Stability Analysis And Bifurcation Control Of Generalized Biological Dynamical Model Of Algal Blooms

Since the beginning of the20th century, with continuous exploration and research of many ecologists, biological science has developed by leaps and bounds. In recent years, a series of edge disciplines related to biological science are emerged successively, among which biological mathematics is one of the youngest discipline. This 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 biological dynamical system of algal blooms is studied, in which non-toxic phytoplankton, toxin producing phytoplankton and zooplankton with the economic value are considered. In accordance with 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 arc summarized as follows:(1) Backgrounds of the research work in the paper are introduced. Firstly, the biological dynamical system and its research status are presented, and then the researches of red tides at home and abroad, occurrence characteristics, the mechanism of occurrence, the harm to the human beings and forecasting methods are introduced.(2) Some basic knowledge of this thesis is introduced. Firstly, the related theory of stability in control theory is presented, and then the stability theory and bifurcation theory in nonlinear system are introduced.(3) Generalized biological dynamical system with no time delay is investigated in this dissertation. The effect of harvesting on zooplankton is considered for the first time, and a generalized biological dynamical model of algal blooms is established. In the absence of economic interest of harvesting, i.e., the phenomenon of biological economic equilibrium, the every equilibrium of the system is obtained. Firstly, the differential-algebraic system and related lemmas are established, and then using singular system theory and bifurcation theory, the existence of transcritical bifurcation and singularity induced bifurcation of the system are studied. Finally, a state feedback controller is designed by using control method in singular system.(4) In real life, people are always interested in the situation of positive economic profits, so research work focuses on the influence of delay to the system. The obtained research result is as follows. The system is stable in a certain range of delay, but beyond the threshold, Hopf bifurcation will appear.