脉冲控制在生物数学中的应用

发布时间：2018-07-06 07:34

  本文选题：脉冲微分方程 + 持久性　； 参考：《吉林大学》2015年博士论文

【摘要】：脉冲微分方程的基本理论与方法在近几十年里已经得到了充分的发展和完善.由于一些自然现象会在特定的时刻发生“跳跃”,譬如动物季节性的繁殖或者迁移,所以它们不能用连续或离散的微分方程来建模,但是脉冲方程能够很好的来描述这些生物现象.首次使用脉冲方法是Beverton和Holt(1957)在一个半离散模型的基础上想要用离散模型去逼近一个连续时间的Logistic模型.从此以后,几乎在生命科学的每一个领域都出现了脉冲模型.本文主要研究了脉冲控制在生物数学中的两个应用.第一个应用是通过研究系统的持久性和稳定的非平凡周期解分支,我们在带有Beddington-DeAngelis功能关系的脉冲控制捕食模型中给出了食饵种群密度的完整的控制策略；第二个应用是研究了如何利用不育昆虫繁殖技术(SIT)在连续和脉冲两种类型的捕食模型中成功地杀灭自然昆虫.本论文共分为四章.第一章主要介绍了脉冲微分方程的一些基本概念和基础理论,包括脉冲微分方程的定义,常见分类以及稳定性等概念.我们特别给出了研究脉冲微分方程的两个重要定理：比较定理和线性周期方程的Floquet定理.之后我们详细地介绍了脉冲微分方程的应用.到目前为止,脉冲微分方程已经广泛地应用到传染病学,医药学以及人口模型等各个方面.第二章主要研究了一个带有Beddington-DeAngelis功能关系的脉冲控制捕食模型.在带有Beddington-DeAngelis功能关系的捕食模型中,B-D关系影响了捕食者的捕食能力,从而在营养关系的表达式上表现出来S.Nundlol-la, L.Mailleretb和F.Grognarda(2010)首先研究了带有一种特殊的脉冲控制的Beddington-DeAngelis功能关系的捕食模型,建立了周期性释放捕食者产生的平凡周期解,并得到了平凡周期解的全局稳定性条件.我们着重研究了该系统的持久性并寻找到了非平凡周期解.结果表明,在此类脉冲控制中,当脉冲控制率大于某个特定的临界值或者脉冲释放周期小于对应的临界值时,系统的对应于食饵灭绝的周期解是局部渐近稳定的.我们给出了系统的持久存在性条件,并且得到随着参数变化当食饵灭绝的周期解失去它的稳定性时系统出现了稳定的非平凡的周期解.最终我们根据平凡周期解和非平凡周期解的稳定性给出了食饵种群密度的完整控制策略.第三章我们研究了不育昆虫繁殖技术(SIT)在一般的捕食模型中如何成功的控制或杀灭自然昆虫.我们在连续型模型中讨论了不育昆虫技术在捕食模型中应用的可行性.这个模型是以Murray(2002)的模型为基础.在Murray的著作中,不育昆虫的数量总被保持成一个常数.我们将Murray的模型扩展成为了一个一般的捕食模型并进行了理论分析,讨论了模型的动力学行为并计算出了使自然昆虫灭绝的临界值.最终我们知道,带有不育昆虫繁殖技术(SIT)的捕食模型具有非常丰富的,有趣的以及复杂的动力学性质,其有可能出现多个稳定平衡点,鞍结点分支,Hopf分支等.它具有两个重要的临界值：固定的SIT临界值和捕食临界值,它们的大小比较决定了捕食者在不育昆虫技术中的作用.第四章我们研究了脉冲控制在不育昆虫繁殖技术(SIT)中的应用,为了计算不育昆虫释放的数量,我们引入了这个模型,一个周期或脉冲释放的模型,这个模型是由一个带有脉冲部分的常微分方程组组成.跟连续型系统相比,脉冲的SIT在实际生产中更容易被操作.我们计算出了自然昆虫灭绝的临界值并得到了脉冲SIT模型平凡周期解的全局稳定性条件.
[Abstract]:The basic theories and methods of impulsive differential equations have been fully developed and perfected in recent decades. Because some natural phenomena will "jump" at a specific time, such as the seasonal reproduction or migration of animals, they can not be modeled by continuous or discrete micro equations, but the pulse equation can be very good. To describe these biological phenomena, the first use of pulse method is that Beverton and Holt (1957) want to use discrete models to approach a continuous time Logistic model on the basis of the 1.5 discrete model. From then on, the pulse model has appeared in almost every field of life science. This paper mainly studies the pulse control in the birth. Two applications in physical mathematics. The first application is to study the complete control strategy of the prey population density in the impulsive predator-prey model with the Beddington-DeAngelis function relationship by studying the persistence of the system and the stable non trivial periodic solution. The second application is to study how to make use of the sterile insect propagation. SIT has successfully killed natural insects in two types of predator-prey models in continuous and pulse types. This paper is divided into four chapters. The first chapter introduces some basic concepts and basic theories of impulsive differential equations, including the definition of impulsive differential equations, common classification and the concepts of stability. Two important theorems of differential equations: the comparison theorem and the Floquet theorem of linear periodic equations. After that, we introduce the application of impulsive differential equations in detail. Up to now, the impulsive differential equations have been widely applied to infectious diseases, medicine and population models. The second chapter mainly studies one with Beddi The impulse control model of the ngton-DeAngelis function relationship. In a predator model with Beddington-DeAngelis function, the B-D relationship affects the predator's predatory ability, thus showing S.Nundlol-la in the expression of the nutrition relation, L.Mailleretb and F.Grognarda (2010) first studies with a special pulse control. The predator-prey model of the Beddington-DeAngelis function relationship is made, the periodic solution generated by the periodic release predator is established, and the global stability condition of the ordinary periodic solution is obtained. We focus on the persistence of the system and find the nontrivial periodic solution. The periodic solution corresponding to the extinction of the prey is locally asymptotically stable when a particular critical value or the pulse release period is less than the corresponding critical value. We give the persistent existence condition of the system and obtain the stability of the system as the periodic solution of the extinction of the prey loses its stability. In the end, we give a complete control strategy for the population density of prey on the basis of the stability of the ordinary periodic solution and the nontrivial periodic solution. In the third chapter, we have studied how the sterile insect propagation technique (SIT) successfully controlled or killed the natural insects in the general predator model. This model is based on the model of Murray (2002). In the work of Murray, the number of sterile insects is always kept as a constant. We extend the model of Murray into a general predation model and carry out a theoretical analysis, and discuss the dynamic behavior of the model and discuss the dynamic behavior of the model and discuss the dynamic behavior of the model. We have calculated the critical value of the extinction of natural insects. Finally, we know that the predation model with the sterile insect propagation technique (SIT) is very rich, interesting and complex dynamics, and it may have many stable equilibrium points, the saddle node branch, the Hopf branch, etc. it has two important critical values: fixed SIT presence. In the fourth chapter, we studied the application of pulse control in the sterile insect propagation technology (SIT). In order to calculate the number of sterile insects released, we introduced the model type, a cycle or a pulse release model, which is the model. It is composed of a group of ordinary differential equations with a pulse part. Compared with the continuous system, the SIT of the pulse is easier to be operated in the actual production. We calculate the critical value of the extinction of natural insects and obtain the global stability conditions of the ordinary periodic solution of the pulse SIT model.
【学位授予单位】：吉林大学
【学位级别】：博士
【学位授予年份】：2015
【分类号】：O175
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本文编号：2101987