  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.
