非线性脉冲微分方程模型在害虫治理中的应用研究

Study on the Application of Nonlinear Impulsive Differential Equation Models in Pest Management

摘要: 针对资源约束下的害虫综合治理需要，本文构建了一个基于Holling-II型功能性反应与Leslie-Gower捕食结构的非线性脉冲控制害虫–天敌动力学模型。首先，通过分析害虫灭绝子系统，结合频闪映射与不动点理论，证明了害虫灭绝周期解的存在性；随后，借助Floquet特征乘子与比较定理，系统分析该周期解的局部渐近稳定性与全局吸引性，并推导出害虫灭绝的阈值条件；在此基础上，进一步通过阈值分析与数值模拟，探究天敌投放周期、最大致死率及半饱和常数对阈值的影响，揭示资源有限性对种群动态的调控作用。结果表明，资源约束会使害虫种群趋于稳定正平衡态而非彻底灭绝，天敌种群可维持稳定水平并持续发挥生物控制功能，因此害虫治理目标应聚焦于将害虫密度控制在经济损害阈值以下。

Abstract: To address the requirements of integrated pest management under resource constraints, this paper develops a nonlinear impulsive control dynamic model for pest-natural enemy systems based on Holling-II type functional response and Leslie-Gower predation structure. Firstly, by analyzing the pest-eradication subsystem and combining the stroboscopic map with the fixed point theorem, the existence of the pest-eradication periodic solution is proved. Subsequently, using Floquet characteristic multipliers and the comparison theorem, the local asymptotic stability and global attractivity of the periodic solution are systematically investigated, and the threshold condition for pest eradication is derived. On this basis, further threshold analysis and numerical simulations are carried out to explore the effects of the natural enemy release period, the maximum mortality rate and the half-saturation constant on the threshold, revealing the regulatory effect of resource limitation on population dynamics. The results demonstrate that resource constraints drive the pest population to converge to a stable positive equilibrium instead of complete extinction, while the natural enemy population can maintain a stable level and exert a sustainable biological control effect. Therefore, the objective of pest management should focus on keeping the pest density below the economic injury level.