This paper is devoted to study the spatiotemporal dynamics of a diffusive Leslie–Gower predator–prey model with Michaelis-Menten type harvesting in the prey population. The existence and stability of possible non-negative constant equilibria are investigated. By regarding [Formula: see text] as a bifurcation parameter, the Hopf bifurcation from the positive constant equilibrium solution is investigated. The necessary and sufficient conditions of Turing instability are explicitly obtained. We show that at the critical value of the bifurcation parameter [Formula: see text] a Turing bifurcation occurs (i.e. a pattern arises). The conditions for the stability of the pattern are also derived in detail. Moreover, the global steady state bifurcation from the positive constant equilibrium solution is investigated. In particular, the local steady state bifurcation from double zero eigenvalues is also obtained by the techniques of space decomposition and the implicit function theorem. Our results show that Michaelis–Menten type harvesting in our model plays a crucial role in the formation of spatiotemporal dynamics, which is a strong contrast to the case without harvesting.

中文翻译：

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弥散型 Leslie-Gower Predator-Prey 模型中 Michaelis-Menten 型猎物捕获诱导的时空动力学

本文致力于研究扩散 Leslie-Gower 捕食者 - 猎物模型在猎物种群中的时空动态，该模型具有 Michaelis-Menten 类型的捕捞。研究了可能的非负常数平衡的存在性和稳定性。以[公式：见正文]为分岔参数，研究正常平衡解的Hopf分岔。明确地获得了图灵不稳定性的充要条件。我们表明，在分岔参数的临界值[公式：见正文]，会发生图灵分岔（即出现模式）。还详细推导出了图案稳定的条件。此外，研究了正常数平衡解的全局稳态分岔。特别是，还通过空间分解和隐函数定理得到了双零特征值的局部稳态分岔。我们的结果表明，我们模型中的 Michaelis-Menten 类型的收获在时空动态的形成中起着至关重要的作用，这与没有收获的情况形成了强烈的对比。