Proc. Natl. Acad. Sci. USA Vol. 76, No. 4, p. 2084-2087, April 1979 Population Bio ogy

Coevolution in structured demes (population genetics/group selection/subdivided populations/mathematical models)

M. SLATKIN* AND D. S. WILSONt *Department of Zoology, NJ-15, University of Washington, Seattle, Washington 98195; and tDivision of Environmental Studies, University of California, Davis, California 95616

Communicated by R. W. Allard, January 11, 1979

ABSTRACT A simple model of coevolution in a subdivided population is considered. It is shown that, when there are frequency- and density-dependent interactions in each site, the sampling variation in numbers in each local site can lead to selection both through the dispersal process and through indirect effects. The model predicts that coevolved relationships between species can result from various interactions other than direct forms of competition and predation. One of the most challenging areas of population genetics theory is the modeling of multispecies coevolution, in which the fitness of a genotype is determined by its interactions with other species that are themselves evolving. Recently, several authors (1-4) have used explicitly genetical models of coevolving species in a single community to show that it is possible to characterize the coevolutionary equilibrium reached under certain restricted conditions. These theories take into account both the immediate effect of genetical changes within each species and the effect that results from the "feedback" through the other species in the community. There is, however, a large and potentially important class of genetical variants within a species that are selectively neutral under the assumptions made in most models of coevolving species. If two genotypes of a species are affected in identical ways by all individuals in the community, including members of their own species, but differ in the way in which they affect one or more other species, those genotypes would have the same fitness despite their different effects on the community. Consequently, the ratio of their frequencies would not change. Two examples will serve to illustrate the nature and scope of this type of interaction. First, consider a species that consists of two genotypes, identical in every way except in their response to a second, competing species. One genotype aggressively excludes the competing species, whereas the other is passive. If both genotypes mix freely throughout the habitat, then the exclusion of the competing species cannot affect them differentially, even though the population at large benefits. Second, consider a genotype that modifies the environment in such a way as to enhance the presence of other species that are beneficial to itself, such as prey, "canopy" species that improve the microclimate, refuge species that protect against predation, detritivores that transform resources from unavailable to available forms (nutrient cycling), and so on. If the benefits of the modification feed back to the population at large, and not differentially to the individuals that caused them, then natural selection cannot discriminate between the modifying genotypes and "freeloader" genotypes that do nothing but receive equal benefit, under the assumptions of the models. Because the benefits of these adaptations result not from the activities themselves, but rather from the consequences of the

activities, Wilson (5, 6) has termed them "indirect effects." He argues that their neutrality in coevolutionary models is an artifact of a simplifying assumption, that no spatial variation in gene frequency exists throughout the population. Wilson then argues that evolution through indirect effects can occur if the interactions between species take place in a large number of local sites, which he calls "trait-groups." Using an approach similar to that taken in considering the evolution of altruistic characters in a single species-the "structured deme" model (5-12)-Wilson (5, 6) presents a verbal argument and some computer simulation results showing that coevolution through indirect effects can occur in a subdivided population. The purpose of this paper is to show that there are two potentially important types of genetic changes that result from species' interactions taking place in local sites, as in the structured deme model. Both types of changes are consequences of the fact that interactions within each site are frequency- and density-dependent, and, because there are assumed to be relatively small numbers of individuals in each site, sampling variation leads to differences between sites in genotypic numbers and frequencies. Consequently, each genotype does not interact with the average numbers of all genotypes in each species but with distributions of those numbers generated by the sampling process. One effect of this is to permit evolution through indirect effects (5), because ode genotype may interact with itself more strongly than in a single population and increase due to its indirect effects on the other species. A second type of genetic change, not described previously, is due to the process of dispersing into the local sites, which can itself lead to selection on the amount of site-to-site variation. THE MODEL We will consider here only the simplest model that will illustrate both types of selection resulting from geographic subdivision and frequency and density dependence. Assume that there is a large number of sites (numbered 1 to T) in which interactions can take place. In each site assume there are only two interacting species: species 1, which is haploid with two genotypes A and B; and species 2, which is monomorphic. Let Mi and mi be the numbers of A and B types in site i and let Nj be the number of individuals of species 2 in site i, with all unprimed variables assumed to be measured after the dispersal stage but before the interactions have changed their numbers. We assume that there is a reproductive pool for each species and that MA, mi, and N, are generated according to the composition of the pools and the model of the dispersal process, to be discussed below. Let M,', mi', and N1' be the contributions to the reproductive pool after the interactions take place in site i and assume they are given by

The publication costs of this article were defrayed in part by page charge payment. This article must therefore be hereby marked "advertisement" in accordance with 18 U. S. C. §1734 solely to indicate this fact.

