The wave speed of intergradation zone in two-species lattice Müllerian mimicry model

A spatially explicit model is studied to analyse the movement of coupled clines in two-species Müllerian mimicry system as exemplified by the comimicking helicoiine butterflies in Central-South America Heliconius erato and Heliconius melpomene. In this system, a pair of comimicking wing patterns of two species (mimicry ring) is found in a geographical region but another pair of wing patterns is found in a different geographical region. The distribution of mimicry rings thus forms a spatial mosaic in a large geographical scale, and the mechanism responsible for their stable maintenance has been a long-standing question in evolutionary biology. We here examine the speed of the movement of boundaries that divide the regions inhabited by different mimetic morphs in each comimicking species, by assuming coupled two-state stochastic cellular automatons where the flipping rate of the site occupied by a mimetic morph depends on the local density of the same morph and of the comimicking morph in the other species. The speed of cline movement shows a complex dependence on the coupling parameter between mimetic species--greater coupling of comimicking morphs between species slows down the cline movement only when the reduction in predation rate exhibits diminishing return to the increase of local mimetic morph density. The analytical predictions are confirmed by the results of Monte Carlo simulations. The speed of advance is quite different from that predicted from the conventional reaction-diffusion model, indicating that demographic stochasticity plays a critical role in determining the speed of cline movement. We also examine if the spatial heterogeneity in migration rate can stably maintain clines.