Bifurcation analysis of pine wilt disease model with both memory-based diffusion and nonlocal effect

Animal movement and nonlocal interspecific competition are key roles in the spatial distribution of population. For the controlling of pine wilt disease which is spread by longhorns, we propose a reaction–diffusion predator–prey model with both nonlocal effect and memory diffusion considering Holling III type functional response in this paper. We explore the well-poesdness of solutions for non-spatial system and spatial system. We analyze the existence conditions of Turing and Hopf bifurcations, and derive the normal form of Hopf bifurcation for the system with both nonlocal effect and memory diffusion by multiple time scales method. Through data analysis, we determine the suitable parameters values to simulate considering biological significance. Results show that only Hopf bifurcation occurs in this system under this set of parameter values. Moreover, we conclude that the joint effect of nonlocal effect and memory diffusion can induce the stable spatially inhomogeneous periodic solutions of Hopf bifurcation. Especially, both memory diffusion and nonlocal effect can maintain the positive constant steady state stable. Furthermore, we also provide some biological explanations.