Mi

2084

=Mijl+ rll

(Mi+mi+ aNi)1

[la]

Proc. Natl. Acad. Sci. USA 76 (1979)

Population Biology: Slatkin and Wilson Mi'

=

mjiI +

-r

r

(Mi + mi + aNi)I

[lb]

Ni'= Ni1I + r2- k22 (Ni + flM2 + f2m0)I, [Ic] which is a discrete time version of the Lotka-Volterra competition model (2, 3) with the additional feature that the interaction terms can depend on the genotype of species 1. Each of the equations in [1] takes both survival and fecundity into account even though the differential fecundities are not manifest until the reproductive stage. We assume there are no differences between the A and B genotypes in their response to species 2, so M,'/m,' = Mi/mi. That is, there are no direct effects and, in the absence of population subdivision, there could be no evolution in species 1 due to the indirect effects present if 2

We can take spatial subdivision into account by first computing M' and mf' by averaging over all i and then complete the model by specifying the way in which the next generation of individuals of both species disperse themselves into the sites. We can compute the changes in the relative frequencies of A and B without yet making any assumptions about the dispersal stage, other than assuming that dispersal does not change the relative frequencies of A and B in the entire population. The condition that A will increase relative to B is

M_>M m

[2]

in

where the bar indicates an average taken over all sites. We can compute the terms on the left-hand side of [2] directly from [1] by averaging over all i to obtain M' = M 1 + -

ri- k

(M

+ m +

aN)j

!(VM + CMm + aCMN)

i' =

+ rik 'Ii[1

-k (Vm

+

CMm

+

(M

+ m +

aCmN),

aN)] [3]

where VM and Vm are the variances of Mi and mi, and the Cs the covariances. There are two important features of [3]. First, the effect of the population subdivision appears in the variance and covariance terms that arise because of the quadratic dependence of MW'and mi' on the unprimed variables. Other functional forms of that dependence would introduce some and possibly all higher moments as well, but, for any model of frequency- and density-dependent interactions, spatial subdivision must introduce some additional terms to the recursion formulae for the average values. The form of [1] was chosen only for analytic convenience, and our main results are true for more general models as well. Selection in species 1 could result from genotypic differences in the variance and covariance terms that would represent differences in the ways that the two genotypes experience their own and other densities. The second important feature of [3] is that there is no dependence on #I and 02, representing the indirect effects. That is because, in the model as formulated so far, there is no time for the numbers of species 2 to change in response to the numbers of the two genotypes in different sites. In order for evolution through indirect effects to occur, there must be an additional generation before the next dispersal stage in which the two genotypes in species 1 can differentially feel the response of species 2. This will be shown later, but we must first discuss the dispersal process.

are

2085

We will consider two models of the dispersal stage that correspond to two types of biological assumptions about the ways in which the genotypes of species 1 distribute themselves in the sites. In both models we will assume the two species disperse independently, so CMN = CmN = 0; but obviously this type of correlation could be considered in the same framework. The first model we consider is the independent dispersion model, in which CMm = 0 by assumption. The dispersal stage is completely described by the relationships between M and VM and between mi and Vm. The second model is the binomial dispersion model, in which the sum M1 + mj has some distribution with a specified variance V. Among all sites with a given value of the sum Mi + mi, the numbers of each genotype are determined by binomial sampling with parameter f = M/ (M + mi). In this model, the dispersal stage is completely determined by V and f, and we can find by direct computation that VM = f(l -f)( + i) + f2V Vm= f(I -f)(M + m-) + (1-f)2V CMm = f(1 - f)(V - M -i). [4] The details for the dispersal process for a particular species would dictate which of these two models is more realistic. If the probability that each individual settles in a site is independent of the numbers of other individuals at the site, then the distributions of Mi, mi and Mi + mi are Poisson with means M, m, and M + im when the number of sites is large (13). If V = M + imi is substituted in [4], VM = M, Vm = m-, and CMm = 0; so the two dispersion models lead to the same equations and predictions. This assumption about dispersal is the simplest possible, and no additional features of the dispersal process are required. This is the assumption made by Wilson (5). However, there may be complicating factors which would lead to differences between the two models. For example, if each individual disperses independently but then produces a number of offspring that is a sample from some given probability distribution (that may depend on genotype), then Mi and mi would have compound Poisson distributions (13). In this case, VM > M and Vm > mi and the ratio of the means to the variances could depend on genotype. Distributions other than a Poisson or compound Poisson could be generated only if there were some kind of density-dependent interactions among individuals settling at a site. Individuals could avoid sites that appear overcrowded and seek less crowded sites, or there could be density-dependent mortality after settling. In any case, the key difference between the independent and binomial dispersion models is whether the interactions depend only on total numbers or on the numbers of each genotype. When the interactions depend on total numbers, the binomial dispersion model should be used, and, if interactions are only within each genotype, then the independent dispersion model would apply. These density-dependent interactions at the time of dispersal are distinct from those that determine the reproductive output of each site, which are modeled by equation [1]. We find the conditions under which A will increase relative to B in a single generation by substituting [3] into [2], which reduces to

