【作者】 韩莉莉；

【导师】 贺明峰；

【作者基本信息】 大连理工大学 ， 应用数学， 2013， 硕士

【摘要】 在生态学领域,生态学家主要研究物种之间的相互作用以及物种数目随时间的变化规律。近几百年来,对物种间的相互依存,相互制约的生存方式的研究得到很大的发展,许多数学家和生态学家对这一现象进行了大量的模拟实验和理论分析。在生态学研究中,最基本的数学模型是Lotka-Volterra方程。在近几十年的研究中,许多数学家和生物学家把捕食被捕食系统和博弈论相结合作为研究对象,通过加入一些限制条件,研究种群的变化规律。本文主要分为三章。第一章主要是对捕食被捕食模型的产生背景,发展,及其现状进行总结。第二章介绍了微分方程与捕食被捕食模型的联系,及其在捕食被捕食模型中的应用,用微分方程定性理论分析系统的局部稳定性和全局稳定性对我一般方法。第三章介绍了一种循环竞争系统下的捕食被捕食模型,并根据已有的层级模式,建立表示各物种种群规模变化规律的常微分方程。通过求解五个物种的循环竞争系统对应的速率方程的平衡点,对其局部稳定性进行分析讨论,并且在一定的条件下构造相应的李雅普诺夫函数,通过李雅普诺夫函数对系统的全局稳定性的讨论,得出系统在稳定状态时会出现共存现象,其中第一个和第五个物种是必定存在的,中间三个物种的存在需要一定的限制条件,并且这三个物种只能是某一个存在或者三个都存在。更多还原

【Abstract】 In the field of ecology, the interaction between species and number of species variation over time are the main research themes of ecologists. In recent centuries, the studies of the interdependence and mutual restraint way in which the species live have been greatly developed, a large number of simulation experiments and theoretical analysis of this phenomenon have been done by ecologists and mathematicians. In the ecological research, the most basic mathematical model is the Lotka-Volterra equations. In recent decades, many mathematicians and biologists discussed the connection between the predator prey system and the game theory, studied the variation of the population by adding some restrictions.This paper is organized as follows:In the first chapter, we give a summary of background, development and status quo for predator prey models.In the second chapter we mainly deal with the predator prey model by differential equation system. Especially, we analyze the local and global stability of this system by the qualitative theory.The third chapter introduces a competitive circulatory predator prey model. We establish the ordinary differential equations based on the level to describe species population size variation. By calculation, we obtain the steady states of the five species ecosystem. We prove that some of these points are locally stable and some are unstable. Under certain conditions, we discuss the global stability of the steady states by constructing the corresponding Lyapunov functions and draw the system coexistence phenomenon on a stable state. And the first and the fifth species must exist, but the three existing species needs certain conditions, and only one or all for the three species exist.更多还原