【摘要】：生物數(shù)學是數(shù)學與生命科學的交叉學科,是研究生命體和生命系統(tǒng)的數(shù)量性質(zhì)與空間格局的科學.種群動力學是生物數(shù)學的重要分支之一.在經(jīng)典的種群動力學研究中:系統(tǒng)狀態(tài)依時間連續(xù).但由于很多種群生態(tài)現(xiàn)象并非是一個連續(xù)過程:其發(fā)展常受短時間擾動的影響.對這類現(xiàn)象,傳統(tǒng)連續(xù)系統(tǒng)已不再適用,需要利用更復雜的脈沖微分方程加以刻畫.脈沖微分方程描述某些運動狀態(tài)在固定或不固定時刻的快速變化或跳躍,對瞬間作用因素給出了一個自然的描述,它兼具離散系統(tǒng)與連續(xù)系統(tǒng)的某些特征,又超出兩者的范疇,給研究工作帶來了不小的難度.近年來,雖然脈沖微分系統(tǒng)在種群動力學研究中取得了大量成果,但亟待解決的問題還有許多.本文主要研究幾類具有脈沖效應的種群模型的動力學性質(zhì),特別是脈沖效應對系統(tǒng)周期解的影響.全文共分為四章.第一章(緒論),簡要概述脈沖微分方程在生物動力學上的研究背景及意義,并介紹論文所涉及的脈沖微分方程的基本概念.第二章,建立了具有固定時刻脈沖效應的Holling Ⅱ型功能性反應的捕食與被捕食系統(tǒng),使新系統(tǒng)能適用于含定期人工放養(yǎng)、收獲或定理噴灑農(nóng)藥等連續(xù)模型不能處理的情形;利用Mawhin重合度理論證明了該系統(tǒng)周期解的存在性,并通過計算機數(shù)值模擬加以驗證.第三章,建立了具有脈沖和強Allee效應的非自治Holling Ⅱ型捕食與食餌模型, 并利用與第二章類似的方法,得到系統(tǒng)周期解存在的充分條件,從理論和數(shù)值模擬兩方面證明了該系統(tǒng)在具有定期收獲(投放)的情況下,可以達到某種生態(tài)平衡.第四章,將第二章所研究的模型中的固定時刻脈沖更改為狀態(tài)反饋脈沖,使系統(tǒng)更符合某些實際情況.利用半連續(xù)動力系統(tǒng)幾何理論,研究了該脈沖狀態(tài)反饋系統(tǒng)周期解的存在性、唯一性和穩(wěn)定性.最后我們對全文進行了總結(jié),并對后續(xù)研究進行展望.
[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 For this kind of phenomenon, the traditional continuous system is no longer applicable and needs to be characterized by more complex impulsive differential equations.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，

本文編號：2297476