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Volume 20,     Number 2,     Summer 2012

 

A CHEMOSTAT MODEL AND ITS DISCRETE
ANALOGUE: OSCILLATORY COEXISTENCE
INDUCED BY DELAYED FEEDBACK CONTROL
HONGYING SHU, LIN WANG AND JUNJIE WEI

Abstract. Criteria are derived to determine the direc- tion and stability of Hopf bifurcation for a chemostat model with delayed feedback control of dilution rate. Due to the delayed feedback control, two organisms can coexist in an oscillatory fashion when they are competing for a single nutrient source. The model's discrete analogue is then derived via the Euler's method. For the resulting discrete model, stability analysis of the positive equilibrium and Neimark-Sacker bifurcation analysis are carried out by analyzing the associated characteristic equation. The direction and stability of Neimark-Sacker bifurcation are also determined by using the normal form method and center manifold theorem. Numerical simulations are carried out to illustrate the analytical results.

 

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