【作者】 王晶囡；

【导师】 蒋卫华；

【作者基本信息】 哈尔滨工业大学 ， 基础数学， 2013， 博士

【摘要】 离散时滞与分布时滞的微分方程被广泛地应用在生物系统当中，用于解释相应系统中具有的稳定性、周期振荡以及混沌等动力学行为。本文主要利用非线性动力学、泛函微分方程等相关理论和方法，研究离散时滞与分布时滞的非线性生物模型的动力学性质。分析了时滞项对非线性生物模型的一些动力学行为的影响，主要包括系统的稳定性、系统的持久性、局部Hopf分支、全局Hopf分支和两类余维二分支等方面的理论结果。并给出了从模型中得到的数学理论结果代表的生物学意义。首先，利用中心流形定理、Hassard规范型理论以及Faria和Magalhaes规范型理论，研究了一个具有休眠特征的离散时滞的捕食-被捕食模型的Hopf及Hopf-fold分支。分析了时滞对系统稳定性、持久性、周期解以及混沌的影响。给出了系统产生Hopf分支和Hopf-fold分支的条件，并从中得到了离散时滞的引入可以使系统产生稳定性切换和多周期轨共存等现象。推证出了带有原系统参数在Hopf-fold临界点附近的规范型及分支性质。利用Lyapunov-Razumikhin有界定理等相关知识证明了时滞系统的持久性质。通过数值模拟展示了周期解的稳定性和大范围存在等性质。其次，研究了具有休眠和脉冲扰动的捕食-被捕食模型的稳定性及分支问题。利用脉冲微分方程理论、小振幅扰动理论和比较原理得到了被捕食者灭绝的周期解的全局渐近稳定条件和系统处于持久状态的条件。利用数值模拟得到了随着脉冲扰动项的增加会导致周期解发生倍周期分支从而导致混沌的现象，分析描述了脉冲扰动项和孵化项对系统固有周期行为的影响。然后，利用规范型理论和拓扑度等相关理论研究了两个具有分布时滞的向日葵模型的局部Hopf分支和全局Hopf分支性质。利用儒歇定理和辐角原理讨论了呈对数生长的向日葵模型的稳定性及产生局部Hopf分支条件，给出了模型的周期解的近似表达形式。将呈对数生长的向日葵模型的Hopf分支性质与已有的呈线性生长的向日葵模型的Hopf分支性质进行了比较，揭示了周期解的周期变化和近似表达形式之间的差别。分别从理论和数值模拟上，揭示了呈线性生长的向日葵模型的周期解可以大范围存在的现象。最后，研究了一个带有离散及分布混合时滞的神经元系统的Hopf-pitchfork分支。通过对特征方程的分析，给出了时滞神经元系统发生余维二Hopf-pitchfork分支的条件。利用中心流形定理与Faria和Magalhaes规范型理论得到了系统在Hopf-pitchfork临界点附近的规范型和带有原系统参数的分支集，细化了分支图。从理论和数值模拟两个方面揭示了当时滞项和系数项在临界点附近扰动时，神经元系统会出现两个稳定平衡点共存以及两个稳定平衡点与一个稳定周期轨共存的现象。更多还原

【Abstract】 The diferential equations with discrete and distributed time delays are widely ap-plied in biological systems for describing biology dynamical behaviors, such as stability,periodic oscillation and chaos of systems.In this thesis, we study the dynamical behaviors of nonlinear biology models withdiscrete and distributed time-delays by using nonlinear dynamics systems, functional dif-ferential equations and other related theories and methods. We analyze the efects of timedelay on the dynamics behavior of nonlinear biology models, including the stability ofthe system, the permanence of the system, local Hopf bifurcation, global Hopf bifurca-tion, and two types of codimension two bifurcation and other related theoretical results.The corresponding biological results, which is represented by the obtained mathematicaltheoretical results from studying the biological system, are revealed.First, we investigate Hopf bifurcation and Hopf-fold bifurcation of predator-preymodel with dormancy of predators and discrete delay by using the center manifold theo-rem, the Hassard normal form theory and the Faria and Magalhaes normal form theory.The efects of time delay on the stability, persistence, periodic solution and chaos ofsystem are analyzed. The occurrence conditions of Hopf bifurcation and Hopf-fold bi-furcation are obtained. It shows that time delay in the prey-species growth can lead tothe phenomena of stability switches and the coexistence of multiple periodic orbits. Wededuce the normal form with original parameters in the model and bifurcation propertiesnear the bifurcation point. By using the Lyapunov-Razumikhin theorem and other relatedknowledge, the uniform persistence of predator-prey model with time delay is proved.Numerical simulation shows the stability as well as a large-scale existence of periodicsolution.Next, we investigate the stability and bifurcation of predator-prey model with dor-mancy of predators and impulsive perturbations. By using the theories of impulsive e-quations, small amplitude perturbation skills and the comparison technique, we get theconditions which guarantee the global asymptotical stability of the prey-eradication peri-odic solution and the permanence of the system. Further, influences of the impulsive per-turbation on the inherent oscillation are studied numerically, which shows rich dynamics,such as period-doubling bifurcation, chaos, and period-halving bifurcation. Moreover, the efects of the impulsive perturbation and hatching rate on the chaos of the system are ob-tained by numerical simulation. In addition, using the theories of the normal form and thetopological degree, we study the local and global Hopf bifurcation properties of two sun-flower equations with distributed delay. Using the Rouche theorem and the principle ofargument, we discuss the occurrence conditions of local Hopf bifurcation of the sunflowermodel with logarithmic growth and give the approximate expression of the periodic so-lution of this model. Local Hopf bifurcation properties between the linear growth modeland the logarithmic growth of the sunflower model are compared. It reveals the diferencebetween period changes of periodic solutions as well as the diference between variousform of approximate expression of the periodic solutions. From theoretical investigationand numerical simulation, we obtain a large-scale existence of periodic solution.Finally, the Hopf-pitchfork bifurcation of a two-neuron system with discrete and dis-tributed delays is investigated. Through analyzing the characteristic equation, we obtainthe occurrence conditions of Hopf-pitchfork bifurcation. Using the center manifold theo-rem and the Faria and Magalhaes normal form theory, we obtain the normal form and theirunfolding with original parameters of the system near the bifurcation point and refine thebifurcation diagram. It is revealed that when the time delay and the parameter changenear the vicinity of the critical point, some dynamical behaviors of neuron system arefound, such as stable periodic orbit, the coexistence of two stable non-trivial equilibrium,and the coexistence of a stable periodic orbit and two stable equilibrium, by theoreticalinvestigation and numerical simulation.更多还原