In this paper, we consider the impacts of the chemotaxis and delay on the dynamics of a diffusive predator–prey system with fear effect under the Neumann boundary conditions. Regarding chemotaxis coefficient and delay as bifurcation parameters, the existence of the codimension-two Turing–Hopf bifurcation is studied by analyzing the associated characteristic equation. We deduce that chemotaxis-driven Turing bifurcation and discrete delay-driven Hopf bifurcation can occur simultaneously. Finally, spatially homogeneous periodic oscillation, spatial patterns and spatiotemporal patterns appear near Turing–Hopf bifurcation by numerical simulations.

中文翻译：

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具有趋化性和恐惧效应的时滞扩散捕食系统的图灵-霍夫分叉

在本文中，我们考虑在诺伊曼边界条件下，趋化性和延迟对具有恐惧效应的扩散捕食－被捕食系统动力学的影响。将趋化系数和延迟作为分叉参数，通过分析相关的特征方程，研究了二维的图灵-霍普夫分叉的存在。我们推论趋化性驱动的图灵分叉和离散延迟驱动的Hopf分叉可以同时发生。最后，通过数值模拟，图灵-霍普夫分叉附近出现了空间均匀的周期性振动，空间模式和时空模式。