Bifurcations and pattern formation in a predator-prey model with memory-based diffusion

New spatial temporal patterns for Holling-Tanner predator-prey model with memory-based diffusion are investigated. Firstly, we show the existence of Turing, Hopf, Turing-Turing, and Turing-Hopf bifurcations, and obtain sufficient and necessary conditions for Turing instability. Secondly, we extend the formulae for coefficients in the normal form of Turing-Hopf bifurcation in Jiang et al. (2020) [1], and provide new formulae applicable to the general reaction-diffusion system with memory-based self-diffusion and cross-diffusion. These formulae are also applicable to the case that the memory delay is absent. From the bifurcation analysis, we finally find that the model can exhibit various complex spatiotemporal patterns, including a pair of stable spatially inhomogeneous periodic solutions with two spatial frequencies and heteroclinic orbits connecting non-constant steady states to a spatially inhomogeneous periodic solution. These results show that large memory-based diffusion coefficient can eliminate spatial patterns driven by Fickian diffusion, and memory delay mainly affects spatiotemporal patterns arising from Hopf and Turing-Hopf bifurcations.