The Necessary and Sufficient Conditions of Density-dependent Evolutionary Stable Strategy (DDESS) in the Two-Phenotype Model

In this paper, we discuss the stability conditions of both the density-dependent pure and mixed strategy models of a polymorphic haploid population with only two pure strategies. Our results show that for the definition of density dependent evolutionary stable strategy DDESS and its stability, Cressmanʹs frequency conditions are necessary and sufficient, and Cressmanʹs density condition is not required.