x E3 + Q x E4

where P and Q are relative frequencies of A and B respectively.  In these simulations we assume that the constant = 10.11	In addition to the pure strategies A and B, there are two mixed strategies, C and D.  These mixed strategies sometimes behave like A, and sometimes like B.

	The dynamics for any strategy are modeled after an asexual population.  For instance:

                     (Old frequency of A) X (Fitness of A) 
New Frequency of A=                                     .
                          Mean Fitness of Population       

This equation is used to determine whether a particular strategy can invade a population consisting primarily of another strategy.3	An ESS is a strategy that cannot be invaded when common. In general, if the payoff matrix is

								A		B

					Strategy A	E1		E2

					Strategy B	E3		E4


then there will be a mixed ESS if E1<E3 and E4<E2, the ESS being to adopt behavior A with probability

E2 - E4
                 .
E2 + E3 - E1 - E4

Try finding the ESS for the default payoff values on the input screen.


Further reading:

J. Maynard Smith, Evolution and the Theory of Games, Cambridge University Press, 1982.
	Use arrow keys to make a menu selection, and press
ENTER to begin the simulation.  When you have finished
exploring a model, press ESC to restore the main menu.
Explanatory help screens can be called up at any point
in the program by pressing F1.  A second press of the
F1 key leads to a Main Help Menu, and an explanation of
all keyboard controls implemented by the program.

	Note that settings for graphics mode, printing,
memory management, stability analysis, and file
handling are accessed through the Populus options menu
(press <Alt-O>).

Press ENTER to begin.
PRINTERS

	To print a plot, narrative introduction page, or any text mode screen (like the parameter input screens), press <Alt-P>.  If an error occurs, make sure that the printer is properly attached to your computer, on-line, and not busy.  Also make sure you have selected the the right printer driver and the right LPT or COM port from the options menu.
	Populus uses Ryle Design's BGI Printer Driver Toolkit which supports a number of printers at various resolutions.  Choose "Printer Setup.../Printer..." on the options menu (called up by pressing <Alt-O>) to select the printer you want to use.  In addition to setting the print resolution, you can set the page orientation, and you can choose to send the output to another printer port, or to the disk.
	You can send any print-out to the disk, for any printer type, whether you have a printer connected to your computer or not.  You can then copy the print-out file to a computer that is connected to the appropriate printer, and print the file using the DOS "print" command.  If you print a graphic to disk using the postscript driver, a standard postscript text file is produced.


VIDEO

	The options menu lists a number of different graphics drivers under the "Graphics Mode..." option.  The highest resolution mode for the selected driver is always used.
	You can choose to use monochrome colors by selecting the "Use Monochrome" option on the options menu.  You might want to do this if you are running Populus on a portable computer that uses an LCD display with VGA color emulation.        This package was produced by the Minnesota evolutionary
ecology group to foster instruction in population biology.  It
is underwritten by IBM, the University of Minnesota, and the NSF
and is not a commercial venture; we encourage you to distribute
it without charge to any colleague or student who will put it to
good use.  Development of the software is continuing, and we
would appreciate suggestions.  If you find a bug please let us
know; we will fix it and provide you with an update.

                          Don Alstad
         Department of Ecology, Evolution & Behavior
                   University of Minnesota
                      318 Church St. SE
               Minneapolis, MN, USA 55455-0302
                         612-625-0488
                      DNA@VX.CIS.UMN.EDU
	The first graph shows expected frequencies (p) of the "A" allele (on the ordinate) against future generations (on the abscissa).  Many different curve shapes are possible, depending on the relative fitnesses specified.  By experimenting with the simulation model, you should gain an understanding of the way in which both allelic frequency and relative fitnesses affect the rate of allelic frequency change.

	By pressing the space bar you can see three additional output screens generated by this model.  One graphs genotypic frequencies in future generations, one shows the change in allelic frequency in a single generation from any starting frequency, and the last graphs the adaptive topography (population mean fitness as a function of allelic frequency).10t+314
Heritability

	Heritability expresses the degree to which phenotypic variation is determined by genetic variation. A heritability of one means complete genetic determination, while a heritability of zero means no genetic determination. This module computes the theoretical heritability of a quantitative trait in a hypothetical infinite population, and also simulates a Monte Carlo breeding experiment to estimate heritability in a finite population.7	The total variance of phenotypic values in a population can be broken down into components: additive variance, dominance variance, epistatic variance, and environmental variance. The first three components added together constitute the total genetic variance.

	Additive variance is the part of the variance that is due to the effects of inidividual alleles. Dominance variance arises from the inter-action of alleles at a single locus. Epistatic variance arises from the interaction of different loci.

	Evolutionary biologists and plant and animal breeders are particularly interested in the additive variance, because it is the "usuable" genetic variance. That is, in a sexually reproducing species parents transmit alleles to their progeny, but not intact genotypes or multi-locus genotypes. For this reason additive genetic variance is the most important quantity for predicting a population's response to selection. With additive variance a population can evolve; without it, there is usually no evolutionary change.6	The usual measure of heritability, which is called narrow-sense heritability, is defined as the proportion of the total variance that is additive, i.e.,

      Va
hn2 =                  
       Va + Vd + Vi + Ve

where h2 is heritability, the subscript n indicates narrow sense, and the subscripts a, d, i, and e are additive, dominance, epistatic, and environmental variances respectively.

	In the Monte Carlo simulation, the heritability is estimated from the regression of offspring phenotypes on the average of the parental phenotypes. The heritability estimate is equal to the slope of the best-fit line.


Further reading:

Falconer, D.S. Introduction to Quantitative Genetics.PKBGI Stroked Font V1.1 - Jul 9, 1992
Copyright (c) 1987,1988 Borland International
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	          5t14+3
Selection and Mutation

	Empirical studies of variability in natural populations often show rare, deleterious alleles that one might expect to be eliminated by natural selection.  Huntington's chorea, cystic fibrosis, Tay-Sachs syndrome, and sickle-cell anemia are human diseases caused by rare deleterious alleles.  While an overdominant advantage affects the frequency of sickle-cell (and possibly Tay-Sachs) in some selective environments, others have no positive effect whatever.  Their frequency probably reflects a balance between the rates at which deleterious mutations appear and are eliminated by selection.  As an example, consider the case of a deleterious recessive.  At low frequencies (small q) its phenotype is seldom expressed because the heterozygous and homozygous genotypes in which it occurs are rare (2pq) and rare-squared (q2), respectively.  On the other hand, as the recessive becomes less common it appears through mutation from the alternative allelic type more frequently, because a larger fraction of the gene pool is mutable.  As a result, we can expect an equilibrial frequency where the rates of selective removal and mutational origin balance.5	Selection-mutation balances are easily simulated with an analog of our autosomal selection model.  We will assume that selection operates against the "a" alleles, and that while p (the frequency of "A" alleles) changes under the influence of selection, mutation converts some "A" alleles to "a" at a rate  per generation.  Since the interesting equilibria occur when "a" alleles are rare and reverse mutations will be unimportant,

       (pwAA + qwAa)(1-)
p' = p                  
                           

where wAA is the relative fitness of "AA" genotypes, etc.

