Coexisting Stable Equilibria in a Multiple-allele Population Genetics Model

In this paper we find and classify all patterns for a single locus threeand four-allele population genetics models in continuous time. A pattern for a k-allele model means all coexisting locally stable equilibria with respect to the flow defined by the equations ṗi = pi(ri − r), i = 1, . . . , k, where pi, 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 patterns depending on the relative fitness between the homozygotes and the heterozygote. It turns out that for the three-allele model there are 14 patterns and for the four-allele model there are 117 patterns. With the help of computer simulations, we find 2351 patterns for the five-allele model. For the six-allele model, there are more than 60, 000 patterns. In addition, for each pattern of the threeallele model, we also determine the asymptotic behavior of solutions of the above system of equations as t → ∞. The problem of finding 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.