Density-Regulated Selection in a Heterogeneous Environment

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Abstract
A model is developed for density-regulated selection in a heterogeneous environment, analogous to Levene's model for selection in a subdivided environment. The population is polymorphic for two alleles at an autosomal locus. Before selection the three genotypes are in Hardy-Weinberg frequencies, and a constant fraction of newborn offspring go to habitat k to live. Genotypes differ in their selective values because of genotypic and habitat differences in intrinsic rates of increase and carrying capacities, and selective values are parameterized in terms of these ecological fitness parameters so that the growth of the whole population is logistic. All selection occurs in the habitats, and at the end of each generation adults emerge from the habitats and join a mating pool to mate at random. In the Levene model, individuals in the mating pool from each habitat make a constant contribution to the gametic pool from which individuals of the next generation develop. In our model, however, the gametic contribution from a habitat is proportional to the number of organisms remaining in the habitat after selection has taken place, and thus to the average selective value of individuals from the habitat. As a consequence, the gametic contributions change with allele frequency. Sufficient conditions for a protected polymorphism under this more realistic assumption are derived, and remarkably enough take a form analogous to Levene's, with the same increased opportunity for genetic polymorphism which is the major result of his model. A protected polymorphism occurs provided that a harmonic mean over habitats of the adjusted carrying capacities of each homozygote are less than a corresponding mean for the heterozygote; the adjusted carrying capacity of a genotype in a habitat is just an extension to the population level of the carrying capacity for a single habitat. The population's carrying capacity, a weighted harmonic mean of the adjusted carrying capacities of the genotypes, is maximized near population size equilibrium. Sufficient conditions for a protected polymorphism under a more general form of density-regulated selection, with the genotypic fitnesses any smooth, strictly decreasing function of population size, are also outlined.