Adaptive dynamics analysis of a predator–prey model with selective disturbance

• Based on predators capture rate functions, we construct an invasion fitness function.
• We use a size-selective disturbance function to study evolutionary dynamics.
• Evolutionarily stable coexistence and branching are explored by numerical simulations.
• Harvesting may drive evolution towards smaller value of phenotype trait.
• Large disturbance can go against evolutionary branching and promote evolutionary stability.

Evolution problem is always a hot topic in the mathematical biology field. In this paper, we investigate the evolutionary effects of selective disturbance on an evolving trait (e.g. body size and maturation age) of the predator individuals in one-predator two-prey community. By using methods of adaptive dynamics and population dynamics we construct an invasion fitness function and obtain the conditions for evolutionary branching and evolutionary stability under selective disturbance in both monomorphic and dimorphic populations. We further conduct a size-selective disturbance function founded on chi-square distribution to study evolutionary stable coexistence, and considering the evolutionary branching and evolutionary stability by using theoretic analysis and numerical simulations. The evolutionary results from a biological point of view show that (1) two strategies could gradually evolve to form a single ancestral strategy, moreover, higher levels of polymorphism cannot build up during evolution, that is, following first evolutionary branching two species will eventually evolve into two generalist species and reach an evolutionary stable coexistence; (2) smaller disturbance could touch off higher levels of dimorphism during evolution, while large disturbance can go against evolutionary branching and advance evolutionary stability.