Evolutionary analysis of adaptive dynamics model under variation of noise environment

This paper proposes a stochastic model for the evolutionary adaptive dynamics of species subject to trait-dependent intrinsic growth rates and the influence of environmental noise. The aim of this paper is twofold: (a) mathematically we make an attempt to investigate the evolutionary adaptive dynamics for models with noises; (b) biologically we investigate how the noises in environment affect the evolutionary stability. We first investigate the extinction and permanence of the population in the presence of environmental noises. Combining evolutionary adaptive dynamics with stochastic dynamics, we then establish a fitness function with stochastic disturbance and obtain the evolutionary conditions for continuously stable strategy and evolutionary branching. Our study finds that under intense competition among species, increasing stochastic disturbance can lead to rapidly stable evolution towards smaller trait values, but there is an opposite effect under weak competition among species. This yields an interesting evolutionary threshold, beyond which any increasing stochastic disturbance can go against evolutionary branching and promote evolutionary stability. We then carry out the evolutionary analysis and numerical simulations to illustrate our theoretical results. Finally, for demonstrating the emergence of high-level polymorphism we perform long-term simulation of evolutionary dynamics.