In this paper, we investigate the qualitative behaviors of a predator–prey system with ratio-dependent function. The system accommodates the diffusion effect to model the migration of individuals and the time delay induced by reproduction. We start with some basic properties of the system. Then the sufficient condition independent of time delay and diffusion effect for global asymptotical stability of the boundary equilibrium is obtained by using the comparison principle. Afterwards, based on the LaSalle’s invariance principle and Lyapunov functional, we investigate the global attractiveness of the positive equilibrium, arriving at its global asymptotical stability. Further, Hopf bifurcation induced by time delay around the positive equilibrium is explored. Finally, numerical examples are listed to verify the corresponding analytical results.

中文翻译：

---

具有比率依赖性功能响应和时间延迟的扩散捕食者 - 猎物系统的动力学

在本文中，我们研究了具有比率依赖函数的捕食者 - 猎物系统的定性行为。该系统适应扩散效应来模拟个体的迁移和繁殖引起的时间延迟。我们从系统的一些基本属性开始。然后利用比较原理得到了边界平衡全局渐近稳定与时滞和扩散效应无关的充分条件。然后，基于LaSalle不变原理和Lyapunov泛函，我们研究了正均衡的全局吸引力，得出了它的全局渐近稳定性。此外，探讨了由正平衡周围的时间延迟引起的 Hopf 分岔。最后通过数值算例验证了相应的解析结果。