Bifurcation from a double eigenvalue in the unstirred chemostat

Abstract

This article concerns a competition model in the unstirred chemostat. The bifurcation solution from a double eigenvalue is obtained. We see that this bifurcation solution connects the positive solution from the semitrivial solution (θ _a , 0) with that from the other semitrivial solution (0, θ _b ). Moreover, the asymptotic stability of the positive solution corresponding to this bifurcation is derived under certain conditions. The method we used here is based on spectral analysis, comparison principle, bifurcation theory and Lyapunov–Schmidt procedure.

Keywords:

• double eigenvalue
• bifurcation
• chemostat
• stability

AMS Subject Classification:

• 35K57

Acknowledgements

The work is supported by the Natural Science Foundation of China (10971124), the PhD specialized grant of Ministry of Education of China (200807180004) and the Foundation of Central University of China.