In this paper, we investigate a predator–prey model with herd behavior and cross-diffusion subject to the zero flux boundary conditions. First, the temporal behavior of the model has been investigated, where Hopf bifurcation has been obtained. Then, by analyzing the characteristic equation it has been proved that the cross-diffusion generate a complex dynamics such as Hopf bifurcation, Turing instability, even Turing–Hopf bifurcation. Further, the impact of the prey herd shape on the spatiotemporal patterns has been discussed. Furthermore, by computing and analyzing the normal form associated with the Turing–Hopf bifurcation point, the spatiotemporal dynamics near the Turing–Hopf bifurcation point has been discussed and also justified by some numerical simulations.

中文翻译：

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具有猎物群形效应的捕食者-猎物模型中交叉扩散诱导的时空模式

在本文中，我们研究了一种在零通量边界条件下具有羊群行为和交叉扩散的捕食者 - 猎物模型。首先，研究了模型的时间行为，获得了 Hopf 分岔。然后，通过对特征方程的分析，证明了交叉扩散产生了复杂的动力学，如 Hopf 分岔、图灵不稳定性，甚至 Turing-Hopf 分岔。此外，已经讨论了猎物群形状对时空模式的影响。此外，通过计算和分析与Turing-Hopf分岔点相关的范式，讨论了Turing-Hopf分岔点附近的时空动力学，并通过一些数值模拟证明了这一点。