Frequency- and Density-dependent Selection in The Diploid Population with Only Two Pure Strategies

In this paper, the dynamical behavior of the frequency- and density-dependent diploid selection system with only two pure strategies is investigated. The results show that: (i) The genetic equilibrium point of the population is locally asymptotically stable if and only if there is heterozygotic advantage at this point. If there is not, then the genetic equilibrium point must be an unstable saddle point. (ii) The phenotypic equilibrium point of the population is locally asymptotically stable if and only if it is a density-dependent evolutionary stable strategy (DDESS). If the phenotypic equilibrium point is not a DDESS, then it must be an unstable saddle point. (iii) The existence of periodic solutions in this system is impossible. (iv) If the density-dependent payoff matrix is symmetric at the genetic and phenotypic equilibrium points, then the dynamical behavior of this system will be completely equivalent to the results of Ginzburg (1983Theory of National Selection and Population Growth, Menlo Park, Benjamin/Cummings). (v) If the matrix is not symmetric at the genetic equilibrium point, then the properties of the genetic equilibrium point are only partly similar to Ginzburgʹs results. If the matrix is not symmetric at the phenotypic equilibrium point, then Ginzburgʹs results cannot completely determine the properties of the phenotypic equilibrium point.