捕食与食饵问题的动力学分析

发布时间：2017-12-30 23:38

本文关键词：捕食与食饵问题的动力学分析　出处：《天津工业大学》2017年硕士论文　论文类型：学位论文

【摘要】：种群生态学是描述生物种群和环境之间的相互作用关系的一门学科.许多生物学家和数学家将这种复杂的相互作用关系建立成数学模型表示,以便用来描述以及预测生物种群的发展过程,进而通过人为的作用进一步调节和控制种群的生存发展,以便达到使得种群持久稳定的状态。本文主要对几类非线性种群系统的动力学行为进行了深入的分析与研究.主要考虑了 Allee效应、捕获、随机噪声等因素对生物系统的稳定性所产生的影响,主要通过构造Lyapunov函数及利用随机过程理论等方法研究了种群系统的动力学行为.本文的主要内容如下:1.研究了一类具有Allee效应的两种群捕食模型,并对该系统的捕食者与食饵施加捕获,通过对模型进行定性分析,证明了正平衡点的存在性和稳定性,进一步通过数值模拟加以验证.结果表明对系统应合理进行捕获,这样才能使种群持久稳定.2.研究了一类具有HollingⅡ功能反应的两种捕食者与一种食饵之间关系的捕食模型,通过分析特征方程,Routh-Hurwitz准则及计算Lyapunov指数,分析了确定性系统平衡点的稳定性,进一步借助数值模拟分析了系统的稳定性.3.建立并研究了一类具有HollingⅡ功能反应函数互惠随机模型.得出,对于任意给定的初值该模型有全局唯一正解以及此解具有随机有界性.另外,经过定性分析,给出了系统唯一正解的随机持久性和全局吸引性的存在条件.同时发现当环境噪声较小时,随机模型与确定性模型的种群衍化情形类似,否则种群将最终灭亡.由此可知考虑环境的随机性是非常必要的.文中每一部分都通过数值模拟证明了结论的正确性.
[Abstract]:Population ecology is a discipline to describe the relationship between population and environment. The interaction between many biologists and mathematicians will establish a mathematical model of this complex that can be used to describe and predict the development of biological population, and then through the people for the role to further regulate and control the survival and development of the population, in order to achieve the permanence stable state. The main dynamical behavior of several nonlinear population system in-depth analysis and research. The main consideration of the Allee effect, capture effect of random noise impact on the stability of biological system, mainly through the construction of Lyapunov function and using the stochastic process theory and method to study the dynamic behavior population system. The main contents of this paper are as follows: 1. of the two species predator-prey model with Allee type effect And, the predator and prey capture is applied in the system, through the qualitative analysis of the model, we prove the existence and stability of positive equilibrium, further verified through numerical simulation. The results show that the system should be reasonable to capture, so as to make the population lasting stable predator-prey model is studied for a class of.2. with Holling II the functional response of two species of predators and one prey relationship, through the analysis of the characteristic equation, Routh-Hurwitz criterion and Lyapunov index calculation, the stability of the equilibrium point of the system uncertainty analysis, further by means of numerical simulation analysis of the stability of.3. system is established and studied a class of Holling functional response function. The reciprocal stochastic model with. For any given initial value of the model has a unique positive solution and the global solution with stochastic boundedness. In addition, through qualitative analysis, gives the system is only The existence conditions of attractive solutions for stochastic permanence and global environment. At the same time that when the noise is small, stochastic models and deterministic models of population derived from a similar situation, otherwise the population will eventually perish. Thus considering random environment is very necessary. In this paper, each part through numerical simulation validates the conclusion.

【学位授予单位】：天津工业大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175

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