	Since gene expression is important in determining whether selection actually operates against an allele when it appears in heterozygotes, dominance has a strong effect on the mutation-selection balance.  It is convenient to define one additional term, "h" such that wAA = 1, wAa = 1 - hs, and waa = 1 - s (where "s" is the selection coefficient against "aa").  When h = 0, "a" is completely recessive relative to "A," and when h = 1 it is completely dominant.  h = 1/2 gives the additive case where heterozygote phenotypes are exactly intermediate between the two homozygotes.7	One of the interesting consequences of the selection-mutation balance is a reduction in population mean fitness of 1 - (1-) = , called the mutational load.  Counterintuitively, this load is independent of s, the selection coefficient against the different mutations, because severely detrimental mutations reach a lower equilibrial frequency than those that suffer a minor penalty and affect a larger fraction of the population.



References

Crow, J. F. and M. Kimura. 1970. An Introduction to Population Genetics Theory.  Harper & Row. New York. pp. 258-262.

Hartl, D. L. and A. G. Clark. 1989. Principles of Population Genetics. Sinauer Associates, Sunderland, MA. pp. 199-201.

Maynard Smith, J. 1989. Evolutionary Genetics. Oxford University Press. pp. 55-60.1014t+3
Population Structure

	Simple population genetic models usually assume that mating is completely random so that frequencies of the different genotypes in each new generation can be estimated from allelic frequencies in the uniting gametes.  In natural populations with a patchy distribution, this assumption will be violated if the probability of within-patch matings is higher than that of between-patch matings.  This "population structuring" has interesting genetic consequences which can be illustrated with a simple example:4	Suppose that mice have a polymorphic enzyme system with two different alleles, and we have determined the genotype of every individual from the population living in a barn.  In addition, suppose that mice seldom move between parts of the barn, so that the population is subdivided ("structured") into partially isolated subpopulations or "demes."  Genetic drift will raise the frequency of "a" alleles in some demes and lower it in others.  Suppose further that the "a" allele frequency in half of the demes drifts to p = 1.0, and in the other half to p = 0.  Overall gene frequency throughout the entire barn is p = 0.5, and the expected frequency of heterozygotes is 2p(1-p), or 1/2.  However, demes where p = 1.0 will produce only "aa" genotypes, and demes where p = 0 will produce only "bb" genotypes, and there are no heterozygotes in the entire barn.  This difference between the observed and expected heterozygote frequencies is evidence of population structure.
	Inbreeding is a second process which reduces heterozygosity and is conceptually related to population structuring.  In the most extreme inbreeding system, hermaphroditic selfing, all of the progeny of homozygotes and half the progeny of heterozygotes will be homozygotes, so population heterozygosity will decline by 1/2 each generation.  Less extreme inbreeding systems will produce a proportionally weaker decline in heterozygosity depending on the mating probability and relatedness of relatives.9	Sewall Wright introduced several related "inbreeding coefficients" which allow us to measure and distinguish the genetic consequences of mating and dispersal patterns.  To define them, we use three different estimates of heterozygosity:

	Hi is the observed frequency of heterozygous individuals in a deme, or subpopulation, averaged among demes.  It is also the probability that one particular gene locus in an individual will be heterozygous.

	Hs is the expected frequency of heterozygous individuals in the deme.  It is calculated as 2p(1-p), where p is the allelic frequency in that deme, and averaged among demes.

	Ht is the expected frequency of heterozygotes in the entire population, calculated as 2p(1-p), where p is the population-wide allelic frequency.9	Wright's three hierarchical inbreeding coefficients are then defined as follows:

	Fis is the heterozygote deficiency caused by nonrandom mating within the demes or subpopulations, calculated as
Fis = (Hs - Hi)/Hs

	Fst is the heterozygote deficiency caused by population subdivision and the divergent drift of allelic frequencies in the separate demes, calculated as
Fst = (Ht - Hs)/Ht

	Fit measures the overall inbreeding coefficient resulting from both causes, and is calculated as
Fit = (Ht - Hi)/Ht
6	Our simulation assumes that a population is subdivided into 6 demes whose size (and drift rate) can be set by the user.  Gametes are chosen randomly to form a new population each generation.  Users can also set a migration rate, which will cause a fraction of the individuals in each deme to be replaced every generation by migrants which are representative of the population-wide allelic frequency.  Two output graphs plot the allelic frequencies in all six demes, and the three inbreeding coefficients, Fis, Fst, and Fit.
	Different combinations of drift and gene flow will affect the equilibrial F-values.  In this simulation F-statistics will also be affected initially by historical disequilibria.  The 6 demes can be initiated independently at different (or similar) frequencies to illustrate this founder effect.
	Our program uses the expressions given above to calculate the F-statistics in a simple and instructive way.  Students should be aware, however, that the algorithms used in real empirical research are more complex, incorporating either sample-size adjustments (Nei and Chesser 1983) or analyses of variance on the allelic frequencies (Weir and Cockerham 1984)10References

Hartl, D. L. 1987.  A Primer of Population Genetics. Sinauer Associates, Sunderland MA.  pp 69-85.

Nei, M, and R. K. Chesser, 1983.  Estimation of fixation indices and gene diversities.  ANN. HUM. GENET. 47:253-259.

Weir, B. and C. C. Cockerham. 1984.  Estimating F-statistics for the analysis of population structure.  EVOLUTION 38:1358-70.

Wright, S. 1968. Evolution and the Genetics of Populations.  University of Chicago Press.PKBGI Stroked Font V1.1 - Jul 24, 1992
Copyright (c) 1987,1988 Borland International
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 $B\n(<b,<JXft    	
		                                                                                                                                                                                                                                                            ~          ~                       ~   ~                                                                ~                                      	                                    		   		                                                                                     ~~                                                                                                                                                                                                                                                  614t+3Theta-Logistic Predator-Prey

	This program is designed to illustrate the population dynamics that occur when a predator population and a prey population interact.  Predator-prey interactions occur in virtually all natural communities.  One of the fundamental properties of most such interactions is an inherent tendency to generate population cycles in both predator and prey.  Cycles do not always occur; it is also possible for predator and prey to approach a constant population size over a long time period.  Whether sustained population cycles occur or not is largely determined by two factors: (1) the population growth of the prey in the absence of the predator, and (2) the relationship between prey population size and the amount of prey eaten by an average predator individual (this relationship is known as the functional response).  All else being equal, the probability of cycles is increased when the prey exhibit little density dependence in their population growth, and the predator's functional response increases rapidly as prey density increases.4
	The predator-prey model which is illustrated in this program has the following form.  The rate of change in the prey population size (denoted by N) with time is described by the following equation:

dN/dt = rN(1 - (N/K)) - fP,

In this equation, P is predator population size, f is the predator's functional response, and the prey population growth rate in the absence of the predator is given by the rN(1 - (N/K)).  This is the familiar logistic model of population growth with an additional term, given by the Greek letter theta, which allows different types of density dependence to be described.  If theta is large, the probabilities of birth and death do not change much until the prey population approaches its carrying capacity.  If theta is small, per capita birth or death rates (or both) decrease rapidly with increasing population size, even at small population densities.  This model of population growth was originally proposed by Gilpin and Ayala (1973), and is discussed in the ecology textbook by Ricklefs (1979).  An illustration of this population growth model may be seen by pressing F1 at this point.7
	The population dynamics of the predator are described by

dP/dt = sP(f - D).

