Dynamics of a pine wilt disease control model with nonlocal competition and memory diffusion.

Pine wilt disease (PWD) is mainly spread by Monochamus alternatus (in short, M. alternatus). Woodpecker, as the natural predator of M. alternatus, is considered for biological prevention and controlling the PWD. In this paper, we propose a new M. alternatus-woodpecker model with nonlocal competition and memory-based diffusion, which makes the model more realistic for the PWD control. We focus on the dynamics and bifurcations of the model with various combinations of the memory diffusion and nonlocal competition. It is shown that the nonlocal competition can only cause the stable constant steady state to lose stability, while the memory-based diffusion can induce unstable spatially inhomogeneous periodic solutions due to Hopf bifurcation. Consequently, we can explain the spatiotemporal heterogeneity problem in ecology by innovatively using mathematical modelling. Normal form theory with the multiple time scales method is applied to particularly consider Hopf bifurcation, showing complex dynamical behaviours involving various oscillating motions. Finally, numerical simulations are presented with the parameter values chosen from the real forest data of Yuan'an County, Hubei Province, China, confirming the theoretical results of the spatiotemporal heterogeneity of forest diseases and pests, as well as the PWD control.