In this paper, we investigate the dynamics of a system in which both prey and predator are harvested with constant rate. Our main objective is to find the effects of harvesting on equilibria, stability, and bifurcations in the system, which may be useful for biological management. The existence and stability of equilibrium points of the model are further investigated. A thorough qualitative analysis has been carried out based on bifurcation theory in dynamical systems and to validate our analytical findings, a large scale numerical simulation has been performed by using plausible values of parameters involved. It is shown that the model can exhibit Hopf bifurcation. The first Lyapunov coefficient is calculated to determine the direction of limit cycle of Hopf bifurcation. Also, it has been proven analytically that the system exhibits Bogdanov–Takens bifurcation of codimension 2. Moreover, discrete-time delay effect has been included due to gestation of the predator species on the same system and observed Hopf bifurcation with respect to the delay parameter. This study renders important tools for investigations of the dynamics of biotic organisms for the management and control of over harvesting. Some phase plane analysis has been carried out to support our analytical results.

中文翻译：

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博格达诺夫-在两个物种持续收获和延迟效应的Gause型模型中对第2维进行分叉的分析

在本文中，我们研究了以恒定速率捕获猎物和捕食者的系统的动力学。我们的主要目标是发现收获对系统平衡、稳定性和分叉的影响，这可能对生物管理有用。进一步研究了模型平衡点的存在性和稳定性。基于动力系统中的分岔理论进行了彻底的定性分析并验证了我们的分析结果，通过使用相关参数的合理值进行了大规模数值模拟。结果表明，该模型可以表现出 Hopf 分岔。计算第一个李雅普诺夫系数以确定 Hopf 分岔的极限环方向。还，分析证明，该系统表现出余维数 2 的 Bogdanov-Takens 分岔。此外，由于捕食者物种在同一系统上的孕育，已包括离散时间延迟效应，并观察到关于延迟参数的 Hopf 分岔。这项研究为研究生物有机体动力学以管理和控制过度捕捞提供了重要工具。已经进行了一些相平面分析以支持我们的分析结果。这项研究为研究生物有机体动力学以管理和控制过度捕捞提供了重要工具。已经进行了一些相平面分析以支持我们的分析结果。这项研究为研究生物有机体动力学以管理和控制过度捕捞提供了重要工具。已经进行了一些相平面分析以支持我们的分析结果。