Gavrilets, S., and Hastings, A. 1998. ``Coevolutionary chase in two-species systems with applications to mimicry.'' Journal of Theoretical Biology 191: 415-427


We study a general dynamical model describing coevolution of two haploid populations with two alleles at a single locus under weak linear symmetric frequency-dependent selection. We apply our results to a series of simple population genetics models describing classical M\"{u}llerian and Batesian mimicries as well as intermediate cases. A novel and more realistic element of our modeling approach is that both species are allowed to evolve. We analyze conditions for ``evolutionary chase'' between two phenotypically similar species in which one species evolves to decrease its resemblance with the other species while this other species evolves to increase its resemblance with the first species. We show that permanent evolutionary chase between Batesian mimic and model or between ``strong'' and ``weak'' M\"{u}llerian mimics can occur under a range of parameter values if within-species interactions are stronger than within-species interactions. The evolutionary chase is represented by periodic solutions of the model equations. These periodic solutions can be characterized by large or small levels of genetic variability and can be simultaneously stable with equilibrium solutions. In the latter case the outcome and pattern of coevolution depend on the initial conditions. Our theoretical results may provide an explanation for the complex patterns of genetic variability observed in systems of mimicry. We show that one of the most important factors influencing the plausibility of non-equilibrium dynamics is the relationship between the strength of between-species and within-species interactions. This indicates that this relationship should be the focus of both experimental and theoretical work. Our results suggest that systematic studies of frequencies of different mimicry morphs through time may be very useful.