Dynamical analysis of a pine wilt disease model with memory-based diffusion and nonlocal effect

Since spatial memory and nonlocal effect have significant impact on animal movement modeling, we construct a pine wilt disease model with memory-based diffusion and nonlocal effect incorporating Holling-II functional response function to study the control of pine wilt disease. We establish the existence conditions for Turing, Hopf and Turing-Hopf bifurcations, and find that the joint effect of memory-based diffusion coefficient and memory delay can give rise to Turing-Hopf bifurcation. Moreover, the multiple time scales method is extended, showing that it is applicable to the reaction-diffusion system with both nonlocal effect and memory-based diffusion, and the methods for eliminating the secular terms and solving high-order terms for such systems are novelly provided as well in the process of derivation. By using this method, the degenerate Hopf-transcritical bifurcation is derived. Via concisely comparing it with the classical Hopf-zero bifurcation, some interesting results are presented, which suggest that the proposed model exhibits various complex spatiotemporal patterns, such as the six-stable phenomenon, that is, a pair of stable spatially inhomogeneous steady states and periodic solutions induced by Turing-Hopf bifurcation and a pair of stable spatially inhomogeneous large amplitude periodic solutions found numerically, which may provide an alternative approach in explaining the periodic outbreaks of pine wilt disease. Numerical simulations are also performed by adopting scientifically obtained parameters with actual biological meaning. Based on this, the corresponding ecological significance and disease control suggestions have more practical reference value.