In this paper, we apply the Z-type control method to a crop-pest–natural enemy model. We consider the indirect Z-controller in the natural enemy population and investigate the mathematical properties of the model. Furthermore, our analytical results are also numerically validated. Our paper supports that the pest population can be controlled by using an indirect Z-control mechanism in the natural enemy population. Investigations on the crop-pest–natural enemy model also highlight how the Z-control method acts in respect to different dynamical regimes of the uncontrolled model.

中文翻译：

---

具有Z型控制的农作物-天敌模型的数学研究

在本文中，我们将Z型控制方法应用于农作物-天敌模型。我们考虑了天敌中的间接Z控制器，并研究了该模型的数学特性。此外，我们的分析结果也得到了数值验证。我们的论文支持可以通过使用天敌种群的间接Z控制机制来控制害虫种群。对农作物-害虫-天敌模型的研究还强调了Z控制方法是如何针对不受控制的模型的不同动力状态起作用的。