A model for population dynamics of the mimetic butterfly papilio polytes in sakishima Islands, Japan (II)

Abstract

Based on recent progresses of both experiment and mathematical analysis, we present an extension of the model for population dynamics of the mimetic swallowtail butterfly Papilio polytes in Sakishima Islands, Japan (Sekimura et al. J Theor Biol 361:133-140, 2014). The model includes four major variables, that is, population densities of three kinds of butterflies (two female forms f. cyrus and f. polytes and the unpalatable butterfly Pachliopta aristolochiae) and their predator. In this extension, we introduce difference in the predation rate between two forms f. cyrus and f. polytes. We still 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 mortality of the mimic by predation decreases with increase in frequency of the model, while the mortality of the model increases as the frequency of the mimic increases. Taking the density-dependent effect by carrying capacity into account, we set up an extended model system consisting of three ordinary differential equations (ODEs), analyze it mathematically, and provide computer simulations that confirm the analytical results.