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Coexisting Stable Equilibria in a Multiple-allele Population Genetics Model

In this paper we find and classify all patterns for a single locus three- and 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 $\dot{p}_i = p_i(r_i-r), i=1,...,k,$ where $p_i, r_i$ are the frequency and marginal fitness of allele $A_i$, 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 three-allele model, we also determine the asymptotic behavior of solutions of the above system of equations as $t \to \infty$. 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.