两类具Michaelis-Menten型收获的捕食系统的稳定性与分支分析
发布时间：2018-07-22 21:33
【摘要】：本文着重研究了两类带有Michaelis-Menten型收获的食饵-捕食系统的稳定性以及分支,其主干内容分别是在第二章与第三章这两章中完成的,主要工作及结论如下:第二章研究了 一类食饵种群带有Michaelis-Menten型收获项的Leslie-Gower型捕食系统,考察了该系统的稳定性以及其中各种分支的存在性等动力学问题.该章的主要理论依据是微分方程的定性与稳定性理论、分支理论和规范性理论等.文章首先探究了系统各类有效平衡点的存在性,通过选取适当的参数作为分析参数,得到了各自平衡点存在时的具体对应条件.其次,研究这些平衡点在其相应存在条件下的稳定性,针对不同情形,分别采取了特征值分析法、线性化法等方法进行分析,得出边界平衡点在其存在条件下都不稳定,而正平衡点在其不同存在条件下则可能分别以汇点、源点、中心、鞍结点、尖点等类型出现的结论.最后,还研究了该系统中的一些分支,包括鞍结分支、Hopf分支和Bogdanov-Takens分支,通过选取恰当的参数作为分支参数,分别计算给出了具体分支点,并通过验证横截条件的方法严格证明了鞍结分支、Hopf分支的存在性,利用第一李雅普诺夫方法给出了 Hopf分支的方向,还通过规范型理论的应用论证了该系统中余维2的Bogdanov-Takens分支的存在性,得到了二阶截断规范型以及分支曲线表达式等.第三章探讨了 一类带有食饵种群成熟时滞和捕食者种群Michaelis-Menten型收获的捕食系统,研究其系统的稳定性和Hopf分支等动力学行为.这章研究的主要理论基础是微分方程的稳定性理论、Hopf分支理论和时滞泛函微分方程理论.文章给出了系统存在惟一正平衡点时的参数条件,并通过分析特征方程,利用Hurwitz判别法,得到了系统正平衡点局部渐近稳定的充要条件.另外,文章还通过选取时滞量作为分析参数,对系统特征方程特征根的实部进行分析,得到了系统保持局部渐近稳定的参数区间,并且给出具体的临界滞量值,论证得出了当该时滞参数穿越该临界值时系统就会发生Hopf分支的结论.本文在主要理论结果得出之后,还借助Matlab进行了相应的数值模拟,验证了其结论的准确性.
[Abstract]:In this paper, the stability and bifurcation of two kinds of predator-prey systems with Michaelis-Menten harvesting are studied. The main work and conclusions are as follows: in chapter 2, we study the Leslie-Gower type predator-prey system with Michaelis-Menten harvesting term, and investigate the stability of the system and the existence of various branches of the system. The main theoretical basis of this chapter is qualitative and stability theory of differential equation, branch theory and normative theory. In this paper, the existence of all kinds of effective equilibrium points in the system is discussed. By selecting appropriate parameters as the analysis parameters, the corresponding conditions for the existence of each equilibrium point are obtained. Secondly, the stability of these equilibrium points under the corresponding conditions is studied. The eigenvalue analysis method and linearization method are used to analyze the stability of these equilibrium points under the conditions of their existence, and the results show that the boundary equilibrium points are unstable under the conditions of their existence. Under different conditions, the positive equilibrium point may appear as meeting point, source point, center, saddle node, tip point and so on. Finally, some branches of the system, including saddle node Hopf bifurcation and Bogdanov-Takens bifurcation, are studied. The existence of Hopf bifurcation of saddle node bifurcation is strictly proved by the method of verifying the transverse condition. The direction of Hopf bifurcation is given by the first Lyapunov method. The existence of Bogdanov-Takens bifurcation of codimension 2 in the system is also proved by the application of normal form theory. The second order truncated canonical form and branch curve expression are obtained. In chapter 3, we study a kind of predator system with prey population maturity delay and predator population Michaelis-Menten harvest, and study its stability and Hopf bifurcation. The main theoretical basis of this chapter is the stability theory of differential equations, Hopf bifurcation theory and delay functional differential equation theory. In this paper, the parameter conditions for the existence of a unique positive equilibrium point are given. By analyzing the characteristic equation and using Hurwitz's criterion, the sufficient and necessary conditions for the local asymptotic stability of the positive equilibrium point of the system are obtained. In addition, by selecting the time-delay as the analysis parameter, the real part of the characteristic root of the system characteristic equation is analyzed, the parameter interval of the system maintaining local asymptotic stability is obtained, and the concrete critical hysteresis value is given. It is proved that Hopf bifurcation will occur when the time-delay parameter crosses the critical value. After the main theoretical results are obtained, the corresponding numerical simulation is carried out with Matlab to verify the accuracy of the conclusions.
【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】：O175
本文编号：2138641
[Abstract]:In this paper, the stability and bifurcation of two kinds of predator-prey systems with Michaelis-Menten harvesting are studied. The main work and conclusions are as follows: in chapter 2, we study the Leslie-Gower type predator-prey system with Michaelis-Menten harvesting term, and investigate the stability of the system and the existence of various branches of the system. The main theoretical basis of this chapter is qualitative and stability theory of differential equation, branch theory and normative theory. In this paper, the existence of all kinds of effective equilibrium points in the system is discussed. By selecting appropriate parameters as the analysis parameters, the corresponding conditions for the existence of each equilibrium point are obtained. Secondly, the stability of these equilibrium points under the corresponding conditions is studied. The eigenvalue analysis method and linearization method are used to analyze the stability of these equilibrium points under the conditions of their existence, and the results show that the boundary equilibrium points are unstable under the conditions of their existence. Under different conditions, the positive equilibrium point may appear as meeting point, source point, center, saddle node, tip point and so on. Finally, some branches of the system, including saddle node Hopf bifurcation and Bogdanov-Takens bifurcation, are studied. The existence of Hopf bifurcation of saddle node bifurcation is strictly proved by the method of verifying the transverse condition. The direction of Hopf bifurcation is given by the first Lyapunov method. The existence of Bogdanov-Takens bifurcation of codimension 2 in the system is also proved by the application of normal form theory. The second order truncated canonical form and branch curve expression are obtained. In chapter 3, we study a kind of predator system with prey population maturity delay and predator population Michaelis-Menten harvest, and study its stability and Hopf bifurcation. The main theoretical basis of this chapter is the stability theory of differential equations, Hopf bifurcation theory and delay functional differential equation theory. In this paper, the parameter conditions for the existence of a unique positive equilibrium point are given. By analyzing the characteristic equation and using Hurwitz's criterion, the sufficient and necessary conditions for the local asymptotic stability of the positive equilibrium point of the system are obtained. In addition, by selecting the time-delay as the analysis parameter, the real part of the characteristic root of the system characteristic equation is analyzed, the parameter interval of the system maintaining local asymptotic stability is obtained, and the concrete critical hysteresis value is given. It is proved that Hopf bifurcation will occur when the time-delay parameter crosses the critical value. After the main theoretical results are obtained, the corresponding numerical simulation is carried out with Matlab to verify the accuracy of the conclusions.
【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】：O175
【参考文献】
相关博士学位论文 前1条
1 袁锐;具时滞和食饵收获的捕食—食饵系统的分支动力学研究[D];哈尔滨工业大学;2015年
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