几类具有自扩散与交叉扩散项的生态模型的空间斑图动力学研究

发布时间：2018-05-12 23:07

本文选题：捕食-食饵模型 + 反应扩散　；参考：《安徽师范大学》2017年硕士论文

【摘要】：生物斑图动力学作为非线性科学的主要分支之一,它的研究非常广泛和丰富.产生生物斑图的机理有很多,最简单的一种是反应扩散系统,最先由Turing在1952年发表的《形态形成的化学基础》中提出,因此通常被称为Turing失稳或由扩散引起的失稳.简单的说,就是由于扩散使得稳定的平衡点变得不稳定.通过构造具有种群动力学特征的数学模型,进行动力学形态分析,可以用来解释种群之间相互作用而形成的空间斑图,同时结合数值模拟的结果,说明种群向时空混沌的转变可以解释种群在空间中的持续、灭绝、进化等问题.本文将利用线性化分析理论、Lyapunov函数方法、Routh-Hurwitz准则以及多重尺度分析方法,研究三类带反应扩散项的捕食-食饵模型,以下是论文的主要研究内容:1.研究了一类具比率依赖功能反应的捕食-食饵模型的Turing斑图生成与选择问题.通过线性化分析,得到Turing空间,利用多重尺度分析方法推导系统的振幅方程并进行斑图选择,在Turing空间中选择合适的参数,得到包括点状、条状以及二者共存的Turing斑图.2.基于自然界中猛兽群体的追捕现象,研究了一类带有负交叉扩散项的一般二维模型,并对一类具比率依赖的捕食-食饵模型进行理论和数值研究,所得结果表明:负交叉扩散项(-d21)影响Turing斑图生成及选择.在其它参数固定情况下d21必须小于某个临界值时,系统才会出现Turing不稳定现象.3.研究一类三种群食物链模型的强耦合交叉扩散系统.首先通过构造Lyapunov函数证明唯一的正平衡点在ODE系统下是全局渐近稳定的,当交叉扩散系数均为零时,唯一的正平衡点仍是全局渐近稳定的,但是,当引入交叉扩散时,正平衡点则变得不稳定.利用Routh-Hurwitz准则和Descartes符号法则证明了大的交叉扩散系数(k21或k32足够大时)可以导致平衡点由原来的稳定变得不稳定.最后利用数学软件Matlab对我们的结果进行数值模拟,得到了不同类型的Turing斑图,包括六边形、条状以及二者共存的斑图.
[Abstract]:As one of the main branches of nonlinear science, biographic dynamics has been widely studied. There are many mechanisms for producing biological patterns, the simplest of which is the reaction-diffusion system, which was first proposed by Turing in the Chemical basis of Morphology published in 1952. Therefore, it is usually referred to as the instability of Turing or the instability caused by diffusion. Simply put, it is because of diffusion that the stable equilibrium becomes unstable. By constructing a mathematical model with the characteristics of population dynamics, the dynamic morphological analysis can be used to explain the spatial pattern formed by the interaction between populations, and at the same time, the results of the numerical simulation can be combined with the results of the numerical simulation. The transformation of population to spatiotemporal chaos can explain the persistence, extinction and evolution of population in space. In this paper, three kinds of predator-prey models with reaction diffusion term are studied by using the Lyapunov function method, Routh-Hurwitz criterion and multi-scale analysis method. The following is the main content of this paper: 1. The problem of Turing pattern generation and selection for a predator-prey model with ratio dependent functional response is studied. Through the linearization analysis, the Turing space is obtained. The amplitude equation of the system is deduced and the pattern selection is carried out by using the multi-scale analysis method. The suitable parameters are selected in the Turing space, and the Turing pattern. 2, which includes the dot shape, the bar shape and the coexistence of the two, is obtained. Based on the hunting phenomenon of predator population in nature, a general two-dimensional model with negative cross-diffusion term is studied, and a kind of predator-prey model with ratio dependence is studied theoretically and numerically. The results show that the negative cross-diffusion term (-d21) affects the generation and selection of Turing patterns. In the case of other fixed parameters D21 must be less than a certain critical value before the system will appear Turing instability. 3. A strong coupling cross diffusion system for a class of three species food chain model is studied. It is proved that the unique positive equilibrium is globally asymptotically stable under the ODE system by constructing the Lyapunov function. When the cross-diffusion coefficients are 00:00, the unique positive equilibrium is globally asymptotically stable. However, when the cross-diffusion is introduced, the unique positive equilibrium is globally asymptotically stable. The positive equilibrium becomes unstable. Using the Routh-Hurwitz criterion and the Descartes sign rule, it is proved that when the cross diffusion coefficient is large enough, the equilibrium point can change from the original stability to the instability. Finally, the numerical simulation of our results is carried out by using the mathematical software Matlab, and different types of Turing patterns are obtained, including hexagonal, bar and coexistence patterns.
【学位授予单位】：安徽师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175

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