The term f is again the functional response.  D represents the intake rate of prey required for a predator to just replace itself in the next generation.  s measures how rapidly the predator population responds to increased or decreased abundance of available prey.  This form of the predator growth equation makes two implicit assumptions; (i) the predator population density does not affect an individual predator's chances of birth or death directly (only indirectly via effects on the prey population size), and (ii) the number of surviving offspring produced by a predator is directly proportional to the amount of prey it consumes.8
	The remaining component of the model is the functional response, which has been denoted by f in the above equations.  The functional responses of many predators have been determined in laboratory experiments in which different numbers of prey are placed in an arena with a predator for a specified amount of time.  The Canadian ecologist, C. S. Holling (1965) categorized the functional responses into 4 possible types.  Three of these have been frequently observed (Hassell 1978) and are discussed in most ecology textbooks.  The type 1 response rises linearly with prey density; the type 2 response rises at a continually decreasing rate, and the type 3 response is sigmoid ('S' - shaped); illustrations of these three types may be found in most ecology textbooks.4
	The type 1 functional response by definition is given by a constant C1 multiplied by prey population density N.  There are many different mathematical formulas for representing type 2 and type 3 responses, but the following two are those that are used most commonly:

   C2N/(1 + h2C2N) for the type 2 response, and

   C3N2/(1 + h3C3N2) for the type 3 functional response.

The parameters C and h can have a number of different possible biological interpretations.  As the number of prey becomes very large, both of the above functional responses approach an asymptotic value of 1/h.  One possible interpretation of h is that it represents the amount of time required to handle a single prey item; at very high prey populations, a predator spends almost all of its time handling (and very little time searching), so the rate at which it catches prey is just
1/h.  Under this interpretation, prey are never captured while another prey item is being handled, and are captured at a rate C2N (type 2) or C3N2 (type 3) while the predator is searching for prey.  If you would like to see how the parameters C and h affect the shape of the functional response, hit the help key, F1. (?)7
	Rosenzweig and MacArthur (1963) developed a general method for determining whether a predator-prey system would undergo population cycles, using what is known as phase-plane analysis.  This is explained in detail in many ecology textbooks; see, for example, Begon and Mortimer (1986).  The phase plane (in this case) is a plot whose x axis is prey population density, and whose y axis is predator density.  The method involves drawing the two lines; the first (the "predator isocline") is a vertical line, representing that prey density which results in zero population growth of the predator.  The second line, the "prey isocline" represents those predator and prey densities which result in zero population growth of the prey.  Rosenzweig and MacArthur's rule was that, if the prey isocline is increasing where the two lines cross, there will be cycles, and if it is decreasing, both species will ultimately reach stable densities.5
Literature Cited
s
Begon, M. and Mortimer, M. 1986. Population ecology: A unified study of animals and plants. 2nd ed. Sinauer.  Sunderland, Mass.

Gilpin, M. E. and Ayala, F. J. 1973. Global models of growth and competition. Proc. Nat. Acad. Sci. 70:3590-3593.

Hassell, M. P. 1978. The dynamics of arthropod predator-prey systems. Princeton University Press. Princeton, N. J.

Holling, C. S. 1965. The functional response of predators to prey density and its role in mimicry and population regulation. Mem. Ent. Soc. Can. 45:3-60.

Rosenzweig, M. L. and MacArthur, R. H. 1963. Graphical representation and stability conditions of predator-prey interactions. American Naturalist 97:209-223.
1.  "N" sets the number of individuals which are randomly placed in each trait group, before individual selection affects within-group frequencies or group sizes.

2.  "G" sets the number of generations of individual selection which take place within the trait groups, between episodes of dispersal, mixis, and reassortment into new groups.  Values can range from 1 to 6.

3.  "b" sets the benefit derived by recipients of the altruism, while "s" sets the cost sustained by the altruists.  Values for "b" range from 0 to 10, while "s" values range from 0 to 1.

4.  As in the interdemic group selection simulation, runs can be initiated with a single altruistic mutant, or with any desired allelic frequency.
General:
  F1    - Help (press twice for the main help menu)
  F2    - Title/Introduction/Background
  Alt-O - Options menu (video, printer, files, and more)
  Esc   - Main Menu or Cancel

Models:
  Enter      - Draw a graph to match the current parameters
  Ctrl-Enter - Force a recalculation, then draw a new graph
  F4         - Plot last data toggle (where available)
  Alt-F1     - Switch to last open model
  Alt-F4     - Return to graph without recalculating
  Alt-F5     - Save the model parameters
  Alt-F6     - Load some previously saved model parameters
  Alt-F7     - Save the data
  Alt-C      - Close model (remove from memory)

Plots:
  Alt-G - Grid toggle (3 levels)
  Alt-L - Log plot toggle (population growth models)
  Alt-Z - Zoom toggle (see Zoom help)
  Alt-S - Stability Analysis toggle (see Stability Analysis help)
  Alt-F - Plot speed toggle

  Cursor keys move through multiple plots.
  (Also, see help screen for stability analysis.)

Data Entry:
  Ins       - Toggles edit mode
  F5        - Restore last entry
  F6        - Restore one default value
  F7        - Restore all defaults
  F8        - Clear entry
  F9/F10    - Min/Max allowed value
  Home/End  - Start/End of entry
  PgUp/PgDn - First/Last entry
  Space Bar - Move through parameters in sequence
  e or E    - Exponential notation8t+314
Density-Independent
Population Growth

	Two sets of counteracting processes affect population density; birth and immigration tend to increase populations while death and emigration decrease them.  For this simulation we will assume that immigration and emigration counteract and balance each other, leaving birth and death as determinants of population growth and density.  We will also assume that all individuals in the population have equal probabilities of dying or producing young, and that environmental resources are in infinite supply.6	First, consider an annual plant, or an insect with non-overlapping generations.  Most individuals in the population are at similar growth stages, and this DISCRETE growth is best described by a finite difference equation.

			If:		Nt = pop size of females at time t
					b = births per female per time interval
					p = probability of surviving the time interval

			Then:	N1= N0 p + p b N0
					Nt+1 = pNt + b p Nt = (p + bp)Nt =  Nt
					Nt =  Nt-1 =  [ Nt-2] = tN0
					Nt = tN0

	"" is a finite rate constant called the "coefficient of geometric increase."  It makes most sense to think of discrete, geometric growth in terms of time intervals equal to one generation because birth and death are not usually evenly distributed through the generation interval.4	Now let's consider organisms like humans, or bacteria in a culture flask, with a continuous breeding season and overlapping generations.  The CONTINUOUS growth of these populations is best described by a differential equation.

			If:		N = population size (the number of females)
					b = instantaneous birth rate per female 
					d = instantaneous death rate per female
					r = b - d = instantaneous growth rate per female

			Then:	dN/dt = (b-d)N
					dN/dt = rN

note that the per capita growth rate is constant:
					dN/Ndt = r

integrating to solve for N:
					N = N0ert

	"r" is the "intrinsic rate of increase" or exponential growth rate of the population.  It is an instantaneous rate.8	The discrete population grows in steps while the continuous model is smooth, but they both show similar dynamic trends; with  > 1 or r > 0 (for discrete and continuous populations growing at the same rate, r = ln), these models give mathematical descriptions of an explosion.  They are directly analogous to the growth of a bank account (N = principal, r or  = interest rate), but biologically unrealistic in assuming an infinite resource base.


