Bifurcation analysis and simulations of a modified Leslie–Gower predator–prey model with constant‐type prey harvesting
Bifurcation analysis and simulations of a modified Leslie–Gower predator–prey model with constant‐type prey harvesting
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DOI：
10.1002/mma.9593
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发表时间：
2023-08
影响因子：
2.9
通讯作者：
Xintian Jia;Ming Zhao;Kunlun Huang
中科院分区：
文献类型：
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作者：
Xintian Jia;Ming Zhao;Kunlun Huang
本文讨论了不断捕获猎物对修改后的莱斯利-高尔捕食者-猎物模型动力学的影响。猎物收获将导致丰富的动态，帮助我们实现生态现象，例如当收获率大于临界水平时，某些具有正初始密度的物种会灭绝。所有这些对于生物管理可能非常有用。基于这些原因，我们首先分析不同平衡点的存在条件和稳定性，以预测给定系统的最终状态。特别是，证明了余维2和余维3尖点的存在。其次，我们成功证明了指定参数值下 Hopf 分岔和鞍节点分岔的存在。为了证明 Hopf 分岔的极限环稳定性，我们用第一个 Lyapunov 系数 K$$ K $$ 来说明。除了计算尖点附近的通用展开式外，还研究了余维 2 和 3 的 Bogdanov-Takens 分岔。如果参数选择得当，系统中将存在一个稳定或不稳定的极限环、一个同宿环、两个极限环或同时存在极限环和同宿环。最后，利用MATLAB进行数值模拟来说明理论结果。
The paper discusses the effects of constant prey harvesting on the dynamics of a modified Leslie–Gower predator–prey model. Prey harvesting will lead to enriched dynamics to help us realize ecological phenomena such as the extinction of some species with a positive initial density when the harvesting rate is larger than a critical level. All these may be very useful for biological management. Based on these reasons, firstly, we analyze the existence conditions and stability of different equilibrium points to predict the eventual state of the given system. In particular, the existence of cusps of Codimensions 2 and 3 is proved. Secondly, we successfully demonstrate the presence of Hopf bifurcation and saddle‐node bifurcation for specified parameter values. To prove the limit cycle stability of Hopf bifurcation, we use the first Lyapunov coefficient K$$ K $$ for illustration. As well as calculating the universal expansion near the cusp, the Bogdanov–Takens bifurcations of Codimensions 2 and 3 are investigated. If the parameters are chosen fittingly, there will be a stable or an unstable limit cycle, a homoclinic loop, two limit cycles, or both a limit cycle and a homoclinic loop simultaneously in the system. Finally, numerical simulations are performed using MATLAB to illustrate the theoretical results.