# American Institute of Mathematical Sciences

• Previous Article
Global analysis of a simple parasite-host model with homoclinic orbits
• MBE Home
• This Issue
• Next Article
A division-dependent compartmental model for computing cell numbers in CFSE-based lymphocyte proliferation assays
2012, 9(4): 737-765. doi: 10.3934/mbe.2012.9.737

## Bacteriophage-resistant and bacteriophage-sensitive bacteria in a chemostat

 1 School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287, United States 2 School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287-1804, United States

Received  December 2011 Revised  June 2012 Published  October 2012

In this paper a mathematical model of the population dynamics of a bacteriophage-sensitive and a bacteriophage-resistant bacteria in a chemostat where the resistant bacteria is an inferior competitor for nutrient is studied. The focus of the study is on persistence and extinction of bacterial strains and bacteriophage.
Citation: 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
##### References:
 [1] E. Beretta and Y. Kuang, Modeling and analysis of a marine bacteriophage infection,, Mathematical Biosciences, 149 (1998), 57.  doi: 10.1016/S0025-5564(97)10015-3.  Google Scholar [2] E. Beretta, H. Sakakibara and Y. Takeuchi, Analysis of a chemostat model for bacteria and bacteriophage,, Vietnam Journal of Mathematics, 30 (2002), 459.   Google Scholar [3] E. Beretta, H. Sakakibara and Y. Takeuchi, Stability analysis of time delayed chemostat models for bacteria and virulent phage,, Dynamical Systems and their Applications in Biology, 36 (2003), 45.   Google Scholar [4] E. Beretta, F. Solimano and Y. Tang, Analysis of a chemostat model for bacteria and virulent bacteriophage,, Discrete and Continuous Dynamical Systems B, 2 (2002), 495.   Google Scholar [5] B. Bohannan and R. Lenski, Effect of resource enrichment on a chemostat community of bacteria and bacteriophage,, Ecology, 78 (1997), 2303.   Google Scholar [6] B. Bohannan and R. Lenski, Linking genetic change to community evolution: insights from studies of bacteria and bacteriophage,, Ecology Letters, 3 (2000), 362.   Google Scholar [7] A. Campbell, Conditions for existence of bacteriophages,, Evolution, 15 (1961), 153.  doi: 10.2307/2406076.  Google Scholar [8] L. Chao, B. Levin and F. Stewart, A complex community in a simple habitat: an experimental study with bacteria and phage,, Ecology, 58 (1977), 369.  doi: 10.2307/1935611.  Google Scholar [9] E. Ellis and M. Delbrück, The growth of bacteriophage,, The Journal of General Physiology, 22 (1939), 365.  doi: 10.1085/jgp.22.3.365.  Google Scholar [10] G. Folland, "Real Analysis: Modern Techniques and Their Applications,'' $2^{nd}$ edition,, Wiley, (1999).   Google Scholar [11] J. Hale and J. Kato, Phase space for retarded equations with infinite delay,, Funkcialaj Ekvacioj, 21 (1978), 11.   Google Scholar [12] J. Hale and S. Verduyn Lunel, "Introduction to Functional Differential Equations,", Springer-Verlag, (1993).   Google Scholar [13] Y. Hino, S. Murakami and T. Naito, "Functional Differential Equations with Infinite Delay,", Springer-Verlag, (1991).   Google Scholar [14] Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics,", Academic Press, (1993).   Google Scholar [15] R. Lenski and B. Levin, Constraints on the coevolution of bacteria and virulent phage: A model, some experiments, and predictions for natural communities,, American Naturalist, 125 (1985), 585.  doi: 10.1086/284364.  Google Scholar [16] B. Levin, F. Stewart and L. Chao, Resource-limited growth, competition, and predation: a model, and experimental studies with bacteria and bacteriophage,, American Naturalist, 111 (1977), 3.   Google Scholar [17] K. Northcott, M. Imran and G. Wolkowicz, Composition in the presence of a virus in an aquatic system: an SIS model in a chemostat,, Journal of Mathematics Biology, 64 (2012), 1043.  doi: 10.1007/s00285-011-0439-z.  Google Scholar [18] W. Ruess and W. Summers, Linearized stability for abstract differential equations with delay,, Journal of Mathematical Analysis and Applications, 198 (1996), 310.  doi: 10.1006/jmaa.1996.0085.  Google Scholar [19] S. Schrag and J. Mittler, Host-parasite coexistence: The role of spatial refuges in stabilizing bacteria-phage interactions,, American Naturalist, 148 (1996), 348.  doi: 10.1086/285929.  Google Scholar [20] H. Smith, "An introduction to Delay Differential Equations with Applications to the Life Sciences,", Springer-Verlag, (2010).   Google Scholar [21] H. Smith and H. Thieme, "Dynamical Systems and Population Persistence,", American Mathematical Society, (2010).   Google Scholar [22] H. Smith and H. Thieme, Persistence of bacteria and phages in a chemostat,, Journal of Mathematical Biology, 64 (2012), 951.  doi: 10.1007/s00285-011-0434-4.  Google Scholar [23] H. Smith and P. Waltman, Perturbation of a globally stable steady state,, Proceeding of the American Mathematical Society, 127 (1999), 447.  doi: 10.1090/S0002-9939-99-04768-1.  Google Scholar [24] H. Smith and P. Waltman, "The Theory of the Chemostat: Dynamics of Microbial Competition,", Cambridge University Press, (1995).   Google Scholar [25] H. Smith and X. Q. Zhao, Robust persistence for semidynamical systems,, Nonlinear Analysis Theory Methods Applications, 47 (2001), 6169.  doi: 10.1016/S0362-546X(01)00678-2.  Google Scholar [26] H. Thieme, Convergence results and a poincaré-bendixson trichotomy for asymptotically autonomous differential equations,, Journal of Mathematical Biology, 30 (1992), 755.   Google Scholar [27] H. Thieme, "Mathematics in Population Biology,", Princeton University Press, (2003).   Google Scholar [28] J. Weitz, H. Hartman and S. Levin, Coevolutionary arms race between bacteria and bacteriophage,, Proceedings of the National Academy of Sciences of the United States of America, 102 (2005), 9535.  doi: 10.1073/pnas.0504062102.  Google Scholar