References

Elton, C. 1958. The Ecology of Invasions by Animals and Plants. Methuen, London.

Ricklefs, R. E. 1990. Ecology (3rd edn). W. H. Freeman and Co. New York. pp. 302-308.

Wilson, E. O. and W. H. Bossert. 1971. A Primer of Population Biology. Sinauer Associates, Sunderland, Massachusetts. see pp. 92-102.t+314
Host, Parasitoid and
Hyperparasitoid

	The final model we have taken from Hassell's 1978 book is a three-trophic-level simulation illustrating host (N) and parasitoid (P) interacting with a hyperparasitoid (Q).  The hyperparasitoid is assumed to reproduce only in hosts that have been previously attacked by the parasitoid.  Hassell's model invokes aggregation for its stability, as in the previous nonrandom searching and competing predator models, using the recursions

Nt+1 = Nt (1+a1Pt/k1)-k1

Pt+1 = Nt [1-(1+a1Pt/k1)-k1](1+a2Qt/k2)-k2

Qt+1 = Nt [1-(1+a1Pt/k1)-k1] [1-(1+a2Qt/k2)-k2]

All terms retain the meanings introduced earlier in this set, and reiterated in the input help screen (press <F1> from the input screen).  In particular, a1 and a2 are the searching efficiencies of the parasitoid and hyperparasitoid, respectively, and k1 and k2 are their dispersion coefficients.10	Again, a thorough stability analysis by May and Hassell (1981) shows local stability of the three-party interaction as long as the parasitoid and hyperparasitoid both aggregate in patches where the density of their respective prey is high.  In particular, such interactions are more likely to be stable when the searching efficiency of the hyperparasitoid is higher than that of the parasitoid (a2/a1 > 1).

References

Hassell, M. P. 1978. The Dynamics of Arthropod Predator-Prey Systems. Monographs in Population Biology, Princeton University Press. Princeton, NJ. pp. 157-164.

May, R. M. and M. P. Hassell.  1981.  The dynamics of multiparasitoid-host interactions. Am. Nat. 117:234-261.
	The graph header lists R0, the net reproductive rate (the number of female offspring expected to accrue to an individual female during her entire life); G, the generation length (the average time elapsing between the birth of a female and her offspring); r, the intrinsic rate of increase determined by successive approximation using Lotka's equation; and (ln R0)/G, an easily calculated estimate of r.  How do the life table values affect the accuracy of this estimate?

	Both the lambda and Nx/Nx plots have a tendency toward damped oscillation, which will vary with the life table values you specify. What are the causes and implications of these patterns?7t+314
Age-Structured Population Growth

	Youngsters and oldsters give birth and die at different rates, and these age-specific survivorship and natality changes can be summarized in a "life table," or "lxmx schedule."  This simulation performs several calculations using life table data, and illustrates the effect of age structure on population growth.

	The Net Reproductive Rate, R0, gives the number of female progeny expected to accrue during the entire lifetime of an individual female.  It is calculated as

R0 = lxmx,

the sum of offspring produced in each age interval, weighted by her probability of surviving to that age.

	The mean generation length, G, is estimated as

   nlxnmxnx
G =         ,
    n   R0   

the weighted average of a female's ages at the time when each of her progeny is born.

	The intrinsic growth rate, r, can be approximated from life table data as

(ln R0)/G,

or determined with any desired precision by successive approximation using Euler's (some-times also called Lotka's) equation,

1 = e-rxlxmx

	The expected value of an organism's future offspring relative to its own current value (called the Reproductive Value, Vx) is calculated as

     nx=ane-rxnmxnlx
Va =             
     n     e-rxnla     

	The simulation requires specification of an initial population composition (the number of individuals in each age class) and an lxmx schedule; it uses the life table data to calculate new compositions over 10 subsequent generations.  This is easily (but tediously) done directly from the life table, but our computer program represents the population composition as a vector whose elements give the number of individuals in each age class.  To produce a new generation, this vector is multiplied by a transformation matrix (the "Leslie Matrix") incorporating lx and mx values from the life table.  Euler's equation giving the intrinsic rate of increase of the population is the characteristic equation of this projection matrix.

References

Leslie, P. H. 1945. On the use of matrices in certain population mathematics. Biometrika 33:183-212.

Emlen, J. M. 1973. Ecology: an evolutionary approach. Addison-Wesley Publishing, Reading, MA. pp. 240-260.

Krebs, C. J. 1985. Ecology: The Experimental Analysis of Distribution and Abundance, Third Edition. Harper & Row. New York. pp. 174-206, 717-720.

Ehrlich, P. R. and J. Roughgarden. 1987. The Science of Ecology. MacMillan Publishing Co. New York. pp. 75-90.
1.  This simulation produces five different output graphs and an output table.  (a) Lambda v. T plots the discrete population growth rate for 10 generations after T0. (b) Nx v. T gives population size (the sum of all individuals in all age classes) v. time.  (c) Nx/Nx v. T shows the proportion of the population currently in age class x, for 10 generations after T0.  (d) Vx v. x demonstrates how reproductive value (the expected value of future offspring relative to the current value) changes with age.  (e) x v. Nx/Nx gives the overall age distribution; if you begin the simulation showing this output, you can increment generations and watch the age structure change; alternatively, if you cycle to this output after a simulation is complete, the final age distribution that will be displayed.  (f) The tabular output option prints the population composition vector and the relative requencies for each age class, by generation, on the text mode parameter input screen.

2.  Specify the number of age classes that you would like to use (up to 50).

3.  Specify which age class you would like plotted for the Nx/Nx v. T graph.

4.  Specify the number of generations (up to 10 you would like the simulation to run.

5.  Values of lx, the probability of surviving from birth to age x, can range from 0 to 1.0.  Values of mx, the number of births in one time interval per adult female aged x to x+1, can be real numbers up to 10E9.  To specify an initial population at time T0, enter a number of individuals in each age class, ranging from 0 to 10E9.
	On the first lines of the screen, specify whether your model will be cast as a set of discrete difference equations or continuous differential equations.
	Next you may choose to view a plot of numbers vs. time or a phase plane (or space) diagram of numbers vs. numbers.  You may also toggle between the two views using the space bar after the first output graph is shown.
	The third line allows you to stop the simulation before it reaches a steady state.  Remember that many models will not have stable equilibria and would cycle indefinitely unless you stopped them by specifying a runtime or pressing <ESC>.
	On line four, specify whether you would like the isoclines or isosurfaces plotted.  Isosurface plots for a complex model are time consuming, but may be terminated at low resolution with <ESC>.
	The parameter screen has two scrolling boxes.  The upper box allows you to specify up to nine equations, and to check which will be run (up to 9) and which will be plotted (up to 3).  Instructions for entering equations are given in the intro screen.
	Numerical values for all parameters and all initial population densities (Ni0) must be entered in the second scrolling box.
	When you type "Enter" the program will begin to simulate the equations and will show you the time and population densities at intervals in the "thinking box" as the calculations proceed.
	Because we cannot anticipate the models that you will run with this engine, we can only suggest here, that the populus utilities for stability analysis, gridding, and zooming are likely to be useful in interpreting your results.  More information about those utilities can be had on the main help menu, press <F1>.3t+314
Interaction Engine

