两类三维系统的稳定性与Hopf分支

【作者】 孙艳艳；

【导师】 窦霁虹；

【作者基本信息】 西北大学 ， 应用数学， 2009， 硕士

【摘要】 本文主要运用常微分方程定性与稳定性理论及分支的方法，研究了两类种群生态学模型．全文内容共分为三章．本文主要作了以下工作：1．第一部分是绪论．先综述了生态学发展的状况，随后介绍了问题的引入以及本文的主要工作．最后介绍了本文所用到的一些有关生态学和稳定性方面的定理和引理．2．第二部分研究了一类具有目标转移强度的三种群捕食-食饵系统，通过Hurwitz判据和Lyapunov方法，得到了系统正平衡点稳定的条件，当两食饵种群的内秉增长率相同时，该捕食系统将有一个周期解．接着讨论了系统的Hopf分支，得到了当取比例常数k₂为分支参数时，系统发生Hopf分支的条件．进一步讨论了当比例常数k₂、k₁满足函数关系时，即k₂=f（k₁）（k₁>0）时，系统发生Hopf分支的必要条件，并相应给出了例子，当k₂=lk₁，k₂=k₁^α（α>1），k₂=k₁^α（0<α<1）时，系统在正平衡点处的Hopf分支．3．第三部分主要研究了一类具有多时滞和阶段结构的捕食-食饵系统，其中食饵具有阶段结构，且捕食者只捕食成年食饵．运用比较定理判断出解的有界性．采用常微分方程稳定性和定性方法，分析了系统的非负不变性、边际平衡点的性质及正平衡点的局部渐近稳定性与Hopf分支，得到了当τ=0时，系统在正平衡点的稳定性的充分条件．同时考虑了时滞对于系统稳定性的影响，当选取时滞τ=τ₁+τ₂作为分支参数时，得到了当时滞τ=τ₁+τ₂由0变化到临界值时，系统在正平衡点附近发生Hopf分支，即当τ增加通过临界值时，从正平衡点分支出周期解，更多还原

【Abstract】 In this paper, by using the qualitative and stability theories and bifurcation method of ordinary differential equations, two population models in ecology are studied .The whole paper consists of three chapters. Details are as follows:1.The first chapter is the introduction. The development of ecology and the main works of the thesis are introduced, then some fundamental theories and lemmas about ecology and stability that can be used in this paper are given.2.A mathematical model of two prey and one predator system which has the switching property of predation is considered. By using the Routh-Hurwitz criteria and Lya-punov method ,the condition of the stable positive equilibrium is obtained. In the special case that two prey species have the same intrinsic growth rates, it is shown that the system asymptotically settles a Volterra’s oscillation in three-dimensional space.Then the Hopf bifurcation of the system is discussed.The conditions of the Hopf bifurcation are obtained when k₂ is parameters.Further ,the necessary conditions of the Hopf bifurcation are obtained when k₂ = f（k₁）（k₁ > 0） ,and several examples are given when k₂ = lk₁, k₂ = k₁^α（α> 1）, k₂ = k₁^α（0 <α< 1）.3.A prey-predator system of two species with stage structure and time delay is investigated . The boundary of the solution to the system is obtained by using summary theory. By using the stability theory of the differential equation ,the invariance of non-negativity, nature of the boundary equilibrium ,the local stability and Hopf bifurcation of the positive equilibrium are analyzed.The sufficient conditions for local asymptotic stability of the positive equilibrium are given when time delayτ= 0.Furthermore, it shows that positive equilibrium is locally asymptotically stable when time delayτ=τ₁ +τ₂ is suitable small, while a loss of stability by a Hopf bifurcation can occur as the delay increases .That is ,a family of periodic solutions bifurcates from positive equilibrium as r passes through the critical value .更多还原