The effect of toxins on the plasmid-bearing and plasmid-free model in the unstirred chemostat

Hua Nie; Jianhua Wu

doi:10.3934/dcds.2012.32.303

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2012, Volume 32, Issue 1: 303-329. Doi: 10.3934/dcds.2012.32.303

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The effect of toxins on the plasmid-bearing and plasmid-free model in the unstirred chemostat

Hua Nie^1, and
Jianhua Wu^2,

1.
College of Mathematics and Information Science, Shaanxi Normal University, Xi’an, Shaanxi 710062
2.
College of Mathematics and Information Science, Shaanxi Normal University, Xi’an, Shaanxi 710119

Received: June 30, 2010

Revised: May 31, 2011

Published: December 2011

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Abstract

This paper deals with an unstirred chemostat model of competition between plasmid-bearing and plasmid-free organisms when the plasmid-bearing organism produces toxins. The toxins are lethal to the plasmid-free organism, which leads to the conservation principle cannot be applied, and the resulting dynamical system is described by three nonlinear partial differential equations and is not monotone. First, the existence and multiplicity of the positive steady-state solutions are determined by bifurcation theory and degree theory. Second, the effects of the toxins are considered by perturbation technique. The results show that if the parameter $r$, which measures the effect of the toxins, is sufficiently large, this model has at least two positive solutions provided that the maximal growth rate $a$ of $u$ lies in a certain range; and has only a unique asymptotically stable positive solution when $a$ belongs to another range.

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Mathematics Subject Classification: Primary: 35K57, 35J55; Secondary: 92B05.

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