Uniqueness and stability for coexistence solutions of the unstirred chemostat model

Abstract

This article deals with the uniqueness and stability of coexistence solutions of a basic N-dimensional competition model in the unstirred chemostat by Lyapunov–Schmidt procedure and perturbation technique. It turns out that if the parameter G ≠ 0, which is given in Theorem 1.1, this model has a unique coexistence solution provided that the maximal growth rates a, b of u, v, respectively, lie in a certain range. Moreover, the unique coexistence solution is globally asymptotically stable if G > 0, while it is unstable if G < 0. In the later case, the semitrivial equilibria are both stable.

Keywords:

• Chemostat
• Uniqueness
• Stability
• Lyapunov–Schmidt procedure
• Perturbation technique

AMS Subject Classifications:

• 35K55
• 35K57
• 35B40

Acknowledgements

The authors would like thank to Prof Sze-Bi Hsu who gave them some important references on this model. They would also like to thank the anonymous referees for their valuable suggestions leading to an improvement of this article. The work is supported by the Natural Science Foundation of China (No. 10971124), the Ph.D. specialized grant of Ministry of Education of China (No. 200807180004) and Natural Science Foundation of Shaanxi Province (No. 2009JQ1007).