A Delayed Chemostat Model with Impulsive Diffusion and Input on Nutrients

Jianjun Jiao,Shaohong Cai

doi:10.1155/2009/514240

Abstract

A chemostat model with delayed response in growth and impulsive diffusion and input on nutrients is considered. Using the discrete dynamical system determined by the stroboscopic map, we obtain a microorganism-extinction periodic solution. Further, it is globally attractive. The permanent condition of the investigated system is also obtained by the theory on impulsive delay differential equation. Finally, numerical analysis is inserted to illustrate dynamical behaviors of the chemostat system. Our results reveal that the impulsive input amount of nutrients plays an important role on the outcome of the chemostat. Our results provide strategy basis for biochemical reaction management.

Highlights

The theory of impulsive differential equation appears as a natural description of several real processes subject to certain perturbations whose duration is negligible in comparison with the duration of the process
By the comparison theorem for impulsive differential equation 1, we know that there exists t1 > t0 τ1 such that the inequality s2 t ≥ v2∗ t − ε holds for t ≥ t1, s2 t ≥ v2∗ − ε for all t ≥ t1
We investigate a delayed chemostat model with impulsive diffusion and input on nutrients

Summary

Introduction

The theory of impulsive differential equation appears as a natural description of several real processes subject to certain perturbations whose duration is negligible in comparison with the duration of the process. Ruan 32 proposed a diffusive plankton-nutrient interaction model with delayed nutrient recycling and delayed growth response and studied Turing instability and the existence of travelling wave solutions They did not investigate a chemostat model with delayed response in growth and impulsive diffusion on nutrients. A chemostat model with delayed response in growth and impulsive diffusion on nutrients is investigated, we will obtain a microorganism-extinction periodic solution.

The Model

The Lemmas

Discussion