时滞脉冲生物动力系统的动力学研究

发布时间：2018-04-19 02:24

  本文选题：时滞 + 脉冲效应　； 参考：《电子科技大学》2015年博士论文

【摘要】：自然界中许多系统状态变量的变化率不仅依赖系统的当前状态,而且与过去某个时刻或过去一段时间的状态有关,对这类系统进行建模时用时滞微分方程或泛函微分方程来代替原来的常微分方程更为合适。另一方面,现实世界中还有许多自然或人为因素会对系统的内在规律带来突然的变化,这些作用时间往往非常短暂,在建模时可视为在某个固定时刻发生,但系统状态在这些时刻却不再连续,因此在对这类系统建模时用半连续的脉冲微分方程来代替连续的动力系统更为合理。此外,现实自然界中许多系统状态变量的变化还会受到环境噪声的影响,这时再用确定的常微分方程来刻画相应的系统也不再合适,而应该用随机微分方程来描述相关问题更为合理。因此,本文正是基于上述背景,分别就四类具有时滞、脉冲效应以及随机扰动的混杂生物动力系统进行研究和讨论：1.对一类脉冲输入营养基和具有分布时滞的营养基再生的恒化器模型进行研究,利用脉冲比较定理、Floquet定理以及微小参数扰动法等技巧得到了系统微生物灭绝周期解全局渐近稳定性的充分判据,另外通过构造合适的Liapunov函数得到了系统的有界性,并在此基础上充分运用分析技巧得到了恒化器中微生物能够连续培养的充分条件。最后对相关理论结果进行数值模拟,并进一步分析了参数变化对微生物灭绝或持久生存的影响。2.对一类具有消化时滞和周期脉冲捕获的食物链系统进行研究,利用脉冲比较定理、微小参数扰动以及微分不等式的技巧等,得到系统具有全局渐近稳定的捕食者灭绝周期解及系统持久生存性的充分条件,并通过数值例子及数值模拟进一步验证了理论结果的正确性和可行性,同时还通过分析及数值实验验证得到了能将害虫数量控制在更低的经济阂值水平之下的控制策略。3.对一类具有多时滞和脉冲效应的非自治概周期捕食系统进行研究,利用多元函数微分中值定理、微分不等式、积分不等式等数学分析技巧,得到了系统持久生存的充分条件。同时,通过构造一系列Liapunov泛函,证明了系统在一定条件下存在唯一的、一致渐近稳定的概周期解。最后通过数值例子及其仿真进一步证实了理论结果的正确性和有效性,并分析了不同脉冲效应和不同时滞对系统动力学行为的影响。4.对一类具有脉冲效应和随机扰动的非自治食物链系统进行研究,利用Ito积分公式、指数鞅不等式、微分不等式等分析技巧得到了系统的灭绝性、非持久生存、均方意义上持久生存、随机持久生存等渐近性质。最后通过一系列数值实验来佐证相关理论结果、观察相关生态学现象,同时还通过数值实验分析讨论了不同强度的脉冲效应和环境噪声对系统的影响。
[Abstract]:The rate of change of many system state variables in nature depends not only on the current state of the system, but also on the state of a past moment or period of time.It is more appropriate to replace the original ordinary differential equation with delay differential equation or functional differential equation when modeling this kind of system.On the other hand, there are also many natural or man-made factors in the real world that can cause sudden changes in the internal laws of the system, which are often very short and can be modeled as occurring at a fixed moment.However, the state of the system is not continuous at these times, so it is more reasonable to use semi-continuous impulsive differential equations instead of continuous dynamical systems in modeling such systems.In addition, many changes of system state variables in real nature will also be affected by environmental noise, and it is no longer appropriate to describe the corresponding system with a definite ordinary differential equation.It is more reasonable to describe the related problems with stochastic differential equations.Therefore, based on the above background, four kinds of hybrid biodynamic systems with time delay, impulsive effect and stochastic disturbance are studied and discussed respectively in this paper.The models of a kind of pulse input nutrient base and nutrient base regeneration with distributed delay are studied.The sufficient criteria for the global asymptotic stability of the periodic solution of microbial extinction are obtained by using the impulsive comparison theorem and the perturbation method of small parameters. In addition, the boundedness of the system is obtained by constructing appropriate Liapunov functions.On the basis of this, the sufficient conditions for the continuous culture of microbes in the chemostat were obtained by using the analytical techniques.Finally, the related theoretical results are numerically simulated, and the effects of parameter changes on the extinction or sustainable survival of microorganisms are further analyzed.In this paper, a class of food chain systems with digestive delay and periodic pulse capture is studied. The comparison theorem of impulses, the perturbation of small parameters and the technique of differential inequality are used.Sufficient conditions for the system to have globally asymptotically stable predator extinction periodic solutions and the persistence of the system are obtained, and the correctness and feasibility of the theoretical results are further verified by numerical examples and numerical simulations.At the same time, through the analysis and numerical experiments, the control strategy. 3. 3, which can keep the pest population under the lower economic threshold level, is obtained.In this paper, a class of nonautonomous almost periodic predator-prey systems with multiple delays and impulsive effects is studied. By using the differential mean value theorem of multivariate functions, differential inequalities, integral inequalities and other mathematical analytical techniques, sufficient conditions for the persistence of the system are obtained.At the same time, by constructing a series of Liapunov Functionals, it is proved that there exists a unique uniformly asymptotically stable almost periodic solution for the system under certain conditions.Finally, the correctness and validity of the theoretical results are further verified by numerical examples and simulations, and the effects of different impulsive effects and different delays on the dynamic behavior of the system are analyzed.In this paper, a class of nonautonomous food chain systems with impulsive effects and stochastic perturbations is studied. By using Ito integral formula, exponential martingale inequality, differential inequality and other analytical techniques, the extinction and non-persistence of the system are obtained.In the sense of mean square persistence, stochastic persistence and other asymptotic properties.Finally, a series of numerical experiments are used to support the related theoretical results and observe the related ecological phenomena. At the same time, the effects of pulse effect and ambient noise on the system are analyzed and discussed by numerical experiments.
【学位授予单位】：电子科技大学
【学位级别】：博士
【学位授予年份】：2015
【分类号】：O175
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本文编号：1771168