脉冲控制理论在害虫综合治理中的应用研究

【作者】 白翠翠；

【导师】 李有文；

【作者基本信息】 中北大学 ， 应用数学， 2009， 硕士

【摘要】 害虫控制在农业生产中具有重要作用,而微分方程数学模型在描述害虫种群动力学行为中起到了非常重要的作用,特别是用脉冲微分方程来描述害虫种群动力学模型能够更合理、更精确地反映各种变化规律,因为现实世界中害虫的繁殖和危害、以及人类的控制行为几乎都是阶段性的。本文基于害虫的综合管理策略,利用脉冲微分方程的相关理论和方法建立并研究了三类具有脉冲效应(即分别在不同的固定时刻分别喷洒杀虫剂和释放染病害虫(或天敌))的生态-流行病模型。它们分别是一类具有脉冲效应和非线性发生率的生态-流行病模型,一类具有脉冲效应和Ivlev功能性反应的生态-流行病模型和一类具有脉冲效应的食饵依赖生态-流行病模型。我们分别证明了各系统的所有解是一致最终有界的。并利用Floquet乘子理论及脉冲比较定理,证明了当脉冲周期小于某个临界值时,系统存在一个全局渐近稳定的易感者害虫根除周期解,否则,系统是持续生存的。最后,为了证实我们的分析结果,借助计算机数值模拟讨论所提出模型的动力学行为,包括系统周期解的存在性、持续生存性、绝灭性和稳定性。所得到的结果,从数学的角度给出在农业中利用害虫综合管理策略控制害虫增长的一个理论依据。更多还原

【Abstract】 Pest control plays an important role in agriculture. Mathematical models of di?erentialequations play an important role in describing pest population dynamic behavior. Especially,impulsive di?erential equations describe pest population dynamic models, which is morereasonable and precise on re?ecting all kinds of change orderliness, since the reproductionand damager of pests, human control behavior are almost periodical in the natural world.In this paper, we investigate and study the dynamic behaviors of three classes ofeco-epidemiological models by means of the theory and method of impulsive di?erentialequations concerning integrated pest management strategy, i.e., periodic releasing infec-tive pests (or natural enemy) and spraying pesticide at di?erent fixed time. They arean eco-epidemiological model with impulsive e?ect and nonlinear incidence rate, an eco-epidemiological model with impulsive e?ect and Ivlev functional response and a class ofprey-dependent eco-epidemiological model with impulsive e?ect, respectively.We prove that all solutions of the investigated system are uniformly ultimately boundedrespectively. By applying the Floquet theorem and comparison theorem, we prove that thereexists a globally asymptotically stable susceptible pest-eradication periodic solution when theimpulsive period is less than some critical value. Otherwise, the system can be permanent.Finally, to substantiate our analytical results, numerical simulations are used to investigatedynamical behaviors including the existence of periodic solution, permanence, extinctionand stability. Mathematically, the theoretical evidence of the controlling pests using theintegrated pest management strategy in agriculture is given by the obtained results.更多还原