Pattern dynamics of a predator–prey system with cross-diffusion, Allee effect and generalized Holling IV functional response

In this paper, a diffusive predator–prey model with Allee effect and functional response of generalized Holling type IV is established. The complex pattern dynamics of and bifurcation phenomena of system are analyzed by using linear stability analyses and bifurcation theory, etc. Corresponding to spiral pattern, spot pattern or spot–stripe mix pattern and chaotic pattern, respectively, the conditions that arise Hopf instability, Turing instability and Hopf–Turing instability of constant positive steady solutions of reaction–diffusion system are presented. The impacts of Allee effect and cross-diffusion on pattern formations of the system are further discussed. Particularly, provided that cross-diffusion is absent, there will not appear the formation of Turing pattern, that is, Turing pattern is only driven by cross-diffusions. Addressed to the different pattern formations involving in cross-diffusion and Allee effect, overall numerical simulations are provided when the spatial dimension is two and the theoretical results are demonstrated. It is revealed that the larger Allee effect constant A and the cross-diffusion coefficient d4 are advantageous for the appearing of formation of Turing pattern, while the cross-diffusion coefficient d3 suppresses the formation of Turing pattern with its increasing.