Bifurcations in diffusive predator-prey systems with Beddington-DeAngelis functional response

Understanding how species' diffusion and environmental heterogeneity affect natural population dynamics is crucial in wildlife management and biodiversity conservation. In this paper, we consider a predation model with Beddington-DeAngelis functional response, where the prey moves between source-sink patches. Using Lyapunov theory and graph method, we demonstrate equilibrium stability, supercritical/subcritical bifurcations and persistence of the system. It is shown that there may exist two limit cycles in the system, and varying a diffusion rate can lead to transition of species' interaction outcomes between coexistence in periodic oscillation, persistence at a steady state, extinction of the predator and extinction of both species in a smooth fashion. Rigorous analysis on the equilibria shows that total population abundance of a diffusing prey may exceed total population abundance of a non-diffusing prey, even when the predator persists. Asymmetry in diffusion is also shown to play a role in the increase.