Bifurcation and turing instability for a predator-prey model with nonlinear reaction cross-diffusion

Abstract

A nonlinear reaction cross-diffusion predator-prey system under Neumann boundary condition is considered. Negative diffusion coefficients with local accumulation effect of prey are introduced. Firstly, the criteria for local asymptotic stability of the positive homogeneous steady state with or without cross-diffusion are discussed. Moreover, the conditions for diffusion-driven instability are obtained and the Turing regions in the plane of cross-diffusion coefficients is achieved. Secondly, the existence and multiplicity of spatially nonhomogeneous/homogeneous steady-state solutions are studied by virtue of the Lyapunov–Schmidt reduction. Finally, to clarify the theoretical results, some numerical simulations are carried out. One of the most interesting finding is that Turing instability in the model is induced by the negative diffusion coefficients.