Bifurcation analysis in a predator-prey model with constant-yield predator harvesting
Bifurcation analysis in a predator-prey model with constant-yield predator harvesting
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DOI：
10.3934/dcdsb.2013.18.2101
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发表时间：
2013-07
影响因子：
1.2
通讯作者：
Jicai Huang;Y. Gong;S. Ruan
中科院分区：
文献类型：
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作者：
Jicai Huang;Y. Gong;S. Ruan
在本文中，我们研究了恒定产量捕食者收获对莱斯利-高尔型捕食者-被捕食者模型动力学的影响。结果表明，对于某些参数值，该模型分别具有余维 3 的 Bogdanov-Takens 奇点（尖点情况）或重数 2 的弱焦点。随着参数值的变化，会出现鞍结分岔、排斥和吸引Bogdanov-Takens分岔、超临界和亚临界Hopf分岔以及简并Hopf分岔。因此，存在不同的参数值，模型具有同宿环或两个极限环。还证明存在一个临界收获值，当收获率大于临界值时，对于两个物种的所有允许的初始密度，捕食者物种都会灭绝。这些结果表明，该模型的动态行为对恒定产量的捕食者收获和两个物种的初始密度非常敏感，并且需要在应用保护和可再生资源背景下进行仔细管理。数值模拟，包括排斥和吸引Bogdanov-Takens分岔图及相应的相图、两个极限环、稳定同宿环与不稳定极限环的共存、以及多重一包围不稳定多重焦点的稳定极限环，不仅支持了理论分析，而且表明了余维3的Bogdanov-Takens分岔（尖点情况）的存在。这些结果揭示了更丰富的内容与没有收获或仅收获恒定产量猎物的模型相比，动力学更加复杂。
In this paper we study the effect of constant-yield predator harvesting on the dynamics of a Leslie-Gower type predator-prey model. It is shown that the model has a Bogdanov-Takens singularity (cusp case) of codimension 3 or a weak focus of multiplicity two for some parameter values, respectively. Saddle-node bifurcation, repelling and attracting Bogdanov-Takens bifurcations, supercritical and subcritical Hopf bifurcations, and degenerate Hopf bifurcation are shown as the values of parameters vary. Hence, there are different parameter values for which the model has a homoclinic loop or two limit cycles. It is also proven that there exists a critical harvesting value such that the predator specie goes extinct for all admissible initial densities of both species when the harvest rate is greater than the critical value. These results indicate that the dynamical behavior of the model is very sensitive to the constant-yield predator harvesting and the initial densities of both species and it requires careful management in the applied conservation and renewable resource contexts. Numerical simulations, including the repelling and attracting Bogdanov-Takens bifurcation diagrams and corresponding phase portraits, two limit cycles, the coexistence of a stable homoclinic loop and an unstable limit cycle, and a stable limit cycle enclosing an unstable multiple focus with multiplicity one, are presented which not only support the theoretical analysis but also indicate the existence of Bogdanov-Takens bifurcation (cusp case) of codimension 3. These results reveal far richer and much more complex dynamics compared to the model without harvesting or with only constant-yield prey harvesting.