Ecogenetic models, competition, and heteropatry

Abstract

We develop a system of equations to analyse the existence of genetic polymorphisms under disruptive selection in heterogeneous environments. These equations have both a genetic and a population density regulation component. In the absence of the genetic component, the equations reduce to a discrete time description of competition between interacting clonal lines or species. We use these equations to demonstrate that different populations, competing along a resource spectrum, are able to dynamically coexist, as asymptotically periodic or chaotic solutions to our system of equations, despite the fact that a coexistence equilibrium—stable or unstable—does not exist. We then extend these results to environments in which several niches are explicitly defined. Our analysis of the ecological component of our model establishes that the answer to questions of coexistence among groups of individuals cannot rely on analyses of the existence of equilibria and their stability properties. In the most general model presented here, we allow for an assortative mating structure that is induced by the spatial heterogeneity of the environment. The level of assortative mating is controlled by a parameter so that at one extreme mating is panmictic, while at the other extreme individuals mate within their natal niches before dispersing to oviposit in other niches. We refer to this spatial mating structure as heteropatry. We investigate, through numerical studies, the properties of a heteropatric model containing both ecological and genetic components. First we address the question of the existence of protected genetic polymorphisms (i.e., the different allelles at a particular locus all increase in frequency when rare) under a wide range of model parameter values in a diallelic one-locus version of our model, assuming panmixis and partial dominance selection of varying direction. We make the point that establishing the instability of monomorphic population equilibria is insufficient to guarantee the existence of a protected polymorphism, since the instability may be in the population size component rather than the genetic frequency component of the monomorphic equilibrium solution. The results indicate that density dependence serves to decrease the likelihood that a protected polymorphism exists, while the degree of selection of natal habitat for oviposition purposes serves to increase this likelihood. The situation is complicated, though, since conditions which promote the existence of protected polymorphisms may actually reduce the possibility that a stable polymorphism exists. Further, the way we introduce density dependence (either by scaling the population interaction coefficients or by altering the shape of the response function) differentially affects the results. Finally, our results suggest that if individuals prefer to oviposit in their best niche (where they are most fit), rather than their natal niche (where they mature), then allele fixation very probably occurs.