Abstract
The study of eco-evolutionary dynamics is based on the idea that ecological and evolutionary processes may operate on the same, or very similar, time scales, and that interactions of ecological and evolutionary processes may have important consequences. Here we develop a model that combines Mendelian population genetics with nonlinear demography to create a truly eco-evolutionary model. We use the vec-permutation matrix approach, classifying individuals by stage and genotype. The demographic component is female dominant and density-dependent. The genetic component includes random mating by stage and genotype, and arbitrary effects of genotype on the demographic phenotype. Mutation is neglected. The result is a nonlinear matrix population model that projects the stage × genotype distribution. We show that the results can include bifurcations of population dynamics driven by the response to selection. We present analytical criteria that determine whether one allele excludes the other or if they persist in a protected polymorphism. The results are based on local stability analysis of the homozygous boundary equilibria.As an example, we use a density-dependent stage-classified model of the flour beetle Tribolium castaneum. Our model permits arbitrary life-cycle complexity and nonlinearity. Tribolium has developed resistance to the pesticide malathion due to a dominant allele at a single autosomal locus. Using parameters reported from laboratory experiments, we show that the model successfully describes the dynamics of both resistant and susceptible homozygotes, and the outcome of a selection experiment containing both alleles. Stability analysis of the boundary equilibria confirms that the resistant allele excludes the susceptible allele, even in the absence of malathion, agreeing with previously reported results.
Highlights
The demographic processes of birth and death drive changes in gene frequencies and changes in population density and structure
Demography is central to understanding ecology and evolution, and eco-evolutionary analyses always strive to incorporate the fundamental demographic processes of birth, death, and development (e.g., Coulson et al, 2006; Metcalf and Pavard, 2007). de Vries and Caswell (2019b) recently introduced an eco-evolutionary framework that combines matrix population models with basic Mendelian genetics
We show how to use that model to project the stage × genotype distribution, and derive analytical conditions that determine whether alleles will coexist in a genetic polymorphism, or if one or another allele will go to fixation
Summary
The demographic processes of birth and death drive changes in gene frequencies and changes in population density and structure. De Vries and Caswell (2019b) recently introduced an eco-evolutionary framework that combines matrix population models with basic Mendelian genetics. We use this framework to explore density-dependent selection. Theoretical work on density-dependent selection (MacArthur, 1962; Roughgarden, 1971) combined population genetics with unstructured ecological models by writing genotype fitnesses as a function of genotypic densities. (1971) extended the logistic equation to multiple competing genotypes, and showed that selection leads to an increase in population density in a constant environment because only alleles with heterozygote advantage in the carrying capacity can invade. Charlesworth (1994) used high-order difference equations to model density-dependent selection in age-structured populations. We investigate the effect of incomplete dominance on the speed of invasion and on the outcome of invasion using evolutionary stability analysis
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