Researches On Complexity For Some Classes Of Biological Dynamical Systems

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.