若干类生物动力系统的复杂性研究
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摘要
本文利用非线性动力系统理论,广义系统理论以及相关控制理论,研究了若干类生物动力系统的复杂性,其中包括正常生物种群动力系统的动态特性,广义生物经济系统的动态特性和混杂生物经济系统的动态特性.研究的主要内容涉及到这些系统的稳定性,各类分岔和混沌现象,脉冲现象以及脉冲状态反馈控制和混沌稳定控制等问题.全文工作包括如下几个方面：
(1)介绍了生物动力系统的研究现状和进展,主要包括传染病动力学的研究进展,浮游生态系统动力学的研究进展和种群动力学的研究进展.特别地,列举出了与本文工作相关的若干类生物种群动力系统的模型,并介绍了目前这些生物模型的研究现状.此外,还介绍了广义生物动力系统的研究现状.
(2)本文研究了两类具有Beddington-DeAngelis功能性反应和收获的捕食生物模型的动态复杂性.功能反应函数表示捕食者与食饵的相互作用关系Beddington-DeAngelis功能反应兼具HollingⅡ类功能性反应和比率型功能性反应的特点,而且避免了它们的缺陷,因而更为接近现实.但目前,还没有关于带有Beddington-DeAngelis功能性反应的离散捕食模型和脉冲状态反馈作用下这类捕食模型的分岔及周期解方面的研究.本文的研究填补了这方面研究的空白.考虑到生物种群世代不重叠性,首先研究了该类离散捕食生物模型的动态复杂性和控制,利用中心流形定理和分岔理论,获得了系统出现Flip分岔和Hopf分岔的分岔条件,通过绘制分岔图和Lyapunov(?)指数图,证明了混沌现象的存在性,进一步地,设计的状态延时反馈控制器有效消除了分岔和混沌现象,并从生物意义角度给出了该控制器的合理解释.另一方面,讨论了该类生物模型在脉冲状态反馈控制下的动态复杂性,应用脉冲控制理论,得到了两个Poincare映射,给出了周期解的存在及其稳定条件,同时分岔图显示出系统复杂的动态行为,包括倍周期分岔,混沌吸引子和周期窗等；此外,通过将固定时刻脉冲控制与脉冲状态反馈控制进行对比,说明脉冲状态反馈控制可以更有效地使生物种群密度保持在稳定区域内.
(3)本文首次将分布式时滞引入到文中的浮游植物-浮游动物-鱼类海洋生态模型中,研究了一类具有分布式时滞的浮游生物-鱼类海洋生态模型的动态复杂性.
一般来说,捕食者种群在捕食结束后并不会立刻将能量转化给下一代,常常有孕育期.数学上往往通过在模型中引进时滞的方法来描述这一现象.目前,大部分有关浮游生态模型的研究都集中考虑了确定性时滞.然而,由于生物的生长过程往往是一个漫长且积累的过程,不是仅仅与过去某一特定时刻有关的,所以分布式时滞的研究是有现实意义和背景的.通过运用中心流形定理和规范形理论,研究结果表明,当平均时滞超过某一临界值时,系统失稳并出现Hopf分岔,且在正平衡点附近出现一族周期解.另外,还得到了分岔周期解的稳定性,方向以及一些相关性质.同时,对相关的生物意义作出了诠释.
(4)本文将时滞和阶段结构,以及扩散行为引入到广义食饵-捕食者生物经济系统中,丰富了广义生物系统的研究.利用广义系统理论和分岔理论,文中主要研究了具有时滞和阶段结构的广义食饵-捕食者生物经济系统,以及具有扩散行为的广义食饵-捕食者生物经济系统的动态复杂性.研究结果表明,当捕获行为的经济利润由负变为正并通过零时,两类系统在正平衡点附近均发生奇异诱导分岔,这意味着某种生物种群密度在短时间内剧增,并导致其对有限生存资源的大量需求和占有,严重破坏了生态的平衡.另一方面,时滞表示捕食者捕获食饵种群后,将这种能量转化为自身能量所需要的时间,研究发现,较长的时间延迟会导致生物种群密度产生波动,破坏生物种群的可持续生长；但是,在幼年种群十分充足的条件下,时滞抑制了捕食者种群的过度繁殖.除此以外,考虑到生物种群在其生存区域间的迁徙活动,本文又研究了扩散行为对系统动态性的影响,结果表明,当扩散系数超过某一阈值时,生物种群系统的稳定性将会发生改变,生物种群密度出现周期振荡现象,且较大的扩散率有利于种群的持久生存.
