Abstract
In this paper, with the method of adaptive dynamics, we investigate the coevolution of phenotypic traits of predator and prey species. The evolutionary model is constructed from a deterministic approximation of the underlying stochastic ecological processes. Firstly, we investigate the ecological and evolutionary conditions that allow for continuously stable strategy and evolutionary branching. We find that evolutionary branching in the prey phenotype will occur when the frequency dependence in the prey carrying capacity is not strong. Furthermore, it is found that if the two prey branches move far away enough, the evolutionary branching in the prey phenotype will induce the secondary branching in the predator phenotype. The final evolutionary outcome contains two prey and two predator species. Secondly, we show that under symmetric interactions the evolutionary model admits a supercritical Hopf bifurcation if the frequency dependence in the prey carrying capacity is very weak. Evolutionary cycle is a likely outcome of the mutation-selection processes. Finally, we find that frequency-dependent selection can drive the predator population to extinction under asymmetric interactions.
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