生物模型的稳定性态研究

发布时间：2019-03-04 17:20

【摘要】：生物数学是一门相对独立且比较完整的学科,对现代科技发展发挥了巨大的作用。它主要是生命科学、公共卫生、医学、生物学和农学等学科与数学相互渗透形成的交叉学科。在研究自然界中生态关系的复杂性时,人们往往通过建立数学模型的办法来研究,这是数学与生物学的交叉学科—生物数学。生物动力系统作为生物数学的一个重要分支,主要是运用动力学的相关知识,来研究已经建立起来的生物数学建模。所得数学结果可以用来解释生物界中已有的现象,也可预测生物界中未来可能发生的事情。这样,人们可以通过选择更为恰当的生活方式,使得人类与自然界能更和谐地生活。本文第一部分主要针对生物动力学以及种群动力学的研究背景以及研究现状做了研究。为了更方便于我们研究,接着介绍了动力系统的一些基本术语和基本定理。本文第二部分主要介绍了几种经典的种群动力学模型:Logistic模型、Lotka-Volterra模型和Leslie-Gower模型。最后主要介绍了食饵种群具有密度制约或非密度制约时三种不同种类的功能性反应函数的动力性态,并介绍了其平衡点处稳定性和极限环发生的条件。本文第三部分主要研究了一类具有常数存放率的Volterra模型的定性分析。这类具有常数存放率的的Volterra模型至少有2个平衡点,运用平面系统的定性理论以及规范型理论分析发现,在不同的参数下,它们可以是稳定结点、不稳定结点、鞍点、弱中心等。通过Hopf分支的规范型理论,利用计算第一Lyapunov系数,得到了系统在弱中心附近会发生超临界Hopf分叉,并从平衡点分支出唯一稳定极限环。通过对这类具有常数存放率的Volterra模型的动力学分析:当npna21+)1(0,)1(010npaa++时,内部平衡点A是系统的稳定结点。如果把n看作系统的稀疏率,则对于任意给定稀疏率n时,当食饵的存放率p足够大,使得npn2)1(+大于食饵的出生率与捕食者的死亡率之比时,此生物系统可以长期共存下去;当)133(1)(nnpnnanp212+++且)n(paak 110++时,系统在平衡点A附近会分支出唯一稳定极限环,这表明这个生物系统以稳定周期解的形式长期共存下去。
[Abstract]:Bio-mathematics is a relatively independent and relatively complete subject, which plays a great role in the development of modern science and technology. It is mainly life science, public health, medicine, biology and agronomy and other disciplines and mathematics to form a cross-discipline. In studying the complexity of ecological relationship in nature, people often study it by establishing mathematical model, which is a cross-discipline of mathematics and biology-bio-mathematics. As an important branch of bio-mathematics, bio-dynamic system mainly uses the knowledge of dynamics to study the established bio-mathematics modeling. The mathematical results can be used to explain the existing phenomena in the biological world and predict what may happen in the biological world in the future. In this way, people can choose a more appropriate way of life, so that man and nature can live in greater harmony. In the first part of this paper, the research background and research status of biodynamics and population dynamics are studied. In order to facilitate our research, some basic terms and theorems of dynamical systems are introduced. The second part of this paper mainly introduces several classical population dynamics models: Logistic model, Lotka-Volterra model and Leslie-Gower model. Finally, the dynamic behavior of three kinds of functional response functions with density or non-density constraints is introduced, and the stability at the equilibrium point and the conditions for the occurrence of limit cycles are also introduced. In the third part of this paper, we mainly study the qualitative analysis of a class of Volterra models with constant retention rate. This kind of Volterra model with constant storage rate has at least two equilibrium points. By using qualitative theory of plane system and normal form theory, it is found that they can be stable nodes, unstable nodes and saddle points under different parameters. Weak center, etc. By using the normal form theory of Hopf bifurcation and the calculation of the first Lyapunov coefficient, the supercritical Hopf bifurcation is obtained near the weak center of the system, and the unique stable limit cycle is subcharged from the equilibrium point. In this paper, the dynamic analysis of the Volterra model with constant storage rate is given: when npna21) 1 (0,) 1 (010npaa), the internal equilibrium point A is the stable node of the system. If n is regarded as the rarefaction rate of the system, then for any given rarity rate n, when the prey storage rate p is large enough to make npn2) 1 (greater than the ratio of the birth rate of the prey to the death rate of the predator, This biological system can coexist for a long time; When) 133 (1) (nnpnnanp212 and) n (paak 110), the system costs a unique stable limit cycle near the equilibrium point A, which indicates that the biological system coexists in the form of stable periodic solutions for a long time.
【学位授予单位】：重庆大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O175

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