Population Dynamics of Some Mutualistic Relationships

Symbiosis or mutualism is one of major ecological interactions between two species, where the both species gain benefits via the interactions. We consider the population dynamics of mutualistic interactions with positive density and frequency dependences. We specifically suppose the dynamics of Müllerian mimicry in butterflies for positive density dependence, where the mortality of both species is reduced depending on the relative frequency of the other species. Positive frequency dependence is studied in contrast with positive density dependence, and supposing the general relationships of symbiosis. We assume that the two species are under the 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. Symbiotic relationships never invoke infinite population growth, because of limited effects on population growth. However, the two models show a distinctive difference for the coexistence. Positive density dependence does not promote coexistence of species. Only when the coexistence is achieved under competition, the equilibrium densities are increased. In contrast, positive frequency dependence always promotes coexistence. This change is also qualitative. Whether Müllerian mimicry is a special case of mutualism, or mutualism does not promote coexistence under competition is an open question.