A model for population dynamics of the mimetic butterfly Papilio polytes in the Sakishima Islands, Japan.

We present a mathematical model for population dynamics of the mimetic swallowtail butterfly Papilio polytes in the Sakishima Islands, Japan. The model includes four major variables, that is, population densities of three kinds of butterflies (two female forms f. cyrus, f. polytes and the unpalatable butterfly Pachliopta aristolochiae) and their predator. It is well-known that the non-mimic f. cyrus resembles and attracts the male most, and the mimic f. polytes mimics the model butterfly P. aristolochiae. Based on experimental evidence, we assume that two forms f. cyrus and f. polytes interact under intraspecific competition for resources including the male, and the growth rate of f. cyrus is higher than that of f. polytes. We further assume that both the benefit of mimicry for the mimic f. polytes and the cost for the model are dependent on their relative frequencies, i.e. the motality of the mimic by predation decreases with increase in frequency of the model, while the motality of the model increases as the frequency of the mimic increases. Taking the density-dependent effect through carrying capacity into account, we set up a model system consisting of three ordinary differential equations (ODEs), analyze it mathematically and provide computer simulations that confirm the analytical results. Our results reproduce field records on population dynamics of P. polytes in the Miyako-jima Island. They also explain the positive dependence of the relative abundance (RA) of the mimic on the advantage index (AI) of the mimicry in the Sakishima Islands defined in Section 2.