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.

中文翻译：

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具有非线性反应交叉扩散的捕食者-猎物模型的分岔和图灵不稳定性

摘要 考虑了Neumann边界条件下的非线性反应交叉扩散捕食-猎物系统。引入具有猎物局部积累效应的负扩散系数。首先，讨论了具有或不具有交叉扩散的正均匀稳态的局部渐近稳定性标准。此外，获得了扩散驱动不稳定性的条件，并实现了交叉扩散系数平面中的图灵区。其次，借助李雅普诺夫-施密特约简，研究了空间非齐次/齐次稳态解的存在性和多样性。最后，为了阐明理论结果，进行了一些数值模拟。