生物动力系统的稳定性和分岔分析

发布时间：2019-01-20 21:00

【摘要】：动力系统是非线性学科的重要组成部分,非线性问题存在于诸多学科和生活的各个领域中,比如,数学、物理学、生物学、医学、工程、力学以及经济学等都可以用非线性动力系统来解释。特别的,生物数学作为生物和数学的交叉学科,近几年得到了快速的发展。为了建立切合实际的数学模型,需要考虑很多因素,比如,时间、空间、时滞、随机、脉冲,阶段等。对此,本文主要研究离散的传染病模型和带有时滞的食饵-捕食者模型以及随机离散的捕食与被捕食者模型的动力学行为,主要内容分为四章如下:1.首先叙述了生物动力系统国内外发展、目的和意义,其次简单叙述了本论文需要用到的一些基本定义和定理,最后介绍了本论文所做的工作。2.主要分析一类离散传染病模型SI系统的动力学行为。首先,根据特征方程的特征根的情况得到了不动点稳定性的条件;其次,根据中心流行定理和分岔理论得到了系统在不动点发生倍周期分岔和Neimark-Sacker分岔的条件;最后,数值模拟验证该结论的正确性。3.主要分析具有双时滞的食饵-捕食者模型的动力学行为。首先根据特征根分布情况判断在平衡点处系统的Hopf分岔的存在性;其次,通过运用泛函微分方程的规范性理论以及中心流行定理对系统的Hopf分岔的方向、周期解进行分析;最后,数值模拟验证理论的正确性。4.主要分析对带有随机滞后食饵-捕食者模型离散系统的渐近稳定性和Hopf分岔问题进行分析。首先,用正交多项式逼近的方法将随机离散系统转化为确定性系统;其次,根据Hopf分岔理论得到了随机系统发生Hopf分岔的临界值并结合中心流行定理分析;最后,数值模拟说明正确性。
[Abstract]:Dynamic systems are an important part of nonlinear disciplines. Nonlinear problems exist in many disciplines and fields of life, such as mathematics, physics, biology, medicine, engineering. Mechanics and economics can be explained by nonlinear dynamic systems. In particular, biological mathematics as an interdisciplinary subject of biology and mathematics, has been rapid development in recent years. In order to establish a practical mathematical model, many factors need to be considered, such as time, space, time delay, randomness, pulse, stage and so on. In this paper, the dynamic behaviors of discrete infectious disease model, prey predator model with time delay and stochastic discrete predator model are studied. The main contents are as follows: 1. This paper first describes the development, purpose and significance of biodynamic system at home and abroad, then briefly describes some basic definitions and theorems that need to be used in this paper, and finally introduces the work done in this paper. 2. The dynamic behavior of a class of discrete infectious disease model (SI) systems is analyzed. Firstly, according to the characteristic root of the characteristic equation, the stability condition of the fixed point is obtained. Secondly, according to the central popular theorem and the bifurcation theory, the condition of the double periodic bifurcation and the Neimark-Sacker bifurcation of the system at the fixed point is obtained. Finally, the numerical simulation verifies the correctness of the conclusion. 3. The dynamic behavior of predator-prey model with two delays is analyzed. Firstly, the existence of Hopf bifurcation of the system at the equilibrium point is judged according to the distribution of the characteristic root, secondly, the direction of the Hopf bifurcation and the periodic solution of the system are analyzed by using the normative theory of functional differential equation and the central popular theorem. Finally, numerical simulation verifies the correctness of the theory. 4. The asymptotic stability and Hopf bifurcation of discrete systems with stochastic lag predator-prey model are analyzed. Firstly, the stochastic discrete system is transformed into a deterministic system by orthogonal polynomial approximation. Secondly, according to the Hopf bifurcation theory, the critical value of Hopf bifurcation for the stochastic system is obtained and the central popular theorem is analyzed. Finally, the numerical simulation shows the correctness.
【学位授予单位】：兰州交通大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O19

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