Bifurcation Analysis And Control Of Generalized Bio-system Based On Environmental Pollution

Since the beginning of the 20th century, with continuous research and exploration of many professors, biological science has got a great achievement. In recent years, a series of disciplines related to biological science are emerged successively. Biological mathematics is a discipline between biology and mathematics, the subject studies and solves the problems in biology from the view of mathematics and mechanics, and combines the dynamical problems and mathematical thinking methods organically.In this thesis, generalized dynamical model based on single species in polluted environment is studied. According to the nonlinear dynamical system theory, singular system theory and control theory, the stability of every equilibrium points in the model is investigated, also the dynamical behavior of the system near the positive equilibrium point and its control problem are studied.The main works of this paper are summarized as follows:(1) Backgrounds of the research work in the paper are introduced. Firstly, we present the current research status of mathematical biology and ecological system. And then, we introduce the development history and research status of generalized systems. Finally, we introduced the research content and research methods of this article.(2) Some basic knowledge of this thesis is introduced. Which include the related theory of stability in control theory, the bifurcation theory in nonlinear system, nonlinear feedback control theory and so on.(3) Generalized dynamical model of single species in polluted environment with no time delay is investigated in this section. For the first time, we extend the single species model to a generalized system. And the dynamic behavior of the system near the positive equilibrium point is analysis, which we also call it bifurcation and control problem. We present the possibility of the occurrence of the singular induced bifurcation (SIB) and transcritical bifurcation near the equilibrium point.(4) Generalized dynamical model of single species in polluted environment with time delay is investigated in this section. We study the dynamic characteristics of the generalized biological dynamical systems described by differential algebraic equations near the positive equilibrium point. Through the analysis, we know that the system is stable if the time delay is limited in a appropriate range, but Hopf bifurcation will occur if the time delay pass through the critical value. Which means the existence of the time delay will lead to the unstability of the bio-system, or even the extinction of the population.(5) we consider the relationship between artificial management and the economic value. We investigate the bifurcation phenomenon of the new generalized system when the economic profit is zero, and design the controller of the system. Finally, we got the conditions and the realistic significance to maintain the stability of the generalized system.