Impulsive Control For A Type Of Chemostat Model

The differential equation model plays a very important role in researches ofthe phenomena in the real world. It enables us to deepen the understanding of theinternal laws in systems from the viewpoint of mathematical theory. In particular,when impulsive phenomenon arises, we can take advantage of the impulsivedifferential equations to reflect the impact of the impulse in the system.Researches on impulsive differential models help a lot on the control andoptimization of such systems.In chapter1, the work of the mathematicians in impulsive differentialequations is described. Moreover, based on the work of the mathematicians anumber of bio-mathematicians carry out a series of theoretical studies onmicrobialculture in Chemostat model.In chapter2, this paper introduces the basic concepts and theorems relatedto impulsive differential systems.In chapter3, some stability theorems on fixed points of the differentialsystem with impulsive control are given, which easily lead to some conclusionsof the Chemostat Model with variable consumption rate. Due to the need ofreality and experiment, it’s necessary to consider the situation where themicrobial concentration is above the threshold. In other words, we need to addthe impulsive control to avoid the unfavorable factors. This paper analyzes theconditions that parameters meet to guarantee the existence of periodic solutions.Periodic solutions mean steady-states where the microbial growth coordinateswith the external impulsive control, ensuring the growth of microorganisms tomeet the experiment and simple implement of the impulsive control.In the end, this paper analyzes the existence of the k-order (k≥2) periodicsolution, since the periodic solution is not necessarily order one. The direction ofthe vector field on the plane helps to analyze the various situations where theimpulsive points may arise. Considering the concept of the order, the k-orderperiodic solution does not exist, as a conclusion.