Complexity Analysis And Control For Some Kinds Of Biological Dynamical Systems

Since 1950s, several subjects have interpreted the essence of motion of life from different perspective of viewpoints. With the involution of mathematics and mechanics into biology, biological dynamics is proposed, which not only investigates the relationship between biological population and environment, but also the mechanism of coexistence and interaction between the given biological population and related population. Generally, biological dynamical system is a typical nonlinear complex system. Although some research results about its dynamical behavior and control problem have been obtained, there are still plenty of problems to be dealt with.In this dissertation, by utilizing the bifurcation theory, chaos theory and its associated control theory, dynamical behavior and control problems for some kinds of biological dynamical systems are investigated. The optimal harvest problem for prey-predator system with time delay and stage structure, complexity analysis and control for prey-predator system with periodic perturbation and stage structure, bifurcation, chaos and its associated control for singular biological dynamical system, etc.. The main work of this dissertation is as follows:(1) The current research status of biological dynamics and some biological dynamical system mainly investigated in this dissertation are introduced, which consist of prey-predator system with stage structure, biological dynamical system with competition, prey-predator system with infective disease in prey population, singular biological dynamical system and chaos in biological dynamical system. Furthermore, the control problem in biological dynamical system and its current research status is introduced.(2) The effect of harvesting on the asymptotical stability of prey-predator system with time delay and stage structure is discussed. With the purpose of guaranteeing the sustainable development of population and maximum economic interest of harvesting, taxation is used to regulate harvest effort. By using Pontryagin maximum principle, the optimal harvest effort amount and its associated taxation rate are obtained. 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. Furthermore, the effect of time delay on stability of biological dynamical system is studied. In the absence of time delay, biological dynamical system is asymptotical stable around interior equilibrium; on the other hand, biological dynamical system becomes unstable when time delay increases across critical value, and then Hopf bifurcation occurs. (3) By virtue of qualitative and quantitative methods, complex dynamical behavior of prey-predator system with periodic perturbation and stage structure is analyzed. The obtained research results are as follows. With the increase of amplitude of periodic perturbation, there are some complex dynamical behavior such as periodicy, quasiperiodicy and chaos. Furthermore, there are also many nonlinear dynamical behavior in the chaotic zone such as tangent bifurcation, periodicy and quasiperiodicy, which reveals that there exist abundant stages and self-similar structure. Based on Pyragas method, a feedback controller is designed. By incorporating control into prey and immature predator population, the chaotic dynamical responses can be stabilized to periodic orbits, and the chaotic state is converted to periodic state.(4) There are some progress on the complex dynamical behavior and control for singular biological dynamical system. However, such progress is far less than those on normal biological system. The above mentioned research is still in its start-up stage, there are quite a lot of problems to be dealt with. Based on particle swarm optimization, a new algorithm is designed to reduce the high dimensional singular biological dynamical system into low dimensional singular biological dynamical system. By using this algorithm, the reduced order singular system can approximate the original singular system as well as reserve some structure property of the original singular system. It is beneficial to reducing the high dimensional singular biological dynamical system and investigating the dynamical behavior and control problem of high dimensional singular biological dynamical system by studying the reduced order singular system.(5) The ultimate goal of harvesting is to acquire the use value of biological resource, and then obtain the economic interest. Influenced by many factors of market economy, harvest effort is generally adjusted based on the change of economic interest of harvesting. Consequently, it is realistic to investigate the dynamical behavior of biological dynamical system with the variation of economic interest of harvesting. By using singular system theory and bifurcation theory, the local bifurcation of the singular biological dynamical system with time delay, stage structure and competition around interior equilibrium is discussed. The research results reveal that when time delay is zero, there is a singularity induced bifurcation when the economic interest of harvesting increases through zero; such system is unstable around interior equilibrium in the case of positive economic interest of harvesting. In order to eliminate the singularity induced bifurcation in the case of zero economic interest of harvesting, a state feedback controller is designed. Moreover, the range for the economic interest of harvesting is studied, which may guarantee the sustainable development of all population in the ecosystem, and then a state feedback controller is also designed to stabilize the singular biological system around the interior equilibrium in the case of given economic interest of harvesting. Furthermore, the stability switch due to the variation of time delay is studied. When the economic interest of harvesting is positive and time delay crosses critical value, there will be a stability switch and a Hopf bifurcation occurs. Compared with the normal biological dynamical system, the critical value for stability switch of singular biological dynamical system becomes less. Finally, the application of the above research into the field of harmful algal bloom and control theory is introduced.(6) By using singular system theory and bifurcation theory, the local bifurcation of the singular biological dynamical system with infective disease in prey population around interior equilibrium is discussed. By virtue of qualitative and quantitative methods, complex dynamical behavior of this system is also analyzed. The research results show that when time delay is zero, there is a singularity induced bifurcation when the economic interest of harvesting increases through zero. On the other hand, in the absence of economic interest of harvesting, s.t. the phenomenon of biological economic equilibrium, there exist complex dynamical behavior with the variation of infective rate of infective disease in the prey population, which consists of periodicy, chaos and quasiperiodicy. The infective rate of infective disease in the prey population should be controlled within certain range, so that the sustainable development of population in such system can be maintained.