Stability and bifurcation analysis of a ratio-dependent community dynamics model on Batesian mimicry

Batesian mimicry is the similarity of coloration and patterns in an unpalatable species (the "model-species") and a palatable species (the "mimic-species"). The resemblance is advantageous for the mimic-species because the mimic-species can deceive predators and avoid predation. While Batesian mimicry is an important subject in ecology as a general phenomenon in nature, previous theoretical studies focus mainly on the evolution of mimicry and the predator learning process. In these mathematical models, the population sizes of the model- and mimic-species are not considered explicitly or are assumed to be constant, but this is not plausible in model-mimic community dynamics. Thus, the model-mimic community has been paid relatively less attention; However, to elucidate problems on Batesian mimicry, it is essential to understand the fundamental characteristics of the model-mimic community dynamics. Here, we construct a basic model-mimic community dynamics model, obtain the existence and stability conditions of its equilibria, and conduct the bifurcation analysis and numerical calculation. The results show that the instability of the model-only population is predicted, and this is consistent with the typical pattern of geographical distribution in Batesian mimicry in the field. We propose three new hypotheses to explain the typical pattern of geographical distribution. Furthermore, we reveal an irreversibility regarding the model-mimic coexistence that is important for the conservation of the model- and mimic-species.