Agent-based diffusion in predation systems with Beddington–DeAngelis response

Understanding dynamical behavior of a spatially distributed population is crucial to conservation and management of endangered species. This paper considers predator–prey systems with Beddington–DeAngelis functional response, where the predator moves between source–sink patches asymmetrically and acts as an agent. Our aim is to show how agent-based diffusion affects dynamics of the system and total population abundance of the species. Using dynamical systems theory, we demonstrate stability of positive equilibria in the system, which implies coexistence of the species and change of abundance by diffusion. Moreover, we show Hopf and Bautin bifurcations with multiple limit cycles, which implies multiple oscillations of populations and even extinction of species. Furthermore, this work demonstrates that diffusion in the system may lead to results reversing those without diffusion. The diffusion could change dynamics of the system between coexistence at a steady state and persistence in periodic oscillation, while evolution in asymmetry of diffusion could make the predator reach a total abundance larger than that without diffusion, even reach the maximal abundance. Our results are consistent with experimental observations and are important in studying conservation of biodiversity.