The Hopf Bifurcation Of The Chemostat With Delay And Simplified Holling Type-IV Response Function

Chemostat is a simplified model of lakes and it has great significance in biology There have been many results about chemostat including many types of equations suchas ODE, PDE, RFDE etc. This paper is devoted to discuss the bifurcation of a chemostat with delay and simplified Holling type-IV response function.This paper is composed of three sections. In the first section i.e. the introduction, we introduce the historical background of the chemostat and the system which will be discussed and the main results of this paper. We also explain why the simplified Holling type-IV response function can be used in the system. In the second section, we consider of the static bifurcation of the system and display the parameter domain partitioned according to the number of interior stationary points. We also give the diagram.In the third section, we are devoted to analyze the characteristic equations of the system. The results about the interior stationary point E₁ are simply stated since they are similar with the results in paper [5] and [6], We mainly discuss the interior stationary point E₂ in three conditions. Two methods are used in the process. One is provided by [6], the other is the soul of this paper. The monotonicity of functions and the properties of inverse function and function of functions are used. These properties are most important for solving the problem. Both of them help us obtain the whole bifurcation diagram about E₂ in parameter plane (r, (3). Furthermore,the second method provides a new idea to the partition of the roots of characteric equations of RFDE. Finally we narrate the feature of the diagram.