Abstract
A new model which allows both the effect of partial selfing selection and an exponential function of the expected payoff is considered. This combines ideas from genetics and evolutionary game theory. The aim of this work is to study the effects of partial selfing selection on the discrete dynamics of population evolution. It is shown that the system undergoes period doubling bifurcation, saddle-node bifurcation, and Neimark-Sacker bifurcation by using center manifold theorem and bifurcation theory. Numerical simulations are presented not only to illustrate our results with the theoretical analysis, but also to exhibit the complex dynamical behaviors, such as the period-3, 6 orbits, cascade of period-doubling bifurcation in period-2, 4, 8, and the chaotic sets. These results reveal richer dynamics of the discrete model compared with the model in Tao et al., 1999. The analysis and results in this paper are interesting in mathematics and biology.
Highlights
Evolutionary game theory was extended to the genetic model
We focus on the bifurcation of system (10)
If there is no partial selfing selection (i.e., β = 0), the saddle-node bifurcation does not occur. This means that the parameter β of partial selfing selection leads to more complex dynamical behavior of the genetic system
Summary
Evolutionary game theory was extended to the genetic model. The notion of an evolutionary stable strategy (ESS) which was proposed by Smith [1] (1982) is important. From [6], each individual can reproduce by selfing or random outcrossing with constant probabilities, respectively, in a population. We consider a one-locus two-allele model where genotypic fitness is an exponential function of the expected payoff and the frequency. If β ≠ 0, one-dimensional discretetime dynamical systems are extended to two-dimensional systems, which lead to great effect on the genetic model It was shown in [7] that a stable polymorphic equilibrium is not an ESS under some condition. Consider a diploid population with nonoverlapping generations and establish a genetic model with one locus and two alleles A1 and A2. According to [6], each individual can reproduce by selfing or random outcrossing with probability β or 1 − β (0 < β < 1), respectively, in the population.
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