This paper deals with the Turing bifurcation and pattern dynamics of a Lotka-Volterra model with the predator-taxis and the homogeneous no-flux boundary conditions. To investigate the pattern dynamics, we first give the occurrence conditions of the Turing bifurcation. It is found that there is no Turing bifurcation when predator-taxis disappears, while the Turing bifurcation occurs as predator-taxis is presented. Next, we establish the amplitude equation by virtue of weakly nonlinear analysis. Our theoretical result suggests the Lotka-Volterra model admits the supercritical or subcritical Turing bifurcation. In this manner, we can determine the stability of the bifurcating solution. Finally, some numerical simulation results verify the validity of the theoretical analysis. The stripe pattern, the mixed patterns, and wave patterns are performed. Interestingly, the stable stripe patterns will be broken and become wave patterns when the predator-taxis parameter is far from the Turing bifurcation critical point.

中文翻译：

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具有滑行机制的 Lotka-Volterra 模型的模式动力学


本文研究了具有捕食者趋向性和齐次无通量边界条件的 Lotka-Volterra 模型的图灵分岔和模式动力学。为了研究模式动力学，我们首先给出图灵分岔的发生条件。研究发现，当捕食者趋向性消失时，不存在图灵分岔；而当捕食者趋向性出现时，则出现图灵分岔。接下来，我们通过弱非线性分析建立振幅方程。我们的理论结果表明 Lotka-Volterra 模型承认超临界或亚临界图灵分岔。通过这种方式，我们可以确定分叉解的稳定性。最后，数值模拟结果验证了理论分析的有效性。执行条纹图案、混合图案和波浪图案。有趣的是，当捕食者趋向性参数远离图灵分岔临界点时，稳定的条纹图案将被打破并变成波浪图案。