Qualitative Analysis Of A Kind Of Differential Equation Of Biochemical Reaction

In the biochemistry dynamics, there is a kind of equationIts mechanism in reaction isHereinto A is original reactant, D and E are ultimate resultant, X and Y are intermediate, k₁, k₂ , k₃, k₄ are reaction rate.There is oscillation in reaction or not depend on the existence of the limit cycle on the system.Also we want to make certain the concentration valve of the original reactant for the oscillation.The first chapter is the introduction, in which we introduce the developing history and the present progress of Biochemistry system, some fundamental definitions and lemmas of dynamics systems such as bifurcation and stability theory that can be used in this paper, and briefly represent the main works of the thesis.The second chapter is the hard core in the paper, which is divided into two sections finished:In the first section, the E_{α,β;p,q} system global behavior with functional response are considered whenα= 0,β> 0, p = 3, q≥3 , the quality of the equilibrium is discussed, the conditions of the existence and nonexistence of the limit cycle are provided.In the second section, the uniquely positive quality of the E_{α,β;p,q} system is considered whenα>0,β>0,p =1, q = 3 , the conditions of the existence and nonexistence of the limit cycle are provided, using Poincare-Bendixson theory and the Hopf bifurcation theory on the system.In the third chapter, the point singularity and existence of limit cycles in cubic system of class are studied. which is the author else other work in graduate study. The point singularity types are judged by using the qualitative theories ,and the existence of limit cycles is given by hopf bifurcation method. Under certain conditions the uniqueness and existence of limit cycles were improved, which improves the previous work.