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

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. (C) 2020 Elsevier Inc. All rights reserved.

Automated summary

In this study, a nonlinear reaction cross-diffusion predator-prey system under Neumann boundary condition is considered, with negative diffusion coefficients introducing local accumulation effect of prey. The paper discusses the criteria for local asymptotic stability of the positive homogeneous steady state with or without cross-diffusion, as well as the conditions for diffusion-driven instability. The existence and multiplicity of spatially nonhomogeneous/homogeneous steady-state solutions are studied using Lyapunov-Schmidt reduction, and numerical simulations are carried out to clarify the theoretical results, revealing that Turing instability in the model is induced by the negative diffusion coefficients.