Nonlinear Frequency-dependent Selection With Partial Selfing

The aim of this work is to construct the discrete model of a type of population evolutions, to analyze the asymptotic behavior of the model and to study the effect of partial selfing selection on the population evolutions.We consider a one-locus model with two alleles. It is assumed that genotypic fitness is an exponential function of the expected payoff in the matrix game, and that each individual of the population can reproduce by selfing with constant probability β. Then the genetic model is proposed that includes the nonlinear frequency-dependent selection and partial selfing.Conditions for the stability of monomorphic equilibria are obtained by the Jury conditions and the center manifold theorem. The existence and stability of polymorphic phenotypic equilibria and polymorphic genotypic equilibria are analyzed. By theoretical analysis, we find that there is much effect of partial selfing selection on the genetic model. ESS isn't the necessary and sufficient conditions of the stability of monomorphic interior phenotypic equilibria. Under certain conditions, the variation of β in (0,1) induces the exchange of stability and the unstability of monomorphic interior phenotypic equilibrium alternately. Furthermore, a stable polymorphic phenotypic equilibrium isn't an ESS under some conditions. For monomorphic genotypic equilibria, we find that the increase of β leads to the unstability of the equilibria under some conditions although the effective fitness of monomorphic allele at the equilibria is greater than that of the absentallele. Moreover, The model admits the stable and unstable period doubling bifurcations, the saddle-node bifurcation and the Naimark-Sacker bifurcation by theoretical analysis and numerical simulations. Among which, the unstable period doubling bifurcation, the saddle-node bifurcation and the Naimark-Sacker bifurcation are all new interesting phenomena if considering partial selfing selection.