Complexity Analysis For Some Classes Of Biological Dynamical Systems

For some classes of biological dynamical systems, this thesis studies their dynamical complexity by utilizing nonlinear dynamical systems theory, differential algebraic systems theory and its associated control theory. These nonlinear dynamical systems include single population systems which have critical depensation characteristic, singular biological dynamical systems with Allee effect in predator and singular biological dynamical systems with Allee effect and infectious disease in prey. The main work is to study stability, bifurcation phenomenon, state feedback control of these systems by the theory we mentioned above. The main work of this thesis is as follows:(1) The current research status of biological dynamical systems, biological dynamical systems mainly investigated in this dissertation and singular biological dynamical systems are introduced. Then introduce some basics knowledge which is used in this thesis.(2) First, investigate single population dynamical systems which have critical depensation, in case of capture, analyze the systems’equilibrium point and stability. When apply tax to control the capture behavior, by analyze, we obtain optimal regulate tax rate.(3) Second, study the prey-predator systems that the predator has Allee effect. In case of capture, analyze the bifurcation behavior of the systems around the positive equilibrium point. Through analysis, we know that the systems occur singularity induced bifurcation around the positive equilibrium point. A state feedback controller is designed in order to eliminate the singularity induced bifurcation, that is to say the systems recover stability.(4) Last, in order to get the affect that Allee effect bring to the systems, construct a prey-predator systems with Allee effect and infectious disease, in case of capture, we study the presence and number of the equilibrium point. The equilibrium points are respectively from the systems that does not have Allee effect and have Allee effect. From the study we obtain that Allee effects can impact the existence of the equilibrium point. We obtain that the systems occur singularity induced bifurcation around the positive equilibrium point in certain effect parameters, that is to say, the stability of the systems is affected. A state feedback controller is designed to eliminate the bifurcation, so the systems recover stability.