In the ecosystem, the chase of the predator with cooperation contributes to fear psychology in the prey, resulting in behavioral changes such as a decrease in the birth rate. We construct a spatially diffusive model with delay to investigate the combined perturbation of these factors. Initially, we establish the existence of positive solutions and examine the stability of steady-state solutions under varying conditions. The bifurcation dynamics of the positive solutions have been analyzed. Turing instability, arising from the random diffusion of the species, generates spatially irregular patterns characterized by patchy distribution of prey and predators in the spatial domain. Hopf bifurcation, resulting from the diffusive rate and delay, contributes to spatially periodic solutions where the number of species will spatially oscillate. The combined influence of diffusion and delay results in the emergence of Turing and Hopf bifurcation phenomena. In this case, the combined effect amplifies the spatially heterogeneous distribution of prey and predator. Our results reveal the heterogeneous behaviors of prey and predator under the coupled influence of cooperation hunting and fear effects. In this paper, we will also study a hybrid control scheme for controlling the generations of Turing patterns and Hopf bifurcation. Our theoretical results and numerical simulations demonstrate that the control scheme can mitigate the negative influence of combined factors and promote the species' stability.

中文翻译：

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空间猎物-捕食者模型中的猎物混合控制，具有协同狩猎和恐惧效应时间滞后


在生态系统中，合作追逐捕食者有助于猎物的恐惧心理，导致出生率下降等行为变化。我们构建了一个延迟的空间扩散模型来研究这些因素的组合扰动。最初，我们确定了正解的存在，并检查了稳态解在不同条件下的稳定性。已经分析了正解的分叉动力学。由物种的随机扩散引起的图灵不稳定性产生了空间不规则的模式，其特征是猎物和捕食者在空间域中的斑驳分布。由扩散速率和延迟引起的 Hopf 分岔有助于空间周期解，其中物种数量将在空间上振荡。扩散和延迟的综合影响导致了图灵和 Hopf 分叉现象的出现。在这种情况下，综合效应放大了猎物和捕食者的空间异质分布。我们的结果揭示了在合作、狩猎和恐惧效应的耦合影响下，猎物和捕食者的行为异质性。在本文中，我们还将研究一种用于控制图灵模式和 Hopf 分岔世代的混合控制方案。理论结果和数值模拟表明，该控制方案可以减轻综合因素的负面影响，促进物种的稳定性。