【作者】 张晓颖；

【导师】 王克；

【作者基本信息】 东北师范大学 ， 应用数学， 2005， 博士

【摘要】 本文主要研究了一阶、二阶脉冲微分方程的周期边值问题的周期解的存在性问题,以及脉冲微分方程应用于具体的生物模型,对生物资源脉冲捕获的最优开发问题,综合了作者在攻读博士学位期间的完成的系统的论文成果。 全文共分成五章: 第一章作为准备知识给出了本文要用到的相关内容,其中包括非线性泛函分析理论和脉冲微分方程理论。 第二章给出了时滞Lotka-Volterra型和具有有限时滞的一阶脉冲微分方程周期边值问题存在周期解的充分条件,并且具体讨论了生态学中所提出的各类时滞脉冲微分方程模型,包括:logistic模型,红细胞再生模型、绿豆蝇模型和多个偏差变元的周期Logistic方程等,得到了一些新成果,推广并改进了已有的相关的成果。 第三章是关于有奇异的二阶脉冲微分方程周期边值问题的周期解的存在性问题,众所周知,二阶脉冲微分方程具有非常重要的物理意义,本章所使用的方法是Leray-Schauder抉择及锥不动点定理。 第四章利用重合度理论,研究了具体的非自治脉冲微分方程互惠模型和捕食者食模型的脉冲周期解的存在性问题。 第五章结合实际的可操作的原则,考虑对资源的开发是间隔性的,是脉冲式进行的,用脉冲开发的假设去研究资源的可持续发展问题,获得脉冲周期解的全局稳定性和最优的捕获策略,推广了Canada学者Clark C W的关于可更新生物资源的最优开发的经典结果。 在论文的最后,总结了论文的创新点提出了论文的改进方向以及研究中所参考的主要文献。更多还原

【Abstract】 This paper discuss mainly the existence of positive solutions to periodic boundary value problems for first or second order impulsive differetial equations. In addition, impulsive differential equations applies to biological system, meanwhile, the optimal harvest policy with impulsive harvest is considered.This paper include most of research work when the author persue her Ph.D. degree.The whole contents is divided into five chapters.Chapter 1, as the beginning of this paper, offers some relative knowledge, such as preliminary qualitative theory of differential equation, basis theory of nonlinear functional analysis, theory of impulsive differential equation.Chapter 2, we obtained the existence positive periodic solutions of periodic boundary value problem for first order Lotka-Volterra integral differential equation with impulse effects and first order impulsive differential equation with delay. We mainly concerns the existence of the nonlinear continuous systems that have previously appeared in the population dyanmics literature such as single species logistic growth models, the model of blood cell production, and so on. New easily verifiable sufficient conditions are derived which improve and generalize related results.Chapter 3, Focuses on the existence positive periodic solution of periodic boundary problem for second order impulsive differential equation. As we all know the second order impulsive differential equation has much important physics significance to study. The method used here is alternative Leray-Schauder type and on the krasnosel-skii fixed point theorem, and studied the periodic solution of repulsive singular differential equation with impulse effects.Chapter 4, using the coincidence degree and the priori estimations, discuss the nonautonomous impulsive differential equations, including a predator-prey system and a cooperative systems.Chapter 5, combined the feasible principle, specially consider supposition of impulsive harvest and study sustainable development of biologic resources under the impulsive harvest, derive the existence and global asymptotically stability of impulsive periodic solution, furthermore, the optimal harvest policy is obtained. The classical re-suits of Clark’s for the management of renewable biological resources are generalized. At the end of the paper, it is proposed the remark of this paper and the further study direction, many related references are listed.更多还原