A new microbial insecticide mathematical model with density dependent for pest is proposed in this paper. First, the system without impulsive state feedback control is considered. The existence and stability of equilibria are investigated and the properties of equilibria under different conditions are verified by using numerical simulation. Since the system without pulse has two positive equilibria under some additional assumptions, the system is not globally asymptotically stable. Based on the stability analysis of equilibria, limit cycle, outer boundary line and Sotomayor’s theorem, the existence of saddle-node bifurcation and global dynamics of the system are obtained. Second, we consider homoclinic bifurcation of the system with impulsive state feedback control. The existence of order-1 homoclinic orbit of the system is studied. When the impulsive function is slightly disturbed, the homoclinic orbit breaks and bifurcates order-1 periodic solution. The existence and stability of order-1 periodic solution are obtained by means of theory of semi-continuous dynamic system.

中文翻译：

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脉冲控制系统的微生物杀虫模型与同宿分岔

本文提出了一种新的害虫密度依赖的微生物杀虫剂数学模型。首先，考虑没有脉冲状态反馈控制的系统。研究了平衡点的存在性和稳定性，并通过数值模拟验证了不同条件下平衡点的性质。由于没有脉冲的系统在一些附加假设下具有两个正平衡，因此该系统不是全局渐近稳定的。基于平衡、极限环、外边界线的稳定性分析和Sotomayor定理，得到系统的鞍节点分岔的存在性和全局动力学。其次，我们考虑了具有脉冲状态反馈控制的系统同宿分岔。研究了系统一阶同宿轨道的存在性。当脉冲函数受到轻微扰动时，同宿轨道就会断裂并分叉出一阶周期解。利用半连续动力系统的理论，得到一阶周期解的存在性和稳定性。