几类具扩散项的种群模型动力学性质分析

发布时间：2018-03-11 10:26

本文选题：反应扩散系统　切入点：稳定性　出处：《哈尔滨工业大学》2015年博士论文　论文类型：学位论文

【摘要】：在生物数学中,研究种群模型的动力学性质已经成为了一个重要内容,而其中对具有扩散项的种群模型的研究受到了许多数学家和生物学家的关注。由于能量在生物个体中的传递、转化的差异,对具有不同功能反应函数以及扩散项的捕食-被捕食系统的长时间动力学性质的研究,如平衡点的稳定性,由扩散引起的Turing不稳定性,以及Hopf分支等问题,具有很强的理论意义和实际意义。本文研究了具有扩散项以及Holling III型和Beddington-DeAngelis型功能反应函数的种群模型的动力学性质。1.在具有Holling III型功能反应函数的捕食模型中,研究了食饵具有的避难项对该模型动力学性质的影响。通过构造Lyapunov函数,建立了正平衡点的全局渐近稳定性定理。以避难系数为分支参数,分析了Hopf分支的存在性,并通过中心流形和规范型理论分析了Hopf分支方向以及分支周期解的稳定性。最后,总结了分析结果,对于避难系数如何影响系统的动力学性质给出了解释并进行了数值模拟。2.在具有Holling III型功能反应函数的捕食模型中,研究了当捕食者的死亡率对系统动力学性质的影响。考虑捕食者内部压力对死亡率的影响,定义捕食者的死亡率是关于捕食者数量的一个递增函数。通过特征根分析,给出了平衡点局部稳定的充分条件,并且讨论了由扩散引起的Turing不稳定性。最后,分析了Hopf分支存在性,以及Hopf分支的性质。3.研究了一类具有Beddington-DeAngelis型功能反应函数的修正LeslieGower种群模型。Beddington-DeAngelis型功能反应函数考虑了捕食者内部作用对捕食效率的影响。另外,该模型假设食饵数量较少时,捕食者会捕食其它的食物,体现在环境对捕食者的最大承受量等于食饵与其他食物的数量和。本文从分支的角度定性的分析了该模型,给出了Turing分支存在条件,Hopf分支存在条件及Hopf分支的性质。4.在一类具有Beddington-DeAngelis型功能反应函数的修正Leslie-Gower种群模型中,分析了时滞对该模型动力学性质的影响。通过特征根分析,给出了正平衡点局部稳定性条件。以时滞为分支参数,讨论了Hopf分支的存在性,并通过偏泛函微分方程的中心流形和规范型理论,给出了决定Hopf分支方向及周期解稳定性的详细计算公式。
[Abstract]:In biological mathematics, it has become an important content to study the dynamic properties of population model. The study of population models with diffusion term has attracted the attention of many mathematicians and biologists. The long-time dynamical properties of predator-prey systems with different functional response functions and diffusion terms are studied, such as the stability of equilibrium points, the Turing instability caused by diffusion, and the Hopf bifurcation, etc. In this paper, the dynamical properties of population model with diffusion term and Holling III type and Beddington-DeAngelis type functional response function are studied. 1. In the predation model with Holling III type functional response function, In this paper, we study the influence of the sheltered term of prey on the dynamic properties of the model. By constructing the Lyapunov function, we establish the global asymptotic stability theorem of the positive equilibrium point. Taking the asylum coefficient as the bifurcation parameter, we analyze the existence of the Hopf bifurcation. The direction of Hopf bifurcation and the stability of the bifurcation periodic solution are analyzed by the theory of center manifold and normal form. Finally, the analysis results are summarized. This paper gives an explanation of how the asylum coefficient affects the dynamic properties of the system and gives a numerical simulation. 2. In the predation model with Holling III type functional response function, The effect of predator mortality on system dynamics is studied. Considering the effect of internal pressure on predator mortality, the predator mortality is defined as an increasing function on the number of predators. The sufficient conditions for local stability of equilibrium point are given, and the Turing instability caused by diffusion is discussed. Finally, the existence of Hopf bifurcation is analyzed. A modified LeslieGower population model with Beddington-DeAngelis functional response function. Beddington-DeAngelis functional response function takes into account the effect of the internal action of predator on the predation efficiency. In addition, the model assumes that the number of prey is small. A predator preys on other foods, which is reflected in the fact that the maximum tolerance of the environment to the predator is equal to the amount of prey and other foods. This paper qualitatively analyzes the model from the perspective of the branch. In this paper, the existence condition of Turing bifurcation and the properties of Hopf bifurcation are given. 4. In a modified Leslie-Gower population model with Beddington-DeAngelis type functional response function, the influence of time delay on the dynamic properties of the model is analyzed. In this paper, the local stability conditions of positive equilibrium point are given. The existence of Hopf bifurcation is discussed with time delay as bifurcation parameter, and the theory of center manifold and normal form of partial functional differential equation is given. A detailed formula for determining the direction of Hopf bifurcation and the stability of periodic solutions is given.
【学位授予单位】：哈尔滨工业大学
【学位级别】：博士
【学位授予年份】：2015
【分类号】：O175

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