(5)本文首次建立了一个由微分-差分-代数方程构成的混杂捕食生物经济模型,其中微分方程是由食饵种群的动态行为确立的,差分方程是由捕食者种群的动态特性确立的,经济利润则确定了代数方程.目前,对正常生物动力系统的复杂动态行为的研究已经取得了很多可喜的成果,同时,也可以找到一些研究广义生物动力系统的相关文献.但是,对于混杂生物系统复杂动态性的研究却很少见.从建模上看,混杂系统比正常系统和广义系统都复杂得多,它主要由微分,差分和代数方程共同构成,另外,研究过程中要分别考虑采样间隔和采样时刻两种情况对系统的影响.这些都给研究混杂生物模型增加了难度.文中针对所提出的混杂捕食生物经济模型,研究了该模型发生鞍结分岔,奇异诱导分岔和Neimark-Sacker分岔的分岔条件.并在此基础上,设计了状态反馈控制器,该控制器可以有效消除分岔现象的发生,并使得系统在给定经济利益条件下能够保持稳定.
By utilizing nonlinear dynamical system theory, differential-algebraic system theory and its associated control theory, this dissertation investigates dynamical complexity for some classes of biological dynamical systems, including normal bio-logical dynamical system, singular biological economic system and hybrid biological economic system. The main content consists of stability, bifurcation, chaos, impul-sive phenomenon, impulsive state feedback control, chaotic stability control and so on. This dissertation is organized as follows:
(1) The current research status and development of biological dynamical sys-tem are introduced, such as epidemic dynamics, plankton dynamics and population dynamics. In particular, some classes of biological dynamical systems relating to this dissertation are enumerated. At the same time, the current research status of these biological models is presented. Furthermore, the current research status of singular biological dynamical systems is also introduced.
(2) This dissertation investigates the complex dynamics of two classes of prey-predator systems with Beddington-DeAngelis functional response and har-vest. Functional response represents the relationship between prey and predator. Beddington-DeAngelis functional response has the advantage of HollingⅡand ratio-dependent models and avoids their disadvantage. Hence, it is close to realistic bi-ological relationship. However, considering the Beddington-DeAngelis functional response, the reports of discrete prey-predator model and biological model with im-pulsive state feedback control are few. At first, since some species have no overlap between successive generations, this dissertation investigates bifurcation and control of a discrete harvested prey-predator system with Beddington-DeAngelis functional response. By using center manifold theorem and bifurcation theory, bifurcation conditions for Flip bifurcation and Hopf bifurcation are obtained. Bifurcation dia-grams and Lyapunov exponent plot show the existence of chaos. Moreover, a state delayed feedback control method is proposed to eliminate bifurcation and chaotic phenomena. And biological implications are discussed. On the other hand, a class of Beddington-DeAngelis prey-predator system with harvest and impulsive state feedback control is studied. Based on impulsive control theory, two Poincare maps are obtained. Conditions for existence and stability of periodic solution are also obtained. Computer simulations show that this impulsive system displays a se-ries of complex phenomena, including period-doubling bifurcation, period window and chaotic bands. Compared with impulsive fixed-time control, the superiority of impulsive state feedback control strategy is also exhibited.
(3) Distributed delay is introduced into phytoplankton-zooplankton-fish model. This dissertation studies a plankton-fish model with distributed delay in the con-text of marine plankton interaction together with predation by planktotrophic fish. Generally speaking, predator need take a period of time to convert the prey into its growth, which is defined with gestation period. This phenomenon is expressed as time delay in theory. At present, most of plankton models are focus on dis-crete time delay. However, biological growth is always an endless and accumulated process, which is related with the entire past history. Therefore, the research on distributed delay is significant. By using the normal form and center manifold the-ory, the research results show that the system occurs Hopf bifurcation and exists periodic solutions when the average time delay increases through critical values. In addition, stability, direction and other properties of bifurcating periodic solutions are derived. The related biological implications are given.
