Abstract

In this article, we consider a predator–prey system with constant rate of harvesting, which exhibits Hopf and Bogdanov–Takens bifurcations under certain parametric conditions. The parametric space under which the system enters into Hopf bifurcation is investigated. By constructing suitable Lyapunov function, global stability results are obtained. Here, death rate and harvesting rate are taken as the Bogdanov–Takens bifurcation parameters. The canonical form of Bogdanov–Takens bifurcation is derived with the use of repeated nonlinear analytic transformation of coordinates. Later, we include the spatiotemporal effect on the same system and observed some relevant outcomes like Turing pattern, Turing–Bogdanov–Takens bifurcation, Turing–Hopf bifurcation and asynchrony of predator and prey in the space. The present study renders important tools for investigations of the dynamics of biotic organisms (predator and prey) for the management and control of overharvesting. Extensive numerical examples are given in support of the physical existence of the model system under consideration.

中文翻译：

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具有Beddington-DeAngelis型响应函数的时空收获-捕食和被捕食系统中Bogdanov-Takens分叉的分析

摘要

在本文中，我们考虑具有恒定收获速度的捕食者－被捕食者系统，在某些参数条件下，该系统具有Hopf和Bogdanov–Takens分支。研究了系统进入Hopf分支的参数空间。通过构造合适的Lyapunov函数，可以获得整体稳定性结果。在这里，死亡率和收成率被视为Bogdanov-Takens分叉参数。Bogdanov–Takens分叉的规范形式是通过使用重复的坐标非线性解析变换得出的。后来，我们将时空效应包括在同一系统中，并观察到了一些相关的结果，例如图灵模式，图灵-博格达诺夫-塔肯斯分叉，图灵-霍普夫分叉以及空间中的捕食者与被捕者的异步性。本研究为调查和管理过度捕捞的生物（捕食者和猎物）的动力学提供了重要的工具。给出了大量的数值示例来支持所考虑的模型系统的物理存在。