Oscillations of frequency in batesian mimics, hawks and doves, and other simple frequency dependent polymorphisms

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Abstract
It is customary to infer the properties of the internal equilibria produced by frequency dependent selection from the properties of the boundary equilibria (often called the "invasion" criterion). This paper demonstrates that there are some circumstances under which a pair of unstable boundary equilibria do permit the conclusion that there is a truly stable, unique, internal equilibrium. For two alleles with complete dominance, if phenotype fitness declines monotonically with increasing frequency, then the internal point of equal phenotype fitness is the unique internal equilibrium of the genes, and is not unstable; this criterion may also be met if the fitness of one phenotype increases with frequency. It must be truly stable, in the sense of not producing oscillations, if the decline of fitness is linear or convex upwards and no phenotype is lethal at any frequency; the hawk-dove game complies with both conditions, and at least the second condition is likely to be met in most of the models encountered in sociobiology. However, an equilibrium which induces damped oscillations, or perhaps even complex limit cycles, is possible if at least one phenotype can be lethal at high frequency, or if the decline in fitness is strongly curvilinear and concave upwards. One case of curvilinear frequency dependence, a dimorphic batesian mimic with a non-mimetic form, is examined in detail. Although oscillations about the equilibrium point are possible, no matter whether the mimicry is dominant, recessive or Y-linked, this will only occur when selection coefficients are very large, and (except for Y-linkage) only if both sexes can be mimetic. As selection is density as well as frequency dependent, such conditions may be produced in the real world by large fluctuations in population size.