(4) Time delay, stage structure and diffusion behavior are introduced into sin-gular biological economic system, which enriches the theory on singular biological system. By using differential-algebraic system theory and bifurcation theory, com-plex dynamical behaviors of two classes of singular biological economic systems are analyzed, which are singular delayed prey-predator economic model with stage structure and singular biological economic model with diffusion. The research re-sult shows that the two systems occur singularity induced bifurcation when the economic interest of harvesting increases through zero. Singularity induced bifur-cation implies that biological population expanses rapidly and struggles for limited nature resource, which destroys ecological balance. On the other hand, time delay implies that the predator takes a period of time to convert the food into its growth when predating behavior is happened. Long time delay can result in population fluctuation and destroy sustainable development of all population in the ecosystem. However, although immature prey is available in abundance, the overflow of preda-tor population will not happen due to time delay. In addition, considering migration behavior of biological population between different survival environment, the effect of diffusion is studied. When the diffusion coefficient crosses a critical value, there will be a stability switch and periodic fluctuation occurs. And greater diffusion rate is favorable for persistence of biological population.
(5) For the first time, this dissertation proposes a hybrid prey-predator eco-nomic model, which is formulated by differential-difference-algebraic equations. The dynamics of prey population is governed by differential equation, predator popula-tion governed by difference equation and economic theory by algebraic equation. At present, there are many results on the complex dynamical behavior for normal biological dynamical systems. At the same time, some literature relating to singu-lar biological dynamical system also can be found. However, the reports of hybrid biological systems are few. From the aspect of model formulation, hybrid systems, consisting of differential equations, difference equations and algebraic equations, are more complex than normal systems and singular systems. In addition, hybrid sys-tems have quite different types of bifurcations at the intersampling instants and sampling instants. These factors increase researching difficult on hybrid biological system. By analysis and computation, bifurcation conditions of saddle-node bifurca-tion, singularity induced bifurcation and Neimark-Sacker bifurcation are obtained. Moreover, a state feedback controller is designed so that bifurcation behavior can be eliminated and biological population can be driven to steady states.
(1)介绍了生物动力系统的研究现状和进展,主要包括传染病动力学的研究进展,浮游生态系统动力学的研究进展和种群动力学的研究进展.特别地,列举出了与本文工作相关的若干类生物种群动力系统的模型,并介绍了目前这些生物模型的研究现状.此外,还介绍了广义生物动力系统的研究现状.
(2)本文研究了两类具有Beddington-DeAngelis功能性反应和收获的捕食生物模型的动态复杂性.功能反应函数表示捕食者与食饵的相互作用关系Beddington-DeAngelis功能反应兼具HollingⅡ类功能性反应和比率型功能性反应的特点,而且避免了它们的缺陷,因而更为接近现实.但目前,还没有关于带有Beddington-DeAngelis功能性反应的离散捕食模型和脉冲状态反馈作用下这类捕食模型的分岔及周期解方面的研究.本文的研究填补了这方面研究的空白.考虑到生物种群世代不重叠性,首先研究了该类离散捕食生物模型的动态复杂性和控制,利用中心流形定理和分岔理论,获得了系统出现Flip分岔和Hopf分岔的分岔条件,通过绘制分岔图和Lyapunov(?)指数图,证明了混沌现象的存在性,进一步地,设计的状态延时反馈控制器有效消除了分岔和混沌现象,并从生物意义角度给出了该控制器的合理解释.另一方面,讨论了该类生物模型在脉冲状态反馈控制下的动态复杂性,应用脉冲控制理论,得到了两个Poincare映射,给出了周期解的存在及其稳定条件,同时分岔图显示出系统复杂的动态行为,包括倍周期分岔,混沌吸引子和周期窗等；此外,通过将固定时刻脉冲控制与脉冲状态反馈控制进行对比,说明脉冲状态反馈控制可以更有效地使生物种群密度保持在稳定区域内.