(VM

CMm)/M > (Vm

CMm)/iii. With the independent dispersion model, [5] becomes VM/!w > Vm/M' +

+

[5]

[6]

and the selection resulting from the dispersal process depends

2086

Population Biology: Slatkin and Wilson

Proc. Natl. Acad. Sci. USA 76 (1979)

on the assumed relationship between the means and variances. If VM and Vm are constants independent of MW and mf, then A will increase in frequency if M < (VM/Vm)iii and decrease otherwise. Thus, there would be an unstable equilibrium, and this type of dispersal would lead to species 1 being monomorphic for whichever genotype had the initial advantage. If, instead, we assume that Mi and mi have Poisson or compound Poisson distributions, the ratios VM/M and Vm/mii are constant, and the frequency of A would either increase, not change, or decrease, depending on the numerical values of those ratios. Finally we could assume VM/M2 and Vm/1fi2 are constant, indicating that the coefficients of variation are constant, in which case this type of selection would lead to a stable polymorphism. In all of these cases, the direction of selection can be found by considering the effect of changes in M and mf on the variance terms in [3], with the covariance terms being zero by assumption. With the binomial dispersion model, the results are much simpler, because, by substituting [4] into [5], the two sides are equal for all values of f and V. Therefore, under the assumptions of this model, the dispersal process will not lead to selection. Furthermore, this can be shown to be true for general iterative equations and not just the special forms used in [1], as long as there are no direct differences between the two genotypes. Thus, the actual course of evolution when there is frequency- and density-dependent selection in a subdivided population depends critically on the exact nature of the dispersal stage. To consider evolution through indirect effects, we must assume that there is at least one generation in which there is no dispersal. That will allow Ni to respond to the different numbers of A and B genotypes. In Wilson's simulation of this process (5), he assumed there were 10 generations between successive generations in which there was dispersal, but 2 are sufficient to obtain the major results. If there were more than two species, more than two generations could be required for the indirect effects to lead to genetic changes. We assume that in the second generation selection in each site proceeds in the same way as in the first, so the numbers of the three types after the second generation, MA", mi , and N1", are obtained from [1] by substituting primed for unprimed variables on the right-hand sides. Now the condition for increase of A relative to B is

1W"/if" > Mf/m,

[71

and a complete evaluation of the left-hand side becomes a nontrivial exercise in elementary algebra, with moments of third and fourth order entering because the expression of M1" and min in terms of the unprimed variables are quartic, even for this simple model. We can, however, make considerable progress without resorting to brute force. Evaluating the left-hand side of [7] in terms of the primed variables, we get M'[1 M"

mef mf'[1 -

+

r1

-

r1(M' + mW' +

agf')/kl]

rl(VM' + CM'm' + aCM'N')/kl

-

+ r1

-

r

(M'

+

W'

