American Institute of Mathematical Sciences

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

Analysis of a chemostat model for bacteria and virulent bacteriophage

Edoardo Beretta 1, , Fortunata Solimano 1, and  Yanbin Tang 2,

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.
Keywords: bacteria-bacteriophage interaction, global stability., delay equations, Chemostat, local stability.
Mathematics Subject Classification: 34K50, 34D23, 92D2.
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
[1]

Zhun Han, Hal L. Smith. Bacteriophage-resistant and bacteriophage-sensitive bacteria in a chemostat. Mathematical Biosciences & Engineering, 2012, 9 (4) : 737-765. doi: 10.3934/mbe.2012.9.737
[2]

Mohammad Akil, Haidar Badawi, Ali Wehbe. Stability results of a singular local interaction elastic/viscoelastic coupled wave equations with time delay. Communications on Pure and Applied Analysis, 2021, 20 (9) : 2991-3028. doi: 10.3934/cpaa.2021092
[3]

Eugen Stumpf. Local stability analysis of differential equations with state-dependent delay. Discrete and Continuous Dynamical Systems, 2016, 36 (6) : 3445-3461. doi: 10.3934/dcds.2016.36.3445
[4]

Anatoli F. Ivanov, Musa A. Mammadov. Global asymptotic stability in a class of nonlinear differential delay equations. Conference Publications, 2011, 2011 (Special) : 727-736. doi: 10.3934/proc.2011.2011.727
[5]

Zhiqi Lu. Global stability for a chemostat-type model with delayed nutrient recycling. Discrete and Continuous Dynamical Systems - B, 2004, 4 (3) : 663-670. doi: 10.3934/dcdsb.2004.4.663
[6]

Luis Barreira, Claudia Valls. Delay equations and nonuniform exponential stability. Discrete and Continuous Dynamical Systems - S, 2008, 1 (2) : 219-223. doi: 10.3934/dcdss.2008.1.219
[7]

Abraham Solar. Stability of non-monotone and backward waves for delay non-local reaction-diffusion equations. Discrete and Continuous Dynamical Systems, 2019, 39 (10) : 5799-5823. doi: 10.3934/dcds.2019255
[8]

Shi-Liang Wu, Wan-Tong Li, San-Yang Liu. Exponential stability of traveling fronts in monostable reaction-advection-diffusion equations with non-local delay. Discrete and Continuous Dynamical Systems - B, 2012, 17 (1) : 347-366. doi: 10.3934/dcdsb.2012.17.347
[9]

Jan Sieber, Matthias Wolfrum, Mark Lichtner, Serhiy Yanchuk. On the stability of periodic orbits in delay equations with large delay. Discrete and Continuous Dynamical Systems, 2013, 33 (7) : 3109-3134. doi: 10.3934/dcds.2013.33.3109
[10]

E. Cabral Balreira, Saber Elaydi, Rafael Luís. Local stability implies global stability for the planar Ricker competition model. Discrete and Continuous Dynamical Systems - B, 2014, 19 (2) : 323-351. doi: 10.3934/dcdsb.2014.19.323
[11]

Frédéric Mazenc, Michael Malisoff, Patrick D. Leenheer. On the stability of periodic solutions in the perturbed chemostat. Mathematical Biosciences & Engineering, 2007, 4 (2) : 319-338. doi: 10.3934/mbe.2007.4.319
[12]

Tomás Caraballo, Leonid Shaikhet. Stability of delay evolution equations with stochastic perturbations. Communications on Pure and Applied Analysis, 2014, 13 (5) : 2095-2113. doi: 10.3934/cpaa.2014.13.2095
[13]

Leonid Berezansky, Elena Braverman. Stability of linear differential equations with a distributed delay. Communications on Pure and Applied Analysis, 2011, 10 (5) : 1361-1375. doi: 10.3934/cpaa.2011.10.1361
[14]

Jan Čermák, Jana Hrabalová. Delay-dependent stability criteria for neutral delay differential and difference equations. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4577-4588. doi: 10.3934/dcds.2014.34.4577
[15]

Elena Braverman, Sergey Zhukovskiy. Absolute and delay-dependent stability of equations with a distributed delay. Discrete and Continuous Dynamical Systems, 2012, 32 (6) : 2041-2061. doi: 10.3934/dcds.2012.32.2041
[16]

Cui-Ping Cheng, Ruo-Fan An. Global stability of traveling wave fronts in a two-dimensional lattice dynamical system with global interaction. Electronic Research Archive, 2021, 29 (5) : 3535-3550. doi: 10.3934/era.2021051
[17]

C. Connell McCluskey. Global stability of an $SIR$ epidemic model with delay and general nonlinear incidence. Mathematical Biosciences & Engineering, 2010, 7 (4) : 837-850. doi: 10.3934/mbe.2010.7.837
[18]

C. Connell McCluskey. Global stability for an SEIR epidemiological model with varying infectivity and infinite delay. Mathematical Biosciences & Engineering, 2009, 6 (3) : 603-610. doi: 10.3934/mbe.2009.6.603
[19]

Bao-Zhu Guo, Li-Ming Cai. A note for the global stability of a delay differential equation of hepatitis B virus infection. Mathematical Biosciences & Engineering, 2011, 8 (3) : 689-694. doi: 10.3934/mbe.2011.8.689
[20]

Yincui Yan, Wendi Wang. Global stability of a five-dimensional model with immune responses and delay. Discrete and Continuous Dynamical Systems - B, 2012, 17 (1) : 401-416. doi: 10.3934/dcdsb.2012.17.401

2021 Impact Factor: 1.497

Tools

Metrics

  • PDF downloads (87)
  • HTML views (0)
  • Cited by (10)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy