Complexity Analysis And Control For Some Kinds Of Biology Systems

This dissertation studied transformation rule of biological population by means of establishing mathematical models based on the methods of dynamics. In combination of the theories and methods of control science and system science, researching the laws of population development and forecasting the trend of the change, have become very important research problems which are of the theoretical and practical meaning. The aims of studying the models are to interpret the structural characteristics of population, in order to control the biological population effectively and offer the evidence to the correlative department for making policy to protect and utilize the biological resource better.Biology dynamical systems are a representative kind of nonlinear complex system. There have been some researches about their dynamical characteristic and control problem, but there also exist many problems that have not been solved. In this dissertation, by the nonlinear dynamical systems theory, the theory for chaos and bifurcation, the main research content consists of the complex dynamical characteristics of biological population systems, bio-economic systems, such as stability, bifurcation, phenomena of chaos, impulse, and state feedback impulsive control which aims at systems for prey-predator models, infect models and bio-economic models.The main work of this dissertation is as follows:(1) The problems of chaotic control are studied for a class of SIR epidemic system with seasonality in this dissertation. The fact that the SIR model possesses chaotic phenomena has been demonstrated by computing the system's Lyapunov exponents. It means the outbreak of the disease is unpredicted. Furthermore, the density of infected individuals is taken as the output of the system, because it may be obtained by observation and medical diagnosis. The output of the system can track an ideal goal by feedback tracking technique. As a result, the density of infected individuals will asymptotically tend to zero. That means the disease will tend to be eliminated. This idea may be realized that susceptible individuals can be free from infection through injecting vaccines or eating prophylactic drug. Finally, numerical simulations are given to illustrate that the feedback control input is effective.(2) In this dissertation, a Beddington-DeAngelis prey-predator system with continuous harvesting and impulsive state feedback control is studied. Based on two Poincare maps which are structured, the predator-free periodic solution is obtained and locally asymptotical stability condition is established. Bifurcation diagram show the complex behaviors including period-doubling bifurcation and cascade, periodic window and chaotic bands. Finally, numerical simulations show that impulsive state feedback control method is more effective than impulsive fix-time control for keeping the population of prey below an appropriate threshold value.(3) This dissertation proposes a prey predator model system with selective harvest effort on predator, where prey refuge is considered and taxation is utilized as a control instrument to protect the population from overexploitation. Conditions which influence positiveness and boundedness of solutions of model system are studied. Local stability analysis around all equilibria of model system is discussed due to variation of prey refuge and taxation. Furthermore, global stability of model system at the positive equilibrium is investigated based on Bendixson criterion, which is theoretically beneficial to studying the coexistence and interaction mechanism of population within harvested ecosystem. By using maximum principle of optimal control method, an optimal taxation policy is derived to ensure the sustainable development of biological resource and prosperous commercial harvesting. Numerical simulations are carried out to show the consistency with theoretical analysis.(4) In this dissertation, a bio-economic model is proposed to investigate dynamics of the effects of a stage-structured prey-predator system with gestation delay. At first, the theoretical analysis reveals that gestation delay is responsible for the stability switch of the system. In absence of gestation delay, the interior equilibrium is locally asymptotical stable around interior equilibrium; In the case of gestation delay, a phenomenon of Hopf bifurcation occurs as the gestation delay increases through a certain threshold. The effect of the economic interest of harvesting on the population density and asymptotical stability of predator system is discussed. An optimal harvesting policy with taxation with which the total cost of harvest per unit effort must be equal to the discounted value of the future price at the steady state level is derived. The research results show that taxation is a constructive measure, which may effectively control the harvest effort and protect the biological resource in the ecosystem from overharvesting. Finally, numerical simulations are carried out to show consistency with theoretical analysis.