捕食系统中疾病传播的数学模型及动力学分析

发布时间：2018-02-25 05:01

本文关键词： 捕食系统传染病模型随机模型庇护效应捕获效应时滞效应传染率功能反应稳定性　出处：《兰州大学》2008年硕士论文　论文类型：学位论文

【摘要】： 近年来，以Lotka和Volterra为代表的种群动力学和以Kermack及McKendrick为代表的流行病动力学，已经有了相当的发展，它们分别在开发利用资源和预防治疗疾病方面都起到了不同程度的指导作用。由于流行病必然在物种之间传播，所以为了更符合实际情况，应该把种群动力学和流行病动力学结合起来考虑，但是这方面的工作至今还寥寥无几。基于此，本文中我们研究了疾病在捕食系统中的传播情况，建立模型，通过数学分析和数值模拟，主要得到以下结论： 1．当食饵具有传染病时，我们建立了食饵有病的生态-流行病模型，讨论了解的有界性，应用特征根法得到了平衡点局部渐近稳定的充分条件。进一步，，分析了平衡点的全局渐近稳定性，得到了边界平衡点和正平衡点全局稳定性的充分条件，并且我们还找到了“基本再生数”R_0，为疾病的控制提供了理论基础。 2．当捕食者有病时，我们建立了捕食者有病的生态-流行病模型，讨论了解的有界性以及平衡点的稳定性。并且在环境扰动为白噪声的情况下，我们建立了随机微分方程模型，并找到了一个Lyapunov函数，结果表明在正平衡点附近的一个随机扰动并不改变此平衡点的稳定性。 3．对于这样一个简单的生态一流行病模型蕴含着复杂的动力学行为，甚至出现混沌现象。 4．对于上述系统中出现的混沌现象，我们提出了三种混沌控制方法：加入庇护效应；捕获效应；时滞效应。通过建立数学模型，并对其进行数学分析和数值模拟，我们发现这三种效应都能阻止种群震荡的发生，增强系统的稳定性。最后，我们详细的讨论了传染率，功能反应和空间因素对系统动力学行为的影响，结合前人的一些工作，提出了今后努力的方向。
[Abstract]:In recent years, population dynamics, represented by Lotka and Volterra, and epidemiological dynamics, represented by Kermack and McKendrick, have developed considerably. They have played a different role in the development and utilization of resources and in the prevention and treatment of diseases. Since epidemics must spread between species, in order to be more realistic, The combination of population dynamics and epidemic dynamics should be considered, but little has been done so far. Based on this, we have studied the spread of disease in predation system and established a model. Through mathematical analysis and numerical simulation, the main conclusions are as follows:. 1. When the prey has infectious disease, we establish the eco-epidemic model of the prey disease, discuss the boundedness of the solution, and obtain the sufficient condition of the local asymptotic stability of the equilibrium point by using the characteristic root method. In this paper, the global asymptotic stability of the equilibrium point is analyzed, and the sufficient conditions for the global stability of the boundary equilibrium point and the positive equilibrium point are obtained. Furthermore, we find the "basic reproducing number" R0, which provides a theoretical basis for disease control. 2. When the predator is ill, we establish the ecological epidemic model of predator disease, discuss the boundedness of solution and the stability of equilibrium point, and establish the stochastic differential equation model when the environmental disturbance is white noise. A Lyapunov function is found. The results show that a random perturbation near the positive equilibrium does not change the stability of the equilibrium. 3. For such a simple ecological epidemic model, there are complex dynamic behaviors and even chaotic phenomena. 4. For the chaotic phenomena in the system mentioned above, we propose three chaos control methods: adding sheltering effect, capturing effect, time-delay effect, mathematical model, mathematical analysis and numerical simulation. We find that these three effects can prevent the occurrence of population oscillation and enhance the stability of the system. Finally, we discuss in detail the effects of infection rate, functional response and spatial factors on the dynamic behavior of the system. Combined with some previous work, we put forward the direction of future efforts.
【学位授予单位】：兰州大学
【学位级别】：硕士
【学位授予年份】：2008
【分类号】：R311

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