November  2010, 9(6): 1639-1651. doi: 10.3934/cpaa.2010.9.1639

Global dynamics of the periodic un-stirred chemostat with a toxin-producing competitor

1. 

Department of Mathematics, Beijing Institute of Technology, Beijing 100081, China

2. 

Department of Mathematics, South China Normal University, Guangzhou, Guangdong, 510631, China

Received  May 2008 Revised  August 2010 Published  August 2010

This paper is concerned with the un-stirred chemostat with a toxin-producing competitor. The novelties of the modified model are the periodicity appearing in the boundary conditions, the different diffusive coefficients of the nutrient and the microorganisms, and some kinds of death rates. Both uniform persistence and global extinction of the microorganisms are established under suitable conditions in terms of principal eigenvalues of scalar periodic parabolic eigenvalue problems. Our result implies that the toxin inhibits the sensitive microorganism indeed. The techniques includes the theories of asymptotic periodic semi-flows, uniform persistence and the perturbation of global attractor.
Citation: Yifu Wang, Jingxue Yin. Global dynamics of the periodic un-stirred chemostat with a toxin-producing competitor. Communications on Pure and Applied Analysis, 2010, 9 (6) : 1639-1651. doi: 10.3934/cpaa.2010.9.1639
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