几类生物动力系统的稳定性和分支问题研究
发布时间：2019-07-04 12:53
【摘要】：生物数学作为一门交叉学科,近些年已经有了飞速的发展.生物动力学是生物数学的一个分支,数学模型在描述生物动力学行为中起到很大的作用.时滞生物动力系统是一个具有丰富实际背景与广泛应用的领域.时滞动力系统的稳定性和分支问题的研究对实际应用领域的发展起着关键作用,其中,稳定性体现了结构平衡性,在无穷维空间上对系统进行稳定性研究,尤其是对全局稳定性的研究会更全面和深入地展示系统的动力学性质.所谓分支,是指当参数发生变化并经过某些临界值时,系统的某些特性发生突变的现象.Hopf分支是一种常见而重要的分支,它主要研究当参数变化时,平衡点的稳定性发生变化,从而在平衡点附近产生小振幅周期解的现象.本文主要应用Lyapunov稳定性理论,LaSalle不变性原理,拓扑度理论,中心流形定理,规范型方法以及全局分支定理等理论和方法对几类生物动力系统的局部和全局稳定性、周期解的存在性以及不动点分支、局部和全局Hopf分支,系统的持久性进行研究.具体内容如下：首先,讨论了一类SIRS模型,选择时滞为参数,得到了无病平衡点的全局渐近稳定性,地方病平衡点的局部渐近稳定性和Hopf分支的存在性,以及系统的持久性.之后研究了一类非自治SIR模型,利用重合度理论,得到了系统正周期解全局存在性,唯一性以及全局稳定性的充分条件.考虑到疾病潜伏期的影响,进而研究了一类SEIRS模型,得到了无病平衡点的全局渐近稳定性,地方病平衡点的局部渐近稳定性和全局Hopf分支的存在性,进而也得到了系统持久的充分条件.其次,研究了一类浮游生态系统的复杂动力学.首先给出了常微分方程系统平衡点的稳定性分析,之后在常微分方程系统中引入时滞,以时滞为参数,得到了边界平衡点全局渐近稳定和不稳定的充分条件,进一步,在一定条件下,系统在正平衡点处出现稳定性开关现象,可能出现周期解,随着时滞的增加周期解继续存在,证实了全局Hopft分支的存在性.同时也发现随着毒素释放率的增加,正平衡点不稳定的区间在缩小,说明毒素有助于系统的稳定.最后,在时滞系统的基础上引入扩散,考察扩散和时滞的共同影响,得到扩散不能改变平衡点的稳定性,即图灵不稳定性不会发生.分别考察了大的扩散和小的扩散对Hopf分支的影响,在一定条件下能够产生空间非齐次周期解,进而给出了算法来决定分支周期解的性质.最后,研究了两类系统的混沌控制策略.首先研究了具有两个时滞的浮游生态系统,以两个时滞为参数得到系统可能出现稳定型开关现象.当参数变化时,发现混沌现象可能发生,进一步,通过数值模拟发现时滞和毒素释放率的增加可以使得系统的混沌现象消失.浮游动物的最大转化率的增加会使得系统从稳定状态经过倍周期分叉,最终导致混沌.其次,研究了一类具有混沌的浮游生态系统的时滞反馈控制,以时滞为参数,得到了控制系统发生Hopf分支的条件,即系统在一定条件下,当时滞取某些参数时可以将系统不稳定的周期解变成稳定周期解或稳定平衡点,说明了控制的有效性.
文内图片：
图片说明：图３＊２：巧（Ｔ）曲线图（如图（ａ）），，邋ｒ邋＝邋０．１邋＜邋ｒ0时系统波图（如图Ｗ）．邋Ｔ邋＝邋２．５邋＞户时逡逑
[Abstract]:As a cross-discipline, biomathematics has developed rapidly in recent years. Biodynamics is a branch of biological mathematics, and the mathematical model plays a very important role in describing the behavior of biological dynamics. The time-delay biodynamic system is a field with rich practical background and wide application. The stability and branch problems of the time-delay power system play a key role in the development of the practical application field, in which the stability shows the structural balance, and the stability of the system is studied in the infinite dimensional space. In particular, the study of global stability is more comprehensive and in-depth show the dynamic nature of the system. The so-called branch refers to the phenomenon that some characteristics of the system change when the parameter changes and passes some critical values. The Hopf bifurcation is a common and important branch, and it mainly studies the phenomenon that the stability of the equilibrium point changes when the parameters change, so as to generate a small-amplitude periodic solution near the equilibrium point. In this paper, we mainly apply the Lyapunov stability theory, the LaSalle invariance principle, the topological degree theory, the central manifold theorem, the standard method and the global branch theorem. The persistence of the local and global Hopf branches and systems is studied. The specific content is as follows: First, we discuss a kind of SIRS model, choose time-delay as the parameter, get the global asymptotic stability of the disease-free equilibrium point, the local asymptotic stability of the local disease equilibrium point and the existence of the Hopf branch, and the persistence of the system. After that, a class of non-self-governing SIR model is studied, and the sufficient conditions for global existence, uniqueness and global stability of the system's positive periodic solution are obtained by using the coincidence degree theory. Taking into account the influence of the latent period of the disease, a kind of SEIRS model is also studied to obtain the global asymptotic stability of the disease-free equilibrium point, the local asymptotic stability of the local disease equilibrium point and the existence of the global Hopf branch, and the sufficient condition of the system is also obtained. Secondly, the complex dynamics of a kind of floating ecosystem were studied. At first, the stability analysis of the equilibrium point of the system of ordinary differential equation is given, and the time-delay is introduced in the system of ordinary differential equation, and the sufficient conditions for global asymptotic stability and instability of the boundary equilibrium point are obtained. The stability switching phenomenon occurs at the positive equilibrium point, and the periodic solution may occur, and the existence of the global Hopft branch is confirmed with the increase of the time delay. It is also found that with the increase of the release rate of the toxin, the interval between the positive equilibrium point and the positive equilibrium point is reduced, indicating that the toxin can contribute to the stability of the system. In the end, diffusion is introduced on the basis of time-delay system, the common influence of diffusion and time-delay is investigated, and the stability of the equilibrium point can not be changed by diffusion, that is, the Turing instability does not occur. The influence of the large diffusion and the small diffusion on the Hopf bifurcation is investigated, and the non-homogeneous periodic solution of the space can be generated under certain conditions, and then the algorithm is given to determine the nature of the solution of the branch period. At last, the hybrid control strategy of two kinds of systems is studied. In this paper, a floating ecosystem with two time-delay is first studied, and a stable switching phenomenon may occur in the system with two time-delay parameters. When the parameters change, it is found that the mixing phenomenon may occur, and further, it is found that the increase of time-delay and the release rate of the toxin can cause the mixing phenomenon of the system to disappear. The increase in the maximum conversion of zooplankton will cause the system to diverge from a steady state through a period of time, resulting in a mixing. Secondly, the time-delay feedback control of a kind of floating ecosystem with hybrid system is studied, with the time-delay as the parameter, the condition of the Hopf bifurcation of the control system is obtained, that is, the system under certain conditions, The unstable periodic solution of the system can be transformed into a stable periodic solution or a stable equilibrium point when certain parameters are delayed at that time, and the effectiveness of the control is explained.
