Abstract
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.
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Notes
Let us recall the unfortunate polysemy of the word “drift”. In the biological literature “genetic drift” corresponds to the noise-induced variations. When using a stochastic model, this is at odds with the “drift” of a diffusion, i.e. the first order term that models a deterministic force.
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Acknowledgments
This paper stems from discussions with the biologists F.-X. Dechaume-Moncharmont and M. Galipaud, who came up with the single generation model for “indirect” fitness. We thank them for stimulating discussions on the biological aspects of the problem. On the mathematical side, our special thanks go to R. Eymard for his input on the inverse numerical analysis problem; we also thank V.-C. Tran and D. Coupier for interesting discussions. Finally we thank J.E. Taylor and an anonymous reviewer for many valuable comments on a preliminary version of this work.
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Goudenège, L., Zitt, PA. A Wright–Fisher model with indirect selection. J. Math. Biol. 71, 1411–1450 (2015). https://doi.org/10.1007/s00285-015-0859-2
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DOI: https://doi.org/10.1007/s00285-015-0859-2