	In ecological communities, large numbers of species interact in a variety of different ways, resulting in a huge range of possible patterns of population dynamics.  Some idea of the range of possibilities may be gained by examining models of the interaction of three to nine species.  This module allows you to simulate the population dynamics of a community consisting of any number of species between one and nine.  In the one- and two-species cases, it can be used to model population growth, competition, and predation, which are explored in more detail in other modules in this package.  Using more species, you can examine the dynamics of communities in which several different pair-wise interactions occur.  Some of the possibilities which you may want to examine are: (1) a linear food chain; (2) a group of species all of which compete with each other; (3) two predator species, each of which eat two prey.  There are no restrictions on the kinds of ecological systems you can simulate with this general-purpose modeling engine.  You may want to consult a textbook or other sections of Populus to get ideas about the form of dynamical equations for interesting cases.5	Differential equation models of three or more interacting populations differ from those with two populations in two important respects: (1) it is possible for the population dynamics to be chaotic when there are three or more species; and (2) indirect effects occur whenever species i affects the per capita growth rate of species j, and j affects the population growth rate of k.  Chaotic dynamics imply that the populations undergo continual change in which a particular set of population levels never occurs more than once, no matter how long the dynamics continue.  In addition, the difference between the population densities in two different systems which are initially very similar in population densities, increases rapidly (exponentially) with time.  For more information on chaos, see the book by Holden listed below, or the article by W. Schaffer.  Chaos cannot occur in systems of two ordinary differential equations, although it can occur in difference equation models with one or two species.  It is not yet known how frequently ecological models exhibit chaos.  Chaos seems to be relatively rare in Lotka-Volterra-type models, in which the per capita growth rate of each species is a linear function of the population density of each other species.  However, Michael Gilpin (see below) has shown that there are parameter values for a Lotka-Volterra one predator-two prey model that result in chaos.
	Indirect effects have received more study than chaos.  One of the simplest possible indirect effects occurs when there are three competing species.  Species 1, for example, has a direct negative effect on each of the other two species by the definition of competition.  However, it also has an indirect positive effect on each of the other two; it benefits species 2 by competing with (and reducing the population density of) competitor 3, and benefits species 3 by competing with species 2.  Thus, it is possible for an increase in the population density of species 1 to increase the equilibrium density of species 3, if the indirect effect is larger than the direct one.  There are a large number of possible indirect effects when there are four or more species.

	This program will allow you to explore these and other aspects of multi-species interactions.  Your models may be phrased either as continuous differential equations or discrete, finite difference equations.  Our engine will parse your expressions, perform a numerical integration or step the difference equations, and plot the resulting dynamics.  The input screen presents you with a number of choices for plotting options, simulation length, and other details, which are explained on the help screen.  The default values represent Gilpin's one-predator-two-prey model, which exhibits chaotic behavior.  Be warned that this model can take a relatively long time to run:  the default time value, t=400, runs about 2 minutes on a 16MHz 386 computer.  If you have a slow machine, you may want to specify short runtimes, or interrupt this simulation (with <ESC>) before its completion.6	The default equations presented may be changed in any way you would like.  The equations may be entered in the same way that you would write them (you do not need to use * for multiplication, for example; aN1 is interpreted as a multiplied by N1; 3b is interpreted as 3 times b).  The four population variables must be represented by a capital N followed by 1, 2, 3, or 4.  The program recognizes any single letter as a distinct parameter, and it recognizes upper and lower case letters as distinct parameters.  If you would like a string of letters or letters and numbers to represent a single parameter (for example, a1), you can make the program recognize this by entering the string (e.g. a1) after the "Define a parameter" input statement.  Each such string must be entered separately.  It will then appear below the double line, and you can enter its numerical value there.

	As a shortcut to assigning values to parameters, you may type the parameter name follow by "=" and an expression (perhaps involving other defined parameters) into the "Define a parameter" input box.  For example, to define the parameter a1 and assign it a value of 3b2, you would type "a1=3b^2" (the "^" symbol denotes exponentiation).12References

Gilpin, M. E. 1979. Spiral chaos in a predator-prey model.  American Naturalist 113:306-308.

Holden, A. V. 1986. Chaos. Princeton Univ. Press.

Schaffer, W. M. 1985. Order and chaos in ecological systems.  Ecology 66:93-106.

Yodzis, P. 1989. An Introduction to Theoretical Ecology.  Harper and Row.
	Programmers joke that there are bugs in any routine longer than three lines.  Populus now compiles from about 70,000 lines of source code; moreover, its entire structure has been rebuilt with object-oriented techniques in the year since version 1.46 appeared, creating bugs and anomalies ad nauseam.  While accepting full responsibility for any problems that remain, I thank the dedicated crew Minnesota grad students and postdocs who have served as beta testers for this new version, especially

               Paul Cabe,                      Brian Farm,
               Hank Fukui,                     Sara Hotchkiss,
               Kevin Johnson,                  Sue Lewis,
               Karen Oberhauser,               Jim Roth,
               Dann Siems,                     Evan Sieman,
               Dave Wedin, and                 Bethany Woodworth.

	For suggesting useful additions to the program, assisting in areas of special expertise, authoring reviews, providing financial support, or donating a forum for disseminating Populus I am also greatful to

                     Lisa Brooks, Brown University
                     Hsin Chi, Chung-Sing University
                     Sean Doolan, Oxford University
                     Aaron Ellison, Mount Holyoke College
                     John Endler, UC Santa Barbara
                     Lou Gross, University of Tennessee
                     Oren Hasson, Hebrew University, Jerusalem
                     John Heywood, SW Missouri State
                     Richard Hoffmann, Iowa State University
                     Sally Jorgensen, University of Minnesota
                     Jeff Karron, University of Wisconsin, Milwaukee
                     Mark Kirkpatrick, University of Texas
                     Bruce Levin, University of Massachusetts
                     Herb Levitan, National Science Foundation
                     Mike Moulton, Georgia Southern University
                     Tim Mousseau, University of South Carolina
                     Mike and Carole Rosenzweig, University of Arizona
                     Bernard Schmid, University of Basel
                     Jon Seger, University of Utah
                     John Thomson, SUNY Stony Brook
                     Don Waller, Universitry of Wisconsin, Madison

	Finally, as a biologist my programming skills are extremely modest.  NONE of this would have been possible without Chris Bratteli, who has worked part time throughout his baccalaureate program at Minnesota, converting my Dick and Jane code and even my fantasies into a coherrent, functioning package.

                                                       Don Alstad
                                                       Minneapolis
                                                       July 1992


                           "popule, vive, precor..."
                                        Ovid, Heroides 5:27]trt~GOCGNC8r8((p(8(0p8pq8B8Cpttrp 	

          ]wpwpwppppppppp                                     

      	   !"#$  *+   ,-  ./012 34567     89:;<       	   !"#$  *+   ,-  ./012 34567     89:;< =   >  	   !"#$  *+   ,-  ./012 34567     89:;< =   >  ?   

