几类带脉冲效应的种群动力学模型研究
发布时间：2018-10-29 10:47
【摘要】：生物数学是数学与生命科学的交叉学科,是研究生命体和生命系统的数量性质与空间格局的科学.种群动力学是生物数学的重要分支之一.在经典的种群动力学研究中:系统状态依时间连续.但由于很多种群生态现象并非是一个连续过程:其发展常受短时间扰动的影响.对这类现象,传统连续系统已不再适用,需要利用更复杂的脉冲微分方程加以刻画.脉冲微分方程描述某些运动状态在固定或不固定时刻的快速变化或跳跃,对瞬间作用因素给出了一个自然的描述,它兼具离散系统与连续系统的某些特征,又超出两者的范畴,给研究工作带来了不小的难度.近年来,虽然脉冲微分系统在种群动力学研究中取得了大量成果,但亟待解决的问题还有许多.本文主要研究几类具有脉冲效应的种群模型的动力学性质,特别是脉冲效应对系统周期解的影响.全文共分为四章.第一章(绪论),简要概述脉冲微分方程在生物动力学上的研究背景及意义,并介绍论文所涉及的脉冲微分方程的基本概念.第二章,建立了具有固定时刻脉冲效应的Holling Ⅱ型功能性反应的捕食与被捕食系统,使新系统能适用于含定期人工放养、收获或定理喷洒农药等连续模型不能处理的情形;利用Mawhin重合度理论证明了该系统周期解的存在性,并通过计算机数值模拟加以验证.第三章,建立了具有脉冲和强Allee效应的非自治Holling Ⅱ型捕食与食饵模型, 并利用与第二章类似的方法,得到系统周期解存在的充分条件,从理论和数值模拟两方面证明了该系统在具有定期收获(投放)的情况下,可以达到某种生态平衡.第四章,将第二章所研究的模型中的固定时刻脉冲更改为状态反馈脉冲,使系统更符合某些实际情况.利用半连续动力系统几何理论,研究了该脉冲状态反馈系统周期解的存在性、唯一性和稳定性.最后我们对全文进行了总结,并对后续研究进行展望.
[Abstract]:Biological mathematics is an interdiscipline between mathematics and life science, and it is also a science to study the quantitative properties and spatial pattern of life body and life system. Population dynamics is one of the important branches of biological mathematics. In the classical study of population dynamics, the state of the system is time-dependent. However, many population ecological phenomena are not a continuous process: their development is often affected by short time disturbances. For this kind of phenomenon, the traditional continuous system is no longer applicable and needs to be characterized by more complex impulsive differential equations. Impulsive differential equations describe the rapid changes or jumps of some moving states at fixed or unfixed times, and give a natural description of the instantaneous action factors. It has some characteristics of both discrete and continuous systems, and goes beyond the scope of both. It brings great difficulty to the research work. In recent years, although a great deal of achievements have been made in the study of population dynamics for impulsive differential systems, there are still many problems to be solved. In this paper, the dynamical properties of several population models with impulsive effects are studied, especially the effects of impulsive effects on the periodic solutions of the systems. The full text is divided into four chapters. In the first chapter (introduction), the research background and significance of impulsive differential equations in biodynamics are briefly summarized, and the basic concepts of impulsive differential equations are introduced. In the second chapter, the prey-prey and prey system of Holling 鈪，
本文编号：2297476
[Abstract]:Biological mathematics is an interdiscipline between mathematics and life science, and it is also a science to study the quantitative properties and spatial pattern of life body and life system. Population dynamics is one of the important branches of biological mathematics. In the classical study of population dynamics, the state of the system is time-dependent. However, many population ecological phenomena are not a continuous process: their development is often affected by short time disturbances. For this kind of phenomenon, the traditional continuous system is no longer applicable and needs to be characterized by more complex impulsive differential equations. Impulsive differential equations describe the rapid changes or jumps of some moving states at fixed or unfixed times, and give a natural description of the instantaneous action factors. It has some characteristics of both discrete and continuous systems, and goes beyond the scope of both. It brings great difficulty to the research work. In recent years, although a great deal of achievements have been made in the study of population dynamics for impulsive differential systems, there are still many problems to be solved. In this paper, the dynamical properties of several population models with impulsive effects are studied, especially the effects of impulsive effects on the periodic solutions of the systems. The full text is divided into four chapters. In the first chapter (introduction), the research background and significance of impulsive differential equations in biodynamics are briefly summarized, and the basic concepts of impulsive differential equations are introduced. In the second chapter, the prey-prey and prey system of Holling 鈪，
本文编号：2297476
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