两类生物种群反应扩散模型的动力学研究

发布时间：2019-05-15 12:38

【摘要】：扩散、空间环境对物种总量有着显著的影响,甚至关乎物种的存亡.本文第一个模型介绍的是经典的Lotka-Volterra竞争扩散系统.主要研究了扩散、空间环境以及竞争力对所描述模型的综合影响.事实上,物种在不同的环境下,竞争力也会发生变化,所以我们将空间相关性并入竞争力中.即:将以往模型中常数竞争力推广至非常数竞争力.我们得到,在弱竞争条件下,两物种的生存环境至少有一个是同质空间时,只要其中一物种竞争力的下确界大于某临界值,该物种总能完全战胜另一物种;当两物种空间环境均为异质时,若其中一物种完全获胜它的竞争者,不仅需要该物种竞争力的下确界大于某临界值,还需其竞争者的竞争力小于某临界值.人群的流动对传染病的消亡和蔓延有着重要影响,另外,及时的治疗有助于阻止传染病的传播,但每个地区对一种传染病的治疗总是有限的,所以我们考虑扩散以及饱和治疗对传染病传播的综合影响,基于此,本文介绍的第二个模型是在异质空间下一类带有饱和治疗项的SIS反应扩散方程.通过分析,我们得到无病平衡解的存在唯一性,继而我们通过主特征值得到传染病流行的阈值,称为基本再生数,用于讨论平衡解的稳定性.当基本再生数小于1时,无病平衡解局部稳定.当基本再生数大于1时,无病平衡解不稳定且存在地方病平衡解.
[Abstract]:Diffusion, space environment has a significant impact on the total number of species, and even related to the survival of species. The first model of this paper introduces the classical Lotka-Volterra competitive diffusion system. The comprehensive effects of diffusion, space environment and competitiveness on the described model are studied. In fact, the competitiveness of species also changes in different environments, so we incorporate spatial correlation into competitiveness. That is, the constant competitiveness in the previous model is extended to extraordinary competitiveness. We get that when at least one of the living environments of the two species is homogeneous under the condition of weak competition, as long as the lower bound of the competitiveness of one species is greater than a certain critical value, the species can always completely defeat the other species. When the spatial environment of the two species is heterogeneous, if one of the species wins its competitors completely, it is necessary not only that the lower bound of the competitiveness of the species is greater than a certain critical value, but also that the competitiveness of its competitors is less than a certain critical value. The flow of people has an important impact on the extinction and spread of infectious diseases. In addition, timely treatment helps to prevent the spread of infectious diseases, but the treatment of an infectious disease is always limited in each region. Therefore, we consider the comprehensive effects of diffusion and saturation therapy on the spread of infectious diseases. Based on this, the second model introduced in this paper is a kind of SIS reaction diffusion equation with saturation treatment term in heterogeneous space. Through the analysis, we obtain the existence and uniqueness of the disease-free equilibrium solution, and then we are worth reaching the threshold of infectious disease epidemic through the main characteristics, which is called the basic regeneration number, which is used to discuss the stability of the equilibrium solution. When the basic regeneration number is less than 1, the disease-free equilibrium solution is locally stable. When the basic regeneration number is greater than 1, the disease-free equilibrium solution is unstable and there is a local equilibrium solution.
【学位授予单位】：上海师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175

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