Population dynamics of Müllerian mimicry under interspecific competition

We ask what the effects of mutualism on population dynamics of two competitive species are. We model the population dynamics of mutualistic interactions with positive density- and frequency-dependences. We specifically assume the dynamics of Müllerian mimicry in butterflies, where the mortality of both species is reduced depending on the relative frequency of the other species. We assume that the two species are under Lotka-Volterra density-dependent competition. The equilibria are compared with the cases of competition alone. Unlike the traditional model of positive density-dependence, population explosion does not appear in the current dynamics, but the new equilibrium is simply achieved. It is because the effects of positive density- or frequency-dependence are restricted to parts of mortality. Both positive density- and frequency-dependences do promote coexistence of the mimetic species. However, the two models show a distinctive difference for coexistence. The effects of positive density-dependence are rather limited. In contrast, positive frequency-dependence always promotes coexistence, irrespective of environmental conditions. The results may imply that the evolutionary origin of Müllerian mimicry may depend on frequency-dependence (and density-dependence), but that its current population dynamics may depend solely on density-dependence. The role of frequency- and density-dependences on evolutionary dynamics is an open question.