Spatial patterns through diffusion-driven instability are stationary structures that appear spontaneously upon breaking the symmetry of the spatial domain, which results only from the coupling between the reaction and the diffusion processes. This paper is concerned with a modified Leslie–Gower-type model with cross-diffusion and indirect predation effect. We first prove the global existence, non-negativity and uniform boundedness for the considered model. Then the linear stability analysis shows that the cross-diffusion is the key mechanism of spatiotemporal pattern formation. Amplitude equations are derived near Turing bifurcation point under nonlinear cross-diffusion to interpret pattern selection among spot pattern, stripe pattern and the mixture of spot and stripe patterns, which reflects the species’s spatially inhomogeneous distribution, and it is also found that the fear factor has great influence on spatially inhomogeneous distribution of the two species under certain cross-diffusivity, that is, high level of fear can induce striped inhomogeneous distribution, low level of fear can induce spotted inhomogeneous distribution, and the intermediate level of fear can induce the mixture of spotted and striped inhomogeneous distribution. Finally, numerical simulations illustrate the effectiveness of all theoretical results.

中文翻译：

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具有恐惧效应的改进的 LESLIE-GOWER PREDATOR-PREY 模型中的交叉扩散驱动模式形成和选择

通过扩散驱动的不稳定性的空间模式是在打破空间域的对称性时自发出现的静止结构，这仅由反应和扩散过程之间的耦合引起。本文关注具有交叉扩散和间接捕食效应的修正 Leslie-Gower 型模型。我们首先证明了所考虑模型的全局存在性、非负性和一致有界性。然后线性稳定性分析表明，交叉扩散是时空格局形成的关键机制。在非线性交叉扩散下推导图灵分岔点附近的幅度方程来解释斑点图案、条纹图案以及斑点和条纹图案混合之间的图案选择，反映了物种在空间上的不均匀分布，并且还发现，在一定的交叉扩散率下，恐惧因素对两个物种的空间不均匀分布有很大的影响，即恐惧程度高会导致条纹不均匀分布，恐惧程度低会导致斑点不均匀分布，中等程度的恐惧会诱发斑点状和条纹状的不均匀分布。最后，数值模拟说明了所有理论结果的有效性。中级恐惧可诱发斑点状和条纹状不均匀分布的混合。最后，数值模拟说明了所有理论结果的有效性。中级恐惧可诱发斑点状和条纹状不均匀分布的混合。最后，数值模拟说明了所有理论结果的有效性。