This paper investigates the spatiotemporal dynamics of a Monod–Haldane type predator–prey interaction system that incorporates: (1) a time delay in the predator response term in the predator equation; and (2) diffusion in both prey and predator. We provide rigorous results of our system including the asymptotic stability of equilibrium solutions and the existence and properties of Hopf bifurcations with or without time delay and diffusion. The effect of diffusion on bifurcated periodic solutions is investigated. We show that analytically and numerically at a certain value the carrying capacity or time-delay-driven stability or instability occurs when the corresponding system has either a unique interior equilibrium or two interior equilibria. Moreover, we illustrate the spatial patterns out of an initially nearly homogeneous state via numerical simulations, which show that the system dynamics exhibits complex pattern replication: spiral wave patterns and chaotic spiral patterns b...

中文翻译：

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具有时滞和空间扩散的捕食系统的时空动力学

本文研究了一个Monod–Haldane型捕食者－被猎物相互作用系统的时空动力学，该系统包括：（1）捕食者方程中捕食者响应项的时间延迟； 2。（2）在猎物和捕食者中的扩散。我们提供了系统的严格结果，包括平衡解的渐近稳定性以及具有或不具有时间延迟和扩散的Hopf分支的存在和性质。研究了扩散对分支周期解的影响。我们表明，当相应系统具有唯一的内部平衡或两个内部平衡时，在一定值上进行分析和数值分析时，就会出现承载力或时滞驱动的稳定性或不稳定性。此外，