Population and evolutionary adaptive dynamics of a stochastic predator-prey model

Abstract

This paper intends to develop a theoretical framework for investigating the evolutionary adaptive dynamics of a stochastic differential system. The key to the question is how to build an evolutionary fitness function. Firstly, we propose a stochastic predator-prey model with disease in the prey and discuss the asymptotic behavior around the positive equilibrium of its deterministic equation. Secondly, by using stochastic population dynamics and adaptive dynamics methods, we propose a fitness function based on stochastic impact and investigate the conditions for evolutionary branching and the evolution of pathogen strains in the infective prey. Our results show that (1) large stochastic impact can lead to rapidly stable evolution towards smaller toxicity of pathogen strains, which implies that stochastic disturbance is beneficial to epidemic control; (2) stochastic disturbance can go against evolutionary branching and promote evolutionary stability. Finally, we carry on the evolutionary analysis and make some numerical simulations to illustrate our main results. The developed methodologies could potentially be used to investigate the evolutionary adaptive dynamics of the stochastic differential systems.