Stochastic reaction–diffusion system modeling predator–prey interactions with prey-taxis and noises

Abstract

In this paper, we are concerned with a new stochastic system of nonlinear partial differential equations modeling the Lotka–Volterra interactions of predators and preys in the presence of prey-taxis, spatial diffusion, and noises. The spatial and temporal variations of the predator’s velocity are determined by the prey gradient. In the first part, we derive a macroscopic model from stochastic kinetic equations by using the micro–macro decomposition method. In the second part, we sketch the proof of the existence of weak martingale solutions by using a Faedo–Galerkin method. In the last part, we develop a one- and two-dimensional finite volume approximation for the stochastic kinetic and macroscopic models, respectively. Our one-dimensional space numerical scheme is uniformly stable along the transition from kinetic to macroscopic regimes. We close with various numerical tests illustrating the convergence of our numerical method and some features of our stochastic macro-scale system.