In this paper, we are concerned with a diffusive predator–prey model where the functional response follows the predator cooperation in hunting and the growth of the prey obeys the Allee effect. Firstly, the existence and stability of the positive equilibrium are explicitly determined by the local system parameters. It is shown that the ability of the hunting cooperation can affect the existence of the positive equilibrium and stability, and the intrinsic growth rate of the predator population does not affect the existence of the positive equilibrium, but affects the stability. Then the diffusion-driven Turing instability is investigated and the Turing bifurcation value is obtained, and we conclude that when the ability of the cooperation in hunting is weaker than some critical value, there is no Turing instability. The standard weakly nonlinear analysis method is employed to derive the amplitude equations of the Turing bifurcation, which is used to predict the types of the spatial patterns. And it is found that in the Turing instability region, with the parameter changing from approaching Turing bifurcation value to approaching Hopf bifurcation value, spatial patterns emerge from spot, spot-stripe to stripe. Finally, the numerical simulations are used to support the analytical results.

中文翻译：

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猎物群体中具有狩猎合作和等位效应的捕食者-猎物模型中的稳定性和图灵模式

在本文中，我们关注一个扩散的捕食者 - 猎物模型，其中功能响应遵循捕食者在狩猎中的合作，猎物的生长服从 Allee 效应。首先，正平衡的存在和稳定性是由局部系统参数明确决定的。结果表明，狩猎合作的能力会影响正平衡的存在和稳定性，捕食者种群的内在增长率不影响正平衡的存在，但会影响稳定性。然后研究了扩散驱动的图灵不稳定性，得到了图灵分岔值，我们得出结论，当狩猎合作能力弱于某个临界值时，不存在图灵不稳定性。采用标准的弱非线性分析方法推导出图灵分岔的幅度方程，用于预测空间模式的类型。并且发现在图灵不稳定区域，随着参数从接近图灵分岔值到接近Hopf分岔值的变化，空间图案出现从点状、点状到条纹状。最后，数值模拟用于支持分析结果。