      	   !"#$  *+   ,-  ./012 34567     89:;< =      !"#$  *+   ,-  ./012 34567     89:;< =   >  ?   @      89:;< =   >  ?   @   %&'()  YZ[\]      ]wpwpwppp   !"#$  *+   ,-  ./012 34567     89:;< =   >  ?   @   %&'()   !"#$  *+   ,-  ./012 34567     89:;< =   >  ?   @   %&'()  YZ[\] %&'()  YZ[\]      ]wpwpwppppppppp             *+   ,-  ./012 34567     89:;< =   >  ?   @   %&'()  YZ[\]      ,-  ./012 34567     89:;< =   >  ?   @   %&'()  YZ[\]      ]wp./012 34567     89:;< =   >  ?   @   %&'()  YZ[\]      ]wpwp34567     89:;< =   >  ?   @   %&'()  YZ[\]      ]wpwpwp89:;< =   >  ?   @   %&'()  YZ[\]      ]wpwpwppp=   >  ?   @   %&'()  YZ[\]      ]wpwpwpppppp>  ?   @   %&'()  YZ[\]      ]wpwpwppppppp?   @   %&'()  YZ[\]      ]wpwpwpppppppp@   %&'()  YZ[\]      ]wpwpwppppppppp     ABCDEFGHIJKLMNOPQRSTUVWX   ]trt~GOCGNC8r8((p(8(0p8pq8B8Cpttrp YZ[\]      ]wpwpwppppppppp                     Main Menu

Population Growth

	1000Density-Independent Growth
	1001Logistic Population Growth
	1002Age-Structured Population Growth

Multi-Species Interactions

	1003Lotka-Volterra Competition
	1004Resource Competition
	1005Lotka-Volterra Predator-Prey
	1006Theta-Logistic Predator-Prey
	1007Interaction Engine
	1008Optimal Diet Choice Based on Energy
	Mutualism

		1042Lotka-Volterra Mutualism
		010433-Party Mutualisms
	
	Arthropod Predator-Prey Dynamics
		APPD.GR

		1033The Nicholson-Bailey Model
		1041Nicholson-Bailey with Spatial Structure
		1034Functional Responses
		1035Non-Random Searching
		1036Interference
		1037Threshold Predator Reproduction
		1038Polyphagous Predators
		1039Competing Predators
		1040Host, Parasite, and Hyperparasite
		1044Insecticides in Host-Parasitoid Interactions
	

Selection

	1009Woozleology
	1010Autosomal Selection
	1011X-Linked Selection
	1012Adaptive Landscapes
	1013Selection on Two Loci
	1014Selection and Mutation
	Group Selection

		1019Interdemic Group Selection
		1020Intrademic Group Selection
	
	Sexual Selection

		1032Kirkpatrick's Haploid Arbitrary Model
	
	Frequency- and Density-Dependent Selection

		1027Frequency-Dependent Selection: Diploid Model
		1028Frequency-Dependent Selection: ESS Model
		1029Density-Dependent Selection with Genetic Variation
	

Genetic Drift

	1015A Monte Carlo Model
	1016A Markov Model
	1017Inbreeding
	1018Drift and Selection

Differentiation Models

	1021Population Structure
	1022Selection, Gene Flow, and Clines
	1026Multiple Niche Polymorphism

Quantitative Genetics

	1023Population and Quantitative Genetics
	1024Heritability
	1025Directional Selection on Quantitative Character

Coevolution

	Host-Parasite Models

		010302-Allele Haploid
		010313-Allele Haploid
	


1Exit Populus

6t14+3Nicholson-Bailey with
Spatial Structure

	This simulation comes not from Hassell's book, but from a paper by Hassell, Comins & May (1991).  It assumes that the environment is subdivided into an array of rectangular patches.  In each generation two processes affect the simulation dynamics.  First, the host and parasitoid populations in each rectangular patch interact according to the Nicholson-Bailey recursions,

Nt+1 = Nte-aPt

Pt+1 = Nt[1 - e-aPt].

Second, there is a dispersal phase in which a fixed fraction of the hosts (N) and parasitoids (P) in each patch are distributed equally among the eight adjoining patches.11	While the simple Nicholson-Bailey recursions give an increasing oscillation which ultimately concludes interaction with the extinction of hosts or parasitoids, this spatially structured arrangement allows more possibilities.  In general, dispersal may cause the global persistence of the locally unstable Nicholson-Bailey interaction.  The probability of global persistence increases with the size and complexity of the spatial array, and decreases as the hosts become more vagile.  When the interaction persists, several different spatial patterns are possible, including spiral waves of changing host and parasitoid density, fixed "crystal lattices," and purely chaotic variation, depending on the parameter values used.  For a more detailed discussion students should consult the primary reference.7	The simulation produces four different output screens.  Three present the spatial array of patches, color-coding local abundance of the interacting parties.  One screen gives the ratio of prey to parasitoids (N/P), one gives prey density (N per patch), and the third gives parasitoid density (P per patch).  On all three screens colors are coded in spectral order, with long wavelengths (red) representing high densities or ratios and short wavelengths (violet) representing low ones.  A legend below the spatial array gives the specific values (which change from generation to generation) represented by each color.  The final output is a graph of average prey and parasitoid densities per patch, across the entire array, as it changes with time.



References

Hassell, M. P., H. N. Comins and R. M. May. 1991. Spatial structure and chaos in insect population dynamics. Nature 353:255-8.
During the dispersal phase migrants from each patch are equally dispersed among the eight adjoining patches.  For patches on the edge of the spatial array, boundaries can be absorbing (migrants are lost off the edge), reflexive (migrants bounce back into the array), or periodic (migrants wrap to the opposite edge of the array).

Options allow you to begin with any of the four output graphs described in the introductory narrative, to begin the simulation from a cell on the edge or in the middle of the spatial array, to assign either real or integer variable types to the host and parasitoid populations, to display the output array every generation within the run interval, and to set the overall runtime.

Parameters are those of the basic Nicholson-Bailey model with a few additions:

N0 and P0 are the initial host and parasitoid population sizes in the starting patch.

N and P are the host and parasitoid migration fractions leaving the patch in each generation.

Lambda is the host growth rate, q sets the number of parasitoids emerging from each parasitized host individual, a is the search efficiency of the parasitoid (the Nicholson-Bailey "area of discovery") and n sets the size of the spatial array of patches.Populus 2.10, August 1992


Overview

	The Populus software contains a set of simulation models that we use in teaching population biology and evolutionary ecology at the University of Minnesota.  All of the simulations share a common format, as follows:  After a model is chosen from the menu, the program displays (optionally) several screens of background material which introduce the mathematics and end with basic references.  Next, there is a screen listing all of the input parameters; students can move among the parameter boxes and change initial defaults to values of their own choosing.  The program maintains a record including permissible maxima and minima for each parameter and filters input values accordingly.  Usually there are several possible outputs (e.g., N vs T graphs and phase planes) which can also be selected from the parameter input screen; alternatively, students view the different outputs in sequence, by pressing the space bar between views.  Context-sensitive help screens are available (press <F1>) from the input and output screens of every model, and by pressing <F1> twice one can see a menu of additional help screens which includes discussions of function and editing key assignments implemented by the program, printing protocols, video options, etc.


