Joint Impact of Diffusion and Advection on Predator-Prey Dynamics with Delay and Fear Effect

Abstract. We study a two-species predator-prey model that incorporates predator maturation delay and the fear effect within an advective environment under general boundary conditions. First, we derive a variational expression for the principal eigenvalue associated with a general eigenvalue problem. Next, we classify the system’s dynamics using basic reproduction numbers, demonstrating the global asymptotic stability of both the trivial solution and the predator-free equilibrium as well as the uniform persistence of the system and the coexistence of predator and prey. Additionally, we analyze how diffusion and advection influence the principal eigenvalues to better understand their impact on predator-prey dynamics. Our results imply that variations in the advection and diffusion rates of both predators and prey can drive multiple dynamic transitions, including successful invasions, failed invasions, and stable coexistence. Finally, by using the generalized maximum principle, we establish the uniqueness of positive steady-state solutions for a general predation system in an advective, heterogeneous environment with general boundary conditions.