Turing bifurcation of solutions in a predator-prey system with nonlinear diffusion: formation of spatially ordered patterns

This paper investigates the Turing bifurcation of Hopf bifurcating periodic solutions in a Holling-Tanner type population system with nonlinear diffusion. It focuses on how diffusion (including self-diffusion, nonlinear diffusion and cross-diffusion) destabilizes periodic solutions and induces the generation of new and abundant spatially ordered patterns. By employing the local Hopf bifurcation theorem, perturbation theory, implicit function theorem and Floquet theory, a diffusion rate formula is derived to determine the conditions for Turing bifurcation of the stable periodic solutions induced by diffusion. Finally, numerical simulations are carried out to verify the theoretical analysis results, and new phenomena of spatial Turing patterns in populations are revealed.