In this paper, we study a diffusive predator–prey system with the Allee effect and threshold hunting. First, the number of interior equilibrium points is determined by discussing the relation of parameters. Then, preliminary analysis on the local asymptotic stability and bifurcations of non-spatial system based on ordinary differential equations is presented. It is noted that four stable equilibrium points coexist due to the Allee effect and threshold hunting. The stability of interior equilibrium points and the existence of Turing instability induced by the diffusion, spatially homogeneous and inhomogeneous Hopf bifurcation, Turing–Hopf bifurcation are studied by analyzing the corresponding characteristic equation for spatial system. By constructing generalized Jacobian matrix, we analyze the stability of interior equilibrium point where u-component is equal to the threshold of functional response. These results show that the Allee effect, threshold hunting and diffusion have significant impacts on the dynamics. Last, we present some numerical simulations that supplement the analytic results.

中文翻译：

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具有Allee效应和阈值狩猎的捕食者-食饵扩散系统的时空动力学。

在本文中，我们研究了具有Allee效应和阈值搜寻的扩散捕食－被捕食系统。首先，通过讨论参数之间的关系来确定内部平衡点的数量。然后，基于常微分方程，对非空间系统的局部渐近稳定性和分支进行了初步分析。注意，由于阿利效应和阈值搜寻，四个稳定的平衡点共存。通过分析空间系统的相应特征方程，研究了内部平衡点的稳定性以及因扩散，空间均匀和不均匀的Hopf分支，Turing-Hopf分支引起的图灵不稳定性的存在。通过构造广义雅可比矩阵，我们分析了内部平衡点的稳定性，其中u分量等于功能响应的阈值。这些结果表明，Allee效应，阈值搜寻和扩散对动力学有重大影响。最后，我们提出一些数值模拟，以补充分析结果。