Hardware Requirements

	The program is written in Turbo Pascal to run under DOS on IBM and compatible computers.  To implement the full range of Populus features, about 500KB of RAM are necessary, free of memory-resident utilities and drivers.  The program will use LIM expanded memory, reducing the frequency of disk reads and speeding program execution.  Although the code will emulate an xx87 coprocessor if one is not present, many of the models are computationally intensive and will run much more rapidly on a machine with the floating-point chip.  Since overlays are read during program execution, the program files (populus.exe, populus.ovr and populus.txt) should be copied to a hard disk (or a RAM disk emulator).  If this is not possible, the program will run from two 720K (or larger) diskette drives, and will prompt users for a path to the Populus.ovr file.

	Version 2.10 will run on any IBM-compatible video system which permits graphic displays, including "Hercules," "CGA," "EGA," "VGA," and "IBM8514."  In most cases, the software will recognize and accommodate your video system automatically.  If your display emulates one of the color systems in shades of gray (as many LCD screens do, with varying success), consider using the option which forces output to the monochrome mode.  The IBM8514 system is detected as VGA by Turbo Pascal; IBM8514 users who desire full 1024x768 resolution (beautiful!) can select that video mode from the Populus Options Menu (Alt-O) and save a populus.cfg file to disk so that their preferred configuration loads automatically each time the program is run.

	Populus 2.10 contains a set of commercially produced drivers by Ryle
Design that supports a wide variety of printers, including 9-pin and 24-pin dot matrix printers, HP laserjets and deskjets, the HP paintjet, and printers implementing the Postscript page-description language.  Print files may also be saved to disk and sent to the printer later using the DOS Print command.  Set the appropriate printing configuration on the Populus options menu.  Note that Postscript supports graphics-mode printing only.  Our introductory narratives and all outputs are rendered in graphics mode, but the parameter inputs and help windows are text mode screens.  To print them on a postscript laser printer you will have to set it up as an HP emulator.


Bells and Whistles

	For many problems and exercises it is desirable that students be able to read numerical values accurately from the graphical output.  We have included several new video routines for this purpose.  A gridding function (press <Alt-G>) plots horizontal and vertical lines from the major axis ticks, and a second <Alt-G> grids the minor ticks.  A third press toggles the feature off.  Our video zooming function (press <Alt-Z>) pops up a rectangle whose corners can be moved (with the cursor keys) to any point on screen.  Pressing <Enter> then zooms this new rectangle to full-screen size.  By zooming in on an interesting equilibrium region and gridding the output, final frequencies can be read with any desired degree of accuracy.

	For many time trajectories of ecological dynamics, comparison of arithmetic and logarithmic plots has pedagogical value.  Populus plots arithmetic time trajectories by default but allows you to toggle a semi-logarithmic plot (press <Alt-L>).  Like other video utilities in Populus, the feature is turned off (and the screen-bottom options line is cleared) with the same <Alt-L> keystrokes that turned it on.

	Populus models that display <F4> in the lower-right corner of their input window include a routine that allows on-screen comparison of outputs resulting from two different sets of parameter values.  To implement this feature, simply toggle <F4>.  Graphical output from the current parameter values will be plotted in the normal colors, while output of the previous run will be shadowed in black.  A second press of <F4> toggles the graphical comparison routine off.

	Stability analyses are important in helping students to understand the dynamics of ecological and evolutionary models, and we have provided a rich set of stability tools in Populus.  Many of the genetical simulations begin from several different starting gene frequencies automatically.  Optional stability analyses are also available with the phase-plane graphs of ecological dynamics.  To implement this feature, press <Alt-S> after a phase diagram is complete; a cursor will pop up and can be moved to any point on the screen.  Pressing <Enter> will then initiate a trajectory from that point, and the dynamics can be run forward or backward.  Alternatively, multiple-starting-point stability analyses may be run by pressing <m> after <Alt-S>.  The program will then initiate trajectories from the perimeter or from a gridded pattern of points (the number and position of starting points is set from the options menu) to illustrate the stability of internal equilibria.

	In addition to these video utilities, Populus now includes several file-handling routines.  Output data from any of the simulations can be saved to disk for analysis using a spreadsheet, statistical package, or graphics editor.  To do so, call up the options menu after running your simulation, and specify a path and name for your data file.  Similarly, Populus can save to disk and reload sets of model parameter values; you might use this feature to save a series of parameter files that show particularly instructive examples, and ask your students to load, run, and analyze them as part of a lab exercise.


Finding Documentation

	This file contains only a brief introduction to Populus.  Full descriptions of each feature and a listing of the keystrokes necessary to call and implement it are built into the program, and are accessible from the Main Help Menu.  To see this menu, press <F1> (or if you are already in one of the model parameter or output screens, press <F1> twice, to get past the context-sensitive help).  We recommend a tour of the Main Help Menu as the quickest way to familiarize yourself with the capabilities of Populus.


Sponsorship and Distribution

	This software has been underwritten by IBM, the University of Minnesota, and the Undergraduate Course and Curriculum Development Program of the National Science Foundation (USE-9150887, USE-919155967); it is not a commercial venture and we encourage you to distribute it without charge to any colleague or student who will put it to good use.  If you find a bug please tell us.  We will fix it and provide you with an update.

Don Alstad
Department of Ecology, Evolution & Behavior
University of Minnesota
318 Church St. SE Minneapolis, MN, USA 55455-0302
612-625-0488
DNA@VX.CIS.UMN.EDU
	The simulation produces four different output screens.  Three present the spatial array of patches, color-coding local abundance of the interacting parties.  One screen gives the ratio of prey to parasitoids (N/P), one gives prey density (N per patch), and the third gives parasitoid density (P per patch).  On all three screens colors are coded in spectral order, with long wavelengths (red) representing high densities or ratios and short wavelengths (violet) representing low ones.  A legend below the spatial array gives the specific values (which change from generation to generation) represented by each color.  The final output is a graph of average prey and parasitoid densities per patch, across the entire array, as it changes with time.


t+E14

t+315Simulations of Population Biology

15
Don Alstad
Jim Curtsinger
Peter Abrams
Dave Tilman
7

Dept of Ecology, Evolution, and Behavior
University of Minnesota

Programming by
Chris Bratteli, Don Alstad, Brian Farm, Jim Curtsinger

Funded by IBM, the University of Minnesota, and the NSF
s-1
Version 2.10414t+3Lotka-Volterra Mutualism

	Note: this model is included in version Populus 2.10 in draft form.  It is not yet completely tested, and the following narrative is only a rough first draft.  Have fun!

