A Wright–Fisher model with indirect selection
We study a generalization of the Wright–Fisher model in which some individuals adopt a behavior that is harmful to others without any direct advantage for themselves. This model is motivated by studies of spiteful behavior in nature, including several species of parasitoid hymenoptera in which sperm-depleted males continue to mate despite not being fertile. We first study a single reproductive season, then use it as a building block for a generalized Wright–Fisher model. In the large population limit, for male-skewed sex ratios, we rigorously derive the convergence of the renormalized process to a diffusion with a frequency-dependent selection and genetic drift. This allows a quantitative comparison of the indirect selective advantage with the direct one classically considered in the Wright–Fisher model. From the mathematical point of view, each season is modeled by a mix between samplings with and without replacement, and analyzed by a sort of “reverse numerical analysis”, viewing a key recurrence relation as a discretization scheme for a PDE. The diffusion approximation is then obtained by classical methods.