Frequency-Dependent Selection: a Two-Phenotype, Two-Allele, Single-Locus, Two-Species Diploid Model

The two-species matrix games developed by Cressman (1996,Theoretical Population Biology49,189–210) are extended to interacting diploid populations. In this paper, a simple two-phenotype, two-allele, single-locus, two-species diploid model is investigated. The main focus of the paper is to illustrate the local stability of all interior positive equilibrium points of the two-species diploid model and the evolutionary significance of the stability. Results show that (i) in the two-species diploid model, three possible types of interior positive equilibrium points can exist, phenotypic equilibrium point, genetic equilibrium point and PG equilibrium point; (ii) if the phenotypic equilibrium point is a two-species ESS, then it must be locally asymptotically stable; (iii) at the genetic equilibrium point, if there are heterozygote advantages in both species, than the genetic equilibrium point is locally asymptotically stable; (iv) at the PG equilibrium point, one species exhibiting phenotypic equilibrium and the other genetic equilibrium, if the genetic equilibrium corresponds to heterozygote advantage and the phenotypic equilibrium is a single-species ESS, then the PG equilibrium point must be locally asymptotically stable.