Multiple stability switches in a reaction-diffusion-advection model with heterogeneous environment and two nonlocal delayed effects

This paper investigates a Logistic type reaction-diffusion-advection model with heterogeneous environment and two nonlocal delay effects. The existence of spatial nonhomogeneous positive steady-state solutions and the associated Hopf bifurcation are studied by analyzing the corresponding eigenvalue problem with delay as a bifurcation parameter. The theoretical results show that multiple stability switching-back phenomenon still exists in system when the populations have a tendency to move up or down along habitat gradients due to environmental heterogeneity. The introduction of advection leads to more complicated stability switch modes of the positive steady-state solution for the delayed reaction-diffusion-advection system. In particular, it is shown that the time-delay thresholds for Hopf bifurcation may be increasing, decreasing, or non-monotonic with respect to advection rate in different situations.