Recently, pest control has become a very interesting research topic because it is closely associated with agricultural and economic loss. Empirical evidence shows that pest insects are responsible for lower crop production and many other adverse effects on the farming sector. There are several biological, physical and chemical control mechanisms. However, the biological control of pest populations by using natural enemies is one of the most important ecosystem services adopted in agriculture around the world. In the present study, we consider an ecological model consisting of prey (pest) and its natural enemy as the predator. Different system equilibria are obtained, their stability is analyzed, and Hopf bifurcation of the system around the interior equilibrium is discussed. The sufficient permanence criteria of the system are also derived. Moreover, we perform bifurcation analysis to explore the existence of limit cycle. We also investigate the stability property of the positive periodic solution when the interior equilibrium loses its stability. Our analytical results are further verified through numerical simulations. Our findings suggest that, in the absence of a super predator, pest and natural enemy show stable coexistence. However, in the presence of super predator, if the natural enemy is killed at a lower rate, both pest and natural enemy coexist. Finally, above a threshold value, the natural enemy is eradicated from the system and pest outbreak occurs.

中文翻译：

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具有害虫管理视角的捕食者-猎物模型研究

最近，害虫防治已成为一个非常有趣的研究课题，因为它与农业和经济损失密切相关。经验证据表明，害虫是造成作物减产和对农业部门的许多其他不利影响的原因。有几种生物、物理和化学控制机制。然而，利用天敌对害虫种群进行生物控制是全球农业采用的最重要的生态系统服务之一。在本研究中，我们考虑一个由猎物（害虫）及其天敌作为捕食者组成的生态模型。得到了不同的系统平衡，分析了它们的稳定性，讨论了系统在内部平衡附近的Hopf分岔。还导出了系统的充分持久性标准。此外，我们进行分岔分析以探索极限环的存在。我们还研究了当内部平衡失去稳定性时正周期解的稳定性。我们的分析结果通过数值模拟得到进一步验证。我们的研究结果表明，在没有超级捕食者的情况下，害虫和天敌会稳定共存。但是，在超级捕食者存在的情况下，如果以较低的速度杀死天敌，则害虫和天敌并存。最后，超过阈值，天敌从系统中被消灭，害虫爆发。我们的分析结果通过数值模拟得到进一步验证。我们的研究结果表明，在没有超级捕食者的情况下，害虫和天敌会稳定共存。但是，在超级捕食者存在的情况下，如果以较低的速度杀死天敌，则害虫和天敌并存。最后，超过阈值，天敌从系统中被消灭，害虫爆发。我们的分析结果通过数值模拟得到进一步验证。我们的研究结果表明，在没有超级捕食者的情况下，害虫和天敌会稳定共存。但是，在超级捕食者存在的情况下，如果以较低的速度杀死天敌，则害虫和天敌并存。最后，超过阈值，天敌从系统中被消灭，害虫爆发。