+

aN')/k1l

ri(Vm' + CM'm' + aCm'N')/kl

and the binomial dispersion model, it can be shown directly that (VM' + CM'm')/M' = (Vm' + CM'm')/1f'; 19] and we have already shown that M'/m' = M/im, so that [8] is equivalent to CM'N'/M < Cm'N;/M. [10] By using [1] to evaluate the Cs, we find that [10] is satisfied when 31 > 32. Thus, for the binomial dispersion model, we have shown that there is no selection arising from the dispersal stage and that selection through indirect effects will favor the genotype with the larger value of 13. For the independent dispersion model, the results are much more complex, because the two types of selection due to spatial subdivision become confounded. The actual direction of selection must be computed by evaluating the variance and covariance terms in [8], which involve moments of third and fourth order in the unprimed variables. The algebraic results in this case show that selection due to dispersal can work in the same or the opposite direction as the selection due to the indirect effects. DISCUSSION We have shown that spatial subdivision can lead to genetical changes either through the dispersal process or through indirect effects on a second species. Selection through indirect effects is caused by the feedback from the differential effects on other species. In our model, the feedback was due to two or more generations spent within the trait-group between dispersal periods [see Hamilton (14) for a biological example]. However, the feedback could also occur due to density-dependent interactions within a single generation. Such a model would require a more detailed analysis of each generation. The strength of selection for indirect effects depends on the magnitude of the between-site variances and covariances. It is strongest when relatively few individuals occupy each traitgroup. However, many species do interact in small local groups, and for these the effects of spatial subdivision must be taken into account to obtain a realistic picture of the evolutionary forces operating upon each species. Colwell (15) provides a well documented example in his study of two species of flower mites, each inhabiting a different species of flower and dispersing between flowers on the bills of hummingbirds. If an individual of one species gets off on the wrong species of flower, it is attacked and killed by individuals of the other mite species. Since these mites are not predatory, the main benefit of their aggressive behavior must be the exclusion of the competing species, a benefit that is shared by the entire population within the flower. Current ecology is biased heavily towards the study of direct interactions among species-predation and standard forms of competition. If indirect effects are routinely selected for in nature, then a much greater diversity of coevolved relationships between species may exist, in which each member of the community attempts to enhance the presence of its allies and inhibit the presence of its enemies through any available pathway (6). The existence of this type of interaction is not yet supported by strong evidence, possibly because it is not often looked for. We would like to encourage its investigation by field

ecologists.

Even if covariance terms are assumed initially to be zero after dispersal, they will be non-zero due to the interactions within each site; so none of the Cs in [8] will, in general, be zero. In particular, indirect effects will lead to differences between CM'N' and Cm'N' that will, in turn, lead to selection. With [1]

We thank J. Felsenstein and M. Turelli for helpful discussions of this topic and comments on an earlier draft of this paper. M.S. was supported by National Institutes of Health Grants KO1-GM00118 and R01-GM22523, and D.S.W. was supported by National Science Foundation Grant DEB7S-03153.

Population Biology: Slatkin and Wilson 1. Levins, R. (1974) Ann. N. Y. Acad. Sci. 231, 123-138. 2. Levins, R. (1975) in Ecology and Evolution of Communitis, eds. Cody, M. L. & Diamond, J. M. (Harvard Univ. Press, Cambridge, MA), pp. 16-50. 3. Roughgarden, J. (1976) Theor. Popul. Biol. 9,388-424. 4. Leon, J. A. & Charlesworth, B. (1978) Ecology 59, 457-464. 5. Wilson, D. S. (1976) Science 192, 1358-1360. 6. Wilson, D. S. (1979) The Natural Selection of Populations and Communities (Benjamin/Cummings, Menlo Park, CA), in press.

7. Wilson, D. S. (1975) Proc. Natl. Acad. Sci. USA 72, 143-146. 8. Wilson, D. S. (1977) Am. Nat. 111, 157-185.

Proc. Nati. Acad. Sci. USA 76 (1979)

2087

9. Matessi, C. & Jayakar, S. D. (1976) Theor. Popul. Biol. 9,360387. 10. Charnov, E. L. & Krebs, J. R. (1975) Am. Nat. 109, 107-112. 11. Cohen, D. & Eshel, I. (1976) Theor. Popul. Biol. 10, 276. 12. Wilson, D. S. (1977) Behav. Ecol. Sociobiol. 2,421. 13. Feller, W. (1975) An Introduction to Probability Theory and its Applications (Wiley, New York), 2nd Ed., pp. 146-148, 270-271. 14. Hamilton, W. D. (1978) in Diversity of Insect Faunas, Symposium of the Royal Entomological Society, eds. Mound, L. A. & Waloff, N. (Blackwell, Oxford), Vol. 9, pp. 154-175. 15. Colwell, R. K. (1973) Am. Nat. 107,737-760.

Coevolution in structured demes.

Proc. Natl. Acad. Sci. USA Vol. 76, No. 4, p. 2084-2087, April 1979 Population Bio ogy Coevolution in structured demes (population genetics/group sel...
759KB Sizes 0 Downloads 0 Views