show all references

##### References:
 [1] E. Beretta and Y. Kuang, Modeling and analysis of a marine bacteriophage infection,, Mathematical Biosciences, 149 (1998), 57.  doi: 10.1016/S0025-5564(97)10015-3.  Google Scholar [2] E. Beretta, H. Sakakibara and Y. Takeuchi, Analysis of a chemostat model for bacteria and bacteriophage,, Vietnam Journal of Mathematics, 30 (2002), 459.   Google Scholar [3] E. Beretta, H. Sakakibara and Y. Takeuchi, Stability analysis of time delayed chemostat models for bacteria and virulent phage,, Dynamical Systems and their Applications in Biology, 36 (2003), 45.   Google Scholar [4] E. Beretta, F. Solimano and Y. Tang, Analysis of a chemostat model for bacteria and virulent bacteriophage,, Discrete and Continuous Dynamical Systems B, 2 (2002), 495.   Google Scholar [5] B. Bohannan and R. Lenski, Effect of resource enrichment on a chemostat community of bacteria and bacteriophage,, Ecology, 78 (1997), 2303.   Google Scholar [6] B. Bohannan and R. Lenski, Linking genetic change to community evolution: insights from studies of bacteria and bacteriophage,, Ecology Letters, 3 (2000), 362.   Google Scholar [7] A. Campbell, Conditions for existence of bacteriophages,, Evolution, 15 (1961), 153.  doi: 10.2307/2406076.  Google Scholar [8] L. Chao, B. Levin and F. Stewart, A complex community in a simple habitat: an experimental study with bacteria and phage,, Ecology, 58 (1977), 369.  doi: 10.2307/1935611.  Google Scholar [9] E. Ellis and M. Delbrück, The growth of bacteriophage,, The Journal of General Physiology, 22 (1939), 365.  doi: 10.1085/jgp.22.3.365.  Google Scholar [10] G. Folland, "Real Analysis: Modern Techniques and Their Applications,'' $2^{nd}$ edition,, Wiley, (1999).   Google Scholar [11] J. Hale and J. Kato, Phase space for retarded equations with infinite delay,, Funkcialaj Ekvacioj, 21 (1978), 11.   Google Scholar [12] J. Hale and S. Verduyn Lunel, "Introduction to Functional Differential Equations,", Springer-Verlag, (1993).   Google Scholar [13] Y. Hino, S. Murakami and T. Naito, "Functional Differential Equations with Infinite Delay,", Springer-Verlag, (1991).   Google Scholar [14] Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics,", Academic Press, (1993).   Google Scholar [15] R. Lenski and B. Levin, Constraints on the coevolution of bacteria and virulent phage: A model, some experiments, and predictions for natural communities,, American Naturalist, 125 (1985), 585.  doi: 10.1086/284364.  Google Scholar [16] B. Levin, F. Stewart and L. Chao, Resource-limited growth, competition, and predation: a model, and experimental studies with bacteria and bacteriophage,, American Naturalist, 111 (1977), 3.   Google Scholar [17] K. Northcott, M. Imran and G. Wolkowicz, Composition in the presence of a virus in an aquatic system: an SIS model in a chemostat,, Journal of Mathematics Biology, 64 (2012), 1043.  doi: 10.1007/s00285-011-0439-z.  Google Scholar [18] W. Ruess and W. Summers, Linearized stability for abstract differential equations with delay,, Journal of Mathematical Analysis and Applications, 198 (1996), 310.  doi: 10.1006/jmaa.1996.0085.  Google Scholar [19] S. Schrag and J. Mittler, Host-parasite coexistence: The role of spatial refuges in stabilizing bacteria-phage interactions,, American Naturalist, 148 (1996), 348.  doi: 10.1086/285929.  Google Scholar [20] H. Smith, "An introduction to Delay Differential Equations with Applications to the Life Sciences,", Springer-Verlag, (2010).   Google Scholar [21] H. Smith and H. Thieme, "Dynamical Systems and Population Persistence,", American Mathematical Society, (2010).   Google Scholar [22] H. Smith and H. Thieme, Persistence of bacteria and phages in a chemostat,, Journal of Mathematical Biology, 64 (2012), 951.  doi: 10.1007/s00285-011-0434-4.  Google Scholar [23] H. Smith and P. Waltman, Perturbation of a globally stable steady state,, Proceeding of the American Mathematical Society, 127 (1999), 447.  doi: 10.1090/S0002-9939-99-04768-1.  Google Scholar [24] H. Smith and P. Waltman, "The Theory of the Chemostat: Dynamics of Microbial Competition,", Cambridge University Press, (1995).   Google Scholar [25] H. Smith and X. Q. Zhao, Robust persistence for semidynamical systems,, Nonlinear Analysis Theory Methods Applications, 47 (2001), 6169.  doi: 10.1016/S0362-546X(01)00678-2.  Google Scholar [26] H. Thieme, Convergence results and a poincaré-bendixson trichotomy for asymptotically autonomous differential equations,, Journal of Mathematical Biology, 30 (1992), 755.   Google Scholar [27] H. Thieme, "Mathematics in Population Biology,", Princeton University Press, (2003).   Google Scholar [28] J. Weitz, H. Hartman and S. Levin, Coevolutionary arms race between bacteria and bacteriophage,, Proceedings of the National Academy of Sciences of the United States of America, 102 (2005), 9535.  doi: 10.1073/pnas.0504062102.  Google Scholar