(3)本文首次将分布式时滞引入到文中的浮游植物-浮游动物-鱼类海洋生态模型中,研究了一类具有分布式时滞的浮游生物-鱼类海洋生态模型的动态复杂性.
一般来说,捕食者种群在捕食结束后并不会立刻将能量转化给下一代,常常有孕育期.数学上往往通过在模型中引进时滞的方法来描述这一现象.目前,大部分有关浮游生态模型的研究都集中考虑了确定性时滞.然而,由于生物的生长过程往往是一个漫长且积累的过程,不是仅仅与过去某一特定时刻有关的,所以分布式时滞的研究是有现实意义和背景的.通过运用中心流形定理和规范形理论,研究结果表明,当平均时滞超过某一临界值时,系统失稳并出现Hopf分岔,且在正平衡点附近出现一族周期解.另外,还得到了分岔周期解的稳定性,方向以及一些相关性质.同时,对相关的生物意义作出了诠释.
(4)本文将时滞和阶段结构,以及扩散行为引入到广义食饵-捕食者生物经济系统中,丰富了广义生物系统的研究.利用广义系统理论和分岔理论,文中主要研究了具有时滞和阶段结构的广义食饵-捕食者生物经济系统,以及具有扩散行为的广义食饵-捕食者生物经济系统的动态复杂性.研究结果表明,当捕获行为的经济利润由负变为正并通过零时,两类系统在正平衡点附近均发生奇异诱导分岔,这意味着某种生物种群密度在短时间内剧增,并导致其对有限生存资源的大量需求和占有,严重破坏了生态的平衡.另一方面,时滞表示捕食者捕获食饵种群后,将这种能量转化为自身能量所需要的时间,研究发现,较长的时间延迟会导致生物种群密度产生波动,破坏生物种群的可持续生长；但是,在幼年种群十分充足的条件下,时滞抑制了捕食者种群的过度繁殖.除此以外,考虑到生物种群在其生存区域间的迁徙活动,本文又研究了扩散行为对系统动态性的影响,结果表明,当扩散系数超过某一阈值时,生物种群系统的稳定性将会发生改变,生物种群密度出现周期振荡现象,且较大的扩散率有利于种群的持久生存.
(5)本文首次建立了一个由微分-差分-代数方程构成的混杂捕食生物经济模型,其中微分方程是由食饵种群的动态行为确立的,差分方程是由捕食者种群的动态特性确立的,经济利润则确定了代数方程.目前,对正常生物动力系统的复杂动态行为的研究已经取得了很多可喜的成果,同时,也可以找到一些研究广义生物动力系统的相关文献.但是,对于混杂生物系统复杂动态性的研究却很少见.从建模上看,混杂系统比正常系统和广义系统都复杂得多,它主要由微分,差分和代数方程共同构成,另外,研究过程中要分别考虑采样间隔和采样时刻两种情况对系统的影响.这些都给研究混杂生物模型增加了难度.文中针对所提出的混杂捕食生物经济模型,研究了该模型发生鞍结分岔,奇异诱导分岔和Neimark-Sacker分岔的分岔条件.并在此基础上,设计了状态反馈控制器,该控制器可以有效消除分岔现象的发生,并使得系统在给定经济利益条件下能够保持稳定.
By utilizing nonlinear dynamical system theory, differential-algebraic system theory and its associated control theory, this dissertation investigates dynamical complexity for some classes of biological dynamical systems, including normal bio-logical dynamical system, singular biological economic system and hybrid biological economic system. The main content consists of stability, bifurcation, chaos, impul-sive phenomenon, impulsive state feedback control, chaotic stability control and so on. This dissertation is organized as follows:
(1) The current research status and development of biological dynamical sys-tem are introduced, such as epidemic dynamics, plankton dynamics and population dynamics. In particular, some classes of biological dynamical systems relating to this dissertation are enumerated. At the same time, the current research status of these biological models is presented. Furthermore, the current research status of singular biological dynamical systems is also introduced.