	In a mutualistic interaction the per capita growth rate of each participant species is augmented by increasing density of the other.  Douglas Boucher (1985) has analyzed a very general model of this sort.  Like the Lotka-Volterra competition and predation models, his equations include only the first-order terms of the related Taylor series, and hence have linear zero isoclines.  Thus the two-species differential equations are

dN1/N1dt = r1 + a11N1 + a12N2

dN2/N2dt = r2 + a22N2 + a21N1

If there are no restrictions on the signs of the coefficients aij, each mutualist species may have either positive or negative density-dependent effects on its own growth.  The slope of the species i isocline is -aij/aii, and if the intraspecific density-dependent feedback is positive, then the species vector will move away from this isocline.8	The dynamics that result from these simulations vary with the signs of a11, a22, and (a12a21 - a11a22) and include stable equilibria, mutualistic saddles, hilltops, and ridge-and-valley topologies.

	Given the linear isoclines of a first-order mutualism model, many sets of parameter values produce infinite population increases.  Using non-linear equations it would be possible to introduce and upper limit to each species' dynamics, intrinsic to the model.  Here, we follow Boucher's suggestion and assume instead that both species are ultimately limited by some extrinsic factors, preserving the simple Lotka-Volterra form.



References

Boucher, Douglas H. 1985. Lotka-Volterra models of mutualism and positive density-dependence.  Ecological Modelling 27:251-70.514t+3Insecticides in
Host-Parasitoid Interactions

Note: this simulation is included in Populus 2.10 in draft form.  It is not completely tested, and the following narrative is incomplete.  Have fun!

	Because parasitoids are often released for biological control purposes in the context of an integrated program that includes pesticide application intended to reduce host densities, it is important to model the effect of insecticides on the dynamics of host-parasitoid interaction.  This set of models was developed by Hassell (1984).  The recursions take the same general form as others in this series of models from his 1978 book, with a host survival function that incorporates negative binomial clumping of the searching parasitoids and a type II functional response.  Because the model is discrete, it is possible that insecticide may have different effects depending on the timing of the application relative to host and parasitoid life cycles.  For this reason, four sub models are presented as follows:6Model 1. In this case, insecticides act before parasitism and kill only hosts.  The recursions are:

Nt+1 = FNtIf(NtI,Pt)
Pt+1 = NtI[1-f(NtI,Pt)]

where

f(NtI,Pt) = [1+aPt/(k(1+aNtI))]-k

Model 2. Here, insecticides act after parasitism and kill only hosts.

Nt+1 = FNtf(Nt,Pt)I
Pt+1 = Nt[1-f(Nt,Pt)]

where

f(Nt,Pt) = [1+aPt/(k(1+aThNt))]-k6Model 3. Next, insecticides act after parasitism and kill both hosts and parasitoids at the same rate.

Nt+1 = FNtf(Nt,Pt)I
Pt+1 = NtI[1-f(Nt,Pt)]

where

f(Nt,Pt) = [1+aPt/(k(1+aThNt))]-k

Model 4. Finally, if insecticides act before parasitism and also kill adult parasitoids at the same rate.

Nt+1 = FNtIf(NtI,PtI')
Pt+1 = NtI[1-f(NtI,PtI')]

where

f(NtI,Pt) = [1+aPtI'/(k(1+aNtI))]-k10	The simulations show that insecticide applications in are likely to lower the equilibrial host density resulting from host-parasitoid interaction alone unless adult parasitoids are affected (Model 4).  In this latter case, equilibrial host densities rise.  The degree of host depression that results from the insecticide application increases with the clumping of the parasitoid and the growth rate of the host.  Model 2 consistently produces the lowest host equilibria, and Model 4 the highest.



References

Hassell, M. P. 1984. Insecticides in Host-Parasitoid Interactions.  Theoretical Population Biology 26:378-86.714t+33-Party Mutualisms

Note: this simulation is included in Populus 2.10 in draft form.  It has not yet been completely tested, and the following narrative is incomplete.  Have fun!

	This module presents a pair of mutualism models that were developed and analyzed by Addicott and Freedman (1984).   The models portray a three-party system in which a mutualist species affects the dynamics of a pair of competitors, or a predator and its prey.  The predator-prey model is that of Rosenzweig and MacArthur, with logistic prey (X) growth, negative exponential predator (Y) growth, and a Type II predator functional response.  The mutualist (U) can affect prey growth (a), the predator functional response (b), and/or predator growth (d), according to the following recursions:

dX/dt = X[1-(aU+X)/K] - XY/(1+mX+bU)
dY/dt = Y[-s-dU+cX/(1+mX+bU)]5The competition model employs the Lotka-Volterra competition equations, with competitors designated X1 and X2; the mutualist (U) is assumed to interaction with X1, the competitor-mutualist.  The competition-model recursions are as follows:

dX1/dt = 1X1/K2[K1-a1U-X1-1X2/(1+b1U)]
dX2/dt = 2X2/K1[K2-a2U-X2-2X1/(1+b2U)]

The simulations allow three general conclusions. (a) The dynamic effects of mutualism vary with the nature of the interaction. (b) The structure of these 3-party models eliminates the unbounded growth of Lotka-Volterra mutualism, through the competitive or predator-prey interaction.  (c) Where there is a cost for predator, prey, or competitor in association with the mutualist, the interaction is not positive at all densities.



References

Addicott, John F. and H. I. Freedman. 1984. On the structure and stability of mutualistic systems: analysis of predator-prey and competition models as modified by the action of a slow-growing mutualist. Theoretical Population Biology 26:320-39.	In addition to this brief read.me, the Populus disk contains 4 files as follows:

1) Populus.doc is an ascii text file that explains features and requirements of the software, many of which are new to this current version.  We suggest that you read it before using the program.  Note that full documentation is an internal part of Populus, accessible from the main help menu (press <F1> once, or twice depending on context), and that many program features are adjusted from the Options Menu (press <Alt-O>).

2) Populus.exe is the binary executable file that runs our program; it should be placed (with populus.ovr and populus.txt) in the hard-drive directory that you will use when running Populus.

3) Populus.ovr contains the overlays that swap different models and routines in and out of computer memory.  This file is required to run the program.

4) Populus.txt contains all of the indexed text screens that are displayed by Populus.  It is the last of three files (also populus.exe, populus.ovr) that are required to run the program.

	Populus allows you to save other files to disk.  For example, if you elect to preserve any of the configuration information chosen from the Populus options menu, the program will save a populus.cfg file and your configuration will be loaded automatically each time Populus starts.  Parameter sets, print files, and data outputs may also be saved with filenames (*.par, *.prn and *.dat) of your choosing, but only the populus.exe, populus.ovr, and populus.txt files are necessary to run the program.

	In today's viral climate I am no longer willing to distribute software that I cannot personally control; for this reason, I have not included a compressed, self-extracting copy of Populus accessible from 360K 5.25" disk drives.  If you need to use a 360K drive, compress the Populus.ovr file to fit, using a compression utility from the public domain or commercial market.

	We encourage you to give this software without charge to any students or colleagues who will put it to good use.  When you do so, please pass on the entire 5-file package intact.  File sizes should be as follows:

populus.doc       8396 bytes
populus.exe     171760 bytes
populus.ovr     720937 bytes
populus.txt     364722 bytes
read.me           2767 bytes

If the file sizes that you receive differ from these values, your disk is corrupt and should be destroyed.  For $10 payable to the University of Minnesota you can obtain a clean, up-to-date copy from Don Alstad, Department of Ecology, Evolution and Behavior, University of Minnesota, 318 Church St. SE, Minneapolis 55455-0302.