Patterns for four-allele population genetics model

In this paper, we find and classify all existing patterns for a single-locus four-allele population genetics models in continuous time. An existing pattern for a k-allele model means a set of all coexisting asymptotically stable equilibria with respect to the flow defined by the system of equations ṗi=pi(ri−r),i=1,…,k, where pi and ri are the frequency and marginal fitness of allele Ai, respectively, and r is the mean fitness of the population. It is well known that for the two-allele model there are only three existing patterns, depending on the relative fitness between the homozygotes and the heterozygote. For the three-allele model there are 14 existing patterns, and we shall show in this paper that for the four-allele model there are 117 existing patterns. We also describe the domains of attraction for coexisting asymptotically stable equilibria. The problem of finding existing patterns has been studied in the past, and it is an important problem because the results can be used to predict the long-term genetic makeup of a population. It should be pointed out that this continuous-time model is only an approximation to the corresponding discrete-time model. However, the set of equilibria and their stability properties are the same for the two models.