This work investigates a prey–predator model with Beddington–DeAngelis functional response and discrete time delay in both theoretical and numerical ways. Firstly, we incorporate into the system a discrete time delay between the capture of the prey by the predator and its conversion to predator biomass. Moreover, by taking the delay as a bifurcation parameter, we analyze the stability of the positive equilibrium in the delayed system. We analytically prove that the local Hopf bifurcation critical values are neatly paired, and each pair is joined by a bounded global Hopf branch. Also, we show that the predator becomes extinct with an increase of the time delay. Finally, before the extinction of the predator, we find the abundance of dynamical complexity, such as supercritical Hopf bifurcation, using the numerical continuation package DDE-BIFTOOL.

中文翻译：

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具有 Beddington-DeAngelis 功能响应的延迟猎物-捕食者模型的动态行为：稳定性和周期性

这项工作以理论和数值方式研究了具有 Beddington-DeAngelis 函数响应和离散时间延迟的猎物 - 捕食者模型。首先，我们将捕食者捕获猎物与其转化为捕食者生物量之间的离散时间延迟纳入系统。此外，我们以时滞为分岔参数，分析了时滞系统中正平衡的稳定性。我们分析证明了局部 Hopf 分岔临界值是整齐配对的，并且每一对都由一个有界全局 Hopf 分支连接。此外，我们表明捕食者随着时间延迟的增加而灭绝。最后，在捕食者灭绝之前，我们使用数值延续包 DDE-BIFTOOL 发现了丰富的动力学复杂性，例如超临界 Hopf 分岔。