Spatial patterns of a reaction-diffusion population system with cross-diffusion and habitat complexity

In this paper, a population system with cross-diffusion and habitat complexity is selected as study object. We investigate that how cross-diffusion and habitat complexity destabilize the otherwise stable periodic solutions of the ODEs to generate the new abundant spatial Turing patterns. By utilizing the local Hopf bifurcation theorem and perturbation theory, we establish a formula to determine the Turing instability of periodic solutions of the population system with cross-diffusion and habitat complexity. Finally, numerical simulations are performed to verify theoretical analysis, simultaneously, we verify the formation process of spatial Turing patterns when the cross-diffusion coefficients and habitat complexity change.