 [1] Stefan Ruschel, Serhiy Yanchuk. The spectrum of delay differential equations with multiple hierarchical large delays. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 151-175. doi: 10.3934/dcdss.2020321 [2] John Mallet-Paret, Roger D. Nussbaum. Asymptotic homogenization for delay-differential equations and a question of analyticity. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3789-3812. doi: 10.3934/dcds.2020044 [3] Mugen Huang, Moxun Tang, Jianshe Yu, Bo Zheng. A stage structured model of delay differential equations for Aedes mosquito population suppression. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3467-3484. doi: 10.3934/dcds.2020042 [4] Fathalla A. Rihan, Hebatallah J. Alsakaji. Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020468 [5] Yukihiko Nakata. Existence of a period two solution of a delay differential equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1103-1110. doi: 10.3934/dcdss.2020392 [6] Hai Huang, Xianlong Fu. Optimal control problems for a neutral integro-differential system with infinite delay. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020107 [7] Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264 [8] Fabio Camilli, Giulia Cavagnari, Raul De Maio, Benedetto Piccoli. Superposition principle and schemes for measure differential equations. Kinetic & Related Models, 2021, 14 (1) : 89-113. doi: 10.3934/krm.2020050 [9] Qingfeng Zhu, Yufeng Shi. Nonzero-sum differential game of backward doubly stochastic systems with delay and applications. Mathematical Control & Related Fields, 2021, 11 (1) : 73-94. doi: 10.3934/mcrf.2020028 [10] Yueyang Zheng, Jingtao Shi. A stackelberg game of backward stochastic differential equations with partial information. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020047 [11] Bixiang Wang. Mean-square random invariant manifolds for stochastic differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1449-1468. doi: 10.3934/dcds.2020324 [12] Rim Bourguiba, Rosana Rodríguez-López. Existence results for fractional differential equations in presence of upper and lower solutions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1723-1747. doi: 10.3934/dcdsb.2020180 [13] Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $q$-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020440 [14] Guangjun Shen, Xueying Wu, Xiuwei Yin. Stabilization of stochastic differential equations driven by G-Lévy process with discrete-time feedback control. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 755-774. doi: 10.3934/dcdsb.2020133 [15] Mohamed Dellal, Bachir Bar. Global analysis of a model of competition in the chemostat with internal inhibitor. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1129-1148. doi: 10.3934/dcdsb.2020156 [16] Sze-Bi Hsu, Yu Jin. The dynamics of a two host-two virus system in a chemostat environment. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 415-441. doi: 10.3934/dcdsb.2020298 [17] Sarra Nouaoura, Radhouane Fekih-Salem, Nahla Abdellatif, Tewfik Sari. Mathematical analysis of a three-tiered food-web in the chemostat. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020369 [18] Simone Göttlich, Elisa Iacomini, Thomas Jung. Properties of the LWR model with time delay. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2020032 [19] Aisling McGlinchey, Oliver Mason. Observations on the bias of nonnegative mechanisms for differential privacy. Foundations of Data Science, 2020, 2 (4) : 429-442. doi: 10.3934/fods.2020020 [20] Giuseppina Guatteri, Federica Masiero. Stochastic maximum principle for problems with delay with dependence on the past through general measures. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020048

2018 Impact Factor: 1.313