两种群竞争系统的大范围性质研究

发布时间：2018-05-03 23:15

本文选题：两种群竞争 + 正平衡点　；参考：《华中师范大学》2017年硕士论文

【摘要】：现在数学与生态学的交叉学科发展越来越来成熟,通过建立数学模型来研究生态问题、揭示生态现象的规律已经成为一种重要的方法,并被众多学者所采用.从马尔萨斯研究人口理论,用单种群人口数量为模型开始,一直到Lotka-Volterra模型,前人这些基本模型都为后人的研究奠定了基础,并形成了一种模式:建立数学模型,运用数学的方法进行计算和数值模拟,最后根据计算的结果来分析生物意义.竞争模型是学者大量研究的模型,本文主要探讨了四类不同的两种群竞争模型的平衡点以及全局稳定性,并剖析了它们的生物学意义:第二章是具有常数输入率的生物竞争模型,经过计算知此模型存在唯一的正平衡点,通过构造环域证明了正平衡点的全局渐近稳定性;第三章是一种群受到密度制约,一种群具有常数输入率的生物竞争模型,经过计算知此模型存在唯一的边界平衡点,计算和讨论了可能存在的正平衡点,当满足不同的条件时,此模型可以存在两个正平衡点、也可以存在一个正平衡点、甚至不存在正平衡点,然后通过构造环域证明了平衡点的全局渐近稳定性;第四章是两种群都受到密度制约的生物竞争模型,也即是Lotka-Volterra竞争模型;计算可知存在三个边界平衡点、存在一个正平衡点,通过构造环域证明了平衡点的全局渐近稳定性;第五章是一种群受到Allee效应,一种群受到密度制约的生物竞争模型,经过计算知此模型存在唯一的边界平衡点,计算和讨论了可能存在的正平衡点,当但满足不同的条件时,此模型可以存在两个正平衡点、也可以存在一个正平衡点、甚至不存在正平衡点,然后通过构造环域证明了平衡点的全局渐近稳定性.
[Abstract]:Nowadays, the interdisciplinary discipline of mathematics and ecology is becoming more and more mature. It has become an important method to establish mathematical models to study ecological problems and reveal the laws of ecological phenomena, which has been adopted by many scholars. From Malthus' study of population theory, using single population as the model and all the way to the Lotka-Volterra model, the previous basic models have laid the foundation for future generations to study, and formed a kind of model: to establish a mathematical model, The mathematical method is used to calculate and simulate, and the biological significance is analyzed according to the result of the calculation. Competition model is a model studied by many scholars. This paper mainly discusses the equilibrium point and global stability of four different two species competition models. In chapter 2, the biological competition model with constant input rate is presented. The unique positive equilibrium point is found by calculation. The global asymptotic stability of the positive equilibrium point is proved by constructing the annular domain. The third chapter is a biological competition model with constant input rate, which is a species restricted by density. It is known that there is a unique boundary equilibrium point in the model, and the possible positive equilibrium point is calculated and discussed when different conditions are satisfied. The model can have two positive equilibrium points or one positive equilibrium point or even no positive equilibrium point. Then the global asymptotic stability of the equilibrium point is proved by constructing the ring domain. The fourth chapter is the biological competition model, that is, Lotka-Volterra competition model, in which both species are restricted by density, and it is found that there are three boundary equilibrium points and a positive equilibrium point, and the global asymptotic stability of the equilibrium point is proved by constructing the annular domain. Chapter 5 is a biological competition model in which a population is subjected to the Allee effect and a population is restricted by density. It is known that there is a unique boundary equilibrium point in the model, and the possible positive equilibrium point is calculated and discussed when different conditions are satisfied. The model can have two positive equilibrium points or one positive equilibrium point or even no positive equilibrium point. Then the global asymptotic stability of the equilibrium point is proved by constructing a ring domain.
【学位授予单位】：华中师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175

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