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

Yifu Wang; Jingxue Yin

doi:10.3934/cpaa.2010.9.1639

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2010, Volume 9, Issue 6: 1639-1651. Doi: 10.3934/cpaa.2010.9.1639

This issue Previous Article Existence and uniqueness of solutions to an aggregation equation with degenerate diffusion Next Article On the upper semicontinuity of pullback attractors with applications to plate equations

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

Yifu Wang^1, and
Jingxue Yin^2,

1.
Department of Mathematics, Beijing Institute of Technology, Beijing 100081
2.
Department of Mathematics, South China Normal University, Guangzhou, Guangdong, 510631

Received: May 2008

Revised: August 2010

Published: November 2010

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Abstract

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.

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Mathematics Subject Classification: Primary: 35K57, 35K40, 35B10; Secondary: 35Q92, 92B05.

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