Evolutionary branching and evolutionarily stable coexistence of predator species: Critical function analysis.

On the ecological timescale, two predator species with linear functional responses can stably coexist on two competing prey species. In this paper, with the methods of adaptive dynamics and critical function analysis, we investigate under what conditions such a coexistence is also evolutionarily stable, and whether the two predator species may evolve from a single ancestor via evolutionary branching. We assume that predator strategies differ in capture rates and a predator with a high capture rate for one prey has a low capture rate for the other and vice versa. First, by using the method of critical function analysis, we identify the general properties of trade-off functions that allow for evolutionary branching in the predator strategy. It is found that if the trade-off curve is weakly convex in the vicinity of the singular strategy and the interspecific prey competition is not strong, then this singular strategy is an evolutionary branching point, near which the resident and mutant predator populations can coexist and diverge in their strategies. Second, we find that after branching has occurred in the predator phenotype, if the trade-off curve is globally convex, the predator population will eventually branch into two extreme specialists, each completely specializing on a particular prey species. However, in the case of smoothed step function-like trade-off, an interior dimorphic singular coalition becomes possible, the predator population will eventually evolve into two generalist species, each feeding on both of the two prey species. The algebraical analysis reveals that an evolutionarily stable dimorphism will always be attractive and that no further branching is possible under this model.