This paper is dedicated to a study of a diffusive one-prey and two-cooperative-predators model with C–M functional response subject to Dirichlet boundary conditions. We first discuss the existence of positive steady states by the fixed point index theory and the degree theory. In the meantime, we analyze the uniqueness and stability of coexistence states under conditions that one predator’s consumer rate is small and the effect of interference intensity of another predator is large. Then, steady-state bifurcations from two strong semi-trivial steady states (provided that they uniquely exist under some conditions) and from one weak semi-trivial steady state are investigated in detail by the Crandall–Rabinowitz bifurcation theorem, the technique of space decomposition and the implicit function theorem. In addition, we study the asymptotic behaviors including the extinction and permanence of the time-dependent system by the comparison principle, upper-lower solution method and monotone iteration scheme. Finally, numerical simulations are done not only to validate the theoretical conclusions, but also to further clarify the impacts of parameters on the three species.

中文翻译：

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具有 C-M 功能响应的扩散单猎物和双合作捕食者模型

本文致力于研究在 Dirichlet 边界条件下具有 C-M 功能响应的扩散单猎物和双合作捕食者模型。我们首先通过不动点指数理论和度理论讨论正稳态的存在。同时，我们分析了在一个捕食者的消费率较小且另一个捕食者的干扰强度影响较大的情况下，共存状态的唯一性和稳定性。然后，通过空间分解技术 Crandall-Rabinowitz 分岔定理，详细研究了来自两个强半平凡稳态（假设它们在某些条件下唯一存在）和一个弱半平凡稳态的稳态分岔和隐函数定理。此外，我们通过比较原理、上下解法和单调迭代方案研究了时变系统的消光和持久性等渐近行为。最后，进行数值模拟不仅验证了理论结论，还进一步阐明了参数对三个物种的影响。