# American Institute of Mathematical Sciences

November  2002, 2(4): 495-520. doi: 10.3934/dcdsb.2002.2.495

## Analysis of a chemostat model for bacteria and virulent bacteriophage

 1 Istituto de Biomatematica, Università di Urbino, I-61029 Urbino, Italy, Italy 2 Department of Mathematics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China

Received  June 2001 Revised  May 2002 Published  August 2002

The purpose of this paper is to study the mathematical properties of the solutions of a model for bacteria and virulent bacteriophage system in a chemostat. A general model was first proposed by Levin, Stewart and Chao [13] and then, a specific one, by Lenski and Levin [12]. The numerical simulations come from the experimental data referred in [12,13]. In our Knowledge the analysis presented herefollowing is the first mathematical attempt to analyse the model of bacteria and virulent bacteriophage and presents two fresh frontiers: 1) the modeling of delay (latency period) incorporating the realistic through time death rate in linear stability analysis brings to characteristic equations with delay dependent parameters for which only recently Beretta and Kuang [5] provided a geometric stability switch criterion which application is presented along the paper; 2) the modelling of the dynamics through three full delay stages can be reduced to two using the integral representation for the density of infected bacteria. The basic properties of the model which are investigated are the existence of equilibria, positive invariance and boundedness of solutions and permanence results. Second, using the geometric stability switch criterion in the delay differential system with delay dependent parameters, we present the local asymptotic stability of the equilibria by analyzing the corresponding characteristic equation which coefficients depend on the time delay (the latency period). Numerical simulations are presented to illustrate the results of local stability. Then, we study the global asymptotic stability of the boundary equilibria via Liapunov functional method. Finally, we give a discussion about the model.
Citation: Edoardo Beretta, Fortunata Solimano, Yanbin Tang. Analysis of a chemostat model for bacteria and virulent bacteriophage. Discrete and Continuous Dynamical Systems - B, 2002, 2 (4) : 495-520. doi: 10.3934/dcdsb.2002.2.495
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