【学位授予单位】：北京科技大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O175
本文编号：2509931
文内图片：
图片说明：图３＊２：巧（Ｔ）曲线图（如图（ａ）），，邋ｒ邋＝邋０．１邋＜邋ｒ0时系统波图（如图Ｗ）．邋Ｔ邋＝邋２．５邋＞户时逡逑
[Abstract]:As a cross-discipline, biomathematics has developed rapidly in recent years. Biodynamics is a branch of biological mathematics, and the mathematical model plays a very important role in describing the behavior of biological dynamics. The time-delay biodynamic system is a field with rich practical background and wide application. The stability and branch problems of the time-delay power system play a key role in the development of the practical application field, in which the stability shows the structural balance, and the stability of the system is studied in the infinite dimensional space. In particular, the study of global stability is more comprehensive and in-depth show the dynamic nature of the system. The so-called branch refers to the phenomenon that some characteristics of the system change when the parameter changes and passes some critical values. The Hopf bifurcation is a common and important branch, and it mainly studies the phenomenon that the stability of the equilibrium point changes when the parameters change, so as to generate a small-amplitude periodic solution near the equilibrium point. In this paper, we mainly apply the Lyapunov stability theory, the LaSalle invariance principle, the topological degree theory, the central manifold theorem, the standard method and the global branch theorem. The persistence of the local and global Hopf branches and systems is studied. The specific content is as follows: First, we discuss a kind of SIRS model, choose time-delay as the parameter, get the global asymptotic stability of the disease-free equilibrium point, the local asymptotic stability of the local disease equilibrium point and the existence of the Hopf branch, and the persistence of the system. After that, a class of non-self-governing SIR model is studied, and the sufficient conditions for global existence, uniqueness and global stability of the system's positive periodic solution are obtained by using the coincidence degree theory. Taking into account the influence of the latent period of the disease, a kind of SEIRS model is also studied to obtain the global asymptotic stability of the disease-free equilibrium point, the local asymptotic stability of the local disease equilibrium point and the existence of the global Hopf branch, and the sufficient condition of the system is also obtained. Secondly, the complex dynamics of a kind of floating ecosystem were studied. At first, the stability analysis of the equilibrium point of the system of ordinary differential equation is given, and the time-delay is introduced in the system of ordinary differential equation, and the sufficient conditions for global asymptotic stability and instability of the boundary equilibrium point are obtained. The stability switching phenomenon occurs at the positive equilibrium point, and the periodic solution may occur, and the existence of the global Hopft branch is confirmed with the increase of the time delay. It is also found that with the increase of the release rate of the toxin, the interval between the positive equilibrium point and the positive equilibrium point is reduced, indicating that the toxin can contribute to the stability of the system. In the end, diffusion is introduced on the basis of time-delay system, the common influence of diffusion and time-delay is investigated, and the stability of the equilibrium point can not be changed by diffusion, that is, the Turing instability does not occur. The influence of the large diffusion and the small diffusion on the Hopf bifurcation is investigated, and the non-homogeneous periodic solution of the space can be generated under certain conditions, and then the algorithm is given to determine the nature of the solution of the branch period. At last, the hybrid control strategy of two kinds of systems is studied. In this paper, a floating ecosystem with two time-delay is first studied, and a stable switching phenomenon may occur in the system with two time-delay parameters. When the parameters change, it is found that the mixing phenomenon may occur, and further, it is found that the increase of time-delay and the release rate of the toxin can cause the mixing phenomenon of the system to disappear. The increase in the maximum conversion of zooplankton will cause the system to diverge from a steady state through a period of time, resulting in a mixing. Secondly, the time-delay feedback control of a kind of floating ecosystem with hybrid system is studied, with the time-delay as the parameter, the condition of the Hopf bifurcation of the control system is obtained, that is, the system under certain conditions, The unstable periodic solution of the system can be transformed into a stable periodic solution or a stable equilibrium point when certain parameters are delayed at that time, and the effectiveness of the control is explained.
【学位授予单位】：北京科技大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O175
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