(2) This dissertation investigates the complex dynamics of two classes of prey-predator systems with Beddington-DeAngelis functional response and har-vest. Functional response represents the relationship between prey and predator. Beddington-DeAngelis functional response has the advantage of HollingⅡand ratio-dependent models and avoids their disadvantage. Hence, it is close to realistic bi-ological relationship. However, considering the Beddington-DeAngelis functional response, the reports of discrete prey-predator model and biological model with im-pulsive state feedback control are few. At first, since some species have no overlap between successive generations, this dissertation investigates bifurcation and control of a discrete harvested prey-predator system with Beddington-DeAngelis functional response. By using center manifold theorem and bifurcation theory, bifurcation conditions for Flip bifurcation and Hopf bifurcation are obtained. Bifurcation dia-grams and Lyapunov exponent plot show the existence of chaos. Moreover, a state delayed feedback control method is proposed to eliminate bifurcation and chaotic phenomena. And biological implications are discussed. On the other hand, a class of Beddington-DeAngelis prey-predator system with harvest and impulsive state feedback control is studied. Based on impulsive control theory, two Poincare maps are obtained. Conditions for existence and stability of periodic solution are also obtained. Computer simulations show that this impulsive system displays a se-ries of complex phenomena, including period-doubling bifurcation, period window and chaotic bands. Compared with impulsive fixed-time control, the superiority of impulsive state feedback control strategy is also exhibited.
(3) Distributed delay is introduced into phytoplankton-zooplankton-fish model. This dissertation studies a plankton-fish model with distributed delay in the con-text of marine plankton interaction together with predation by planktotrophic fish. Generally speaking, predator need take a period of time to convert the prey into its growth, which is defined with gestation period. This phenomenon is expressed as time delay in theory. At present, most of plankton models are focus on dis-crete time delay. However, biological growth is always an endless and accumulated process, which is related with the entire past history. Therefore, the research on distributed delay is significant. By using the normal form and center manifold the-ory, the research results show that the system occurs Hopf bifurcation and exists periodic solutions when the average time delay increases through critical values. In addition, stability, direction and other properties of bifurcating periodic solutions are derived. The related biological implications are given.
(4) Time delay, stage structure and diffusion behavior are introduced into sin-gular biological economic system, which enriches the theory on singular biological system. By using differential-algebraic system theory and bifurcation theory, com-plex dynamical behaviors of two classes of singular biological economic systems are analyzed, which are singular delayed prey-predator economic model with stage structure and singular biological economic model with diffusion. The research re-sult shows that the two systems occur singularity induced bifurcation when the economic interest of harvesting increases through zero. Singularity induced bifur-cation implies that biological population expanses rapidly and struggles for limited nature resource, which destroys ecological balance. On the other hand, time delay implies that the predator takes a period of time to convert the food into its growth when predating behavior is happened. Long time delay can result in population fluctuation and destroy sustainable development of all population in the ecosystem. However, although immature prey is available in abundance, the overflow of preda-tor population will not happen due to time delay. In addition, considering migration behavior of biological population between different survival environment, the effect of diffusion is studied. When the diffusion coefficient crosses a critical value, there will be a stability switch and periodic fluctuation occurs. And greater diffusion rate is favorable for persistence of biological population.
(5) For the first time, this dissertation proposes a hybrid prey-predator eco-nomic model, which is formulated by differential-difference-algebraic equations. The dynamics of prey population is governed by differential equation, predator popula-tion governed by difference equation and economic theory by algebraic equation. At present, there are many results on the complex dynamical behavior for normal biological dynamical systems. At the same time, some literature relating to singu-lar biological dynamical system also can be found. However, the reports of hybrid biological systems are few. From the aspect of model formulation, hybrid systems, consisting of differential equations, difference equations and algebraic equations, are more complex than normal systems and singular systems. In addition, hybrid sys-tems have quite different types of bifurcations at the intersampling instants and sampling instants. These factors increase researching difficult on hybrid biological system. By analysis and computation, bifurcation conditions of saddle-node bifurca-tion, singularity induced bifurcation and Neimark-Sacker bifurcation are obtained. Moreover, a state feedback controller is designed so that bifurcation behavior can be eliminated and biological population can be driven to steady states.
引文
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