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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), 57-76. doi: 10.1016/S0025-5564(97)10015-3. [2] E. Beretta, H. Sakakibara and Y. Takeuchi, Analysis of a chemostat model for bacteria and bacteriophage, Vietnam Journal of Mathematics, 30 (2002), 459-472. [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), 45-48. [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), 495-520. [5] B. Bohannan and R. Lenski, Effect of resource enrichment on a chemostat community of bacteria and bacteriophage, Ecology, 78 (1997), 2303-2315. [6] B. Bohannan and R. Lenski, Linking genetic change to community evolution: insights from studies of bacteria and bacteriophage, Ecology Letters, 3 (2000), 362-377. [7] A. Campbell, Conditions for existence of bacteriophages, Evolution, 15 (1961), 153-165. doi: 10.2307/2406076. [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), 369-378. doi: 10.2307/1935611. [9] E. Ellis and M. Delbrück, The growth of bacteriophage, The Journal of General Physiology, 22 (1939), 365-384. doi: 10.1085/jgp.22.3.365. [10] G. Folland, "Real Analysis: Modern Techniques and Their Applications,'' $2^{nd}$ edition, Wiley, New York, 1999. [11] J. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkcialaj Ekvacioj, 21 (1978), 11-41. [12] J. Hale and S. Verduyn Lunel, "Introduction to Functional Differential Equations," Springer-Verlag, New York, 1993. [13] Y. Hino, S. Murakami and T. Naito, "Functional Differential Equations with Infinite Delay," Springer-Verlag, New York, 1991. [14] Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics," Academic Press, San Diego, 1993. [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), 585-602. doi: 10.1086/284364. [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), 3-24. [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), 1043-1086. doi: 10.1007/s00285-011-0439-z. [18] W. Ruess and W. Summers, Linearized stability for abstract differential equations with delay, Journal of Mathematical Analysis and Applications, 198 (1996), 310-336. doi: 10.1006/jmaa.1996.0085. [19] S. Schrag and J. Mittler, Host-parasite coexistence: The role of spatial refuges in stabilizing bacteria-phage interactions, American Naturalist, 148 (1996), 348-377. doi: 10.1086/285929. [20] H. Smith, "An introduction to Delay Differential Equations with Applications to the Life Sciences," Springer-Verlag, New York, 2010. [21] H. Smith and H. Thieme, "Dynamical Systems and Population Persistence," American Mathematical Society, 2010. [22] H. Smith and H. Thieme, Persistence of bacteria and phages in a chemostat, Journal of Mathematical Biology, 64 (2012), 951-979. doi: 10.1007/s00285-011-0434-4. [23] H. Smith and P. Waltman, Perturbation of a globally stable steady state, Proceeding of the American Mathematical Society, 127 (1999), 447-453. doi: 10.1090/S0002-9939-99-04768-1. [24] H. Smith and P. Waltman, "The Theory of the Chemostat: Dynamics of Microbial Competition," Cambridge University Press, 1995. [25] H. Smith and X. Q. Zhao, Robust persistence for semidynamical systems, Nonlinear Analysis Theory Methods Applications, 47 (2001), 6169-6179. doi: 10.1016/S0362-546X(01)00678-2. [26] H. Thieme, Convergence results and a poincaré-bendixson trichotomy for asymptotically autonomous differential equations, Journal of Mathematical Biology, 30 (1992), 755-763. [27] H. Thieme, "Mathematics in Population Biology," Princeton University Press, 2003. [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), 9535-9540. doi: 10.1073/pnas.0504062102.

show all references

##### References:
 [1] E. Beretta and Y. Kuang, Modeling and analysis of a marine bacteriophage infection, Mathematical Biosciences, 149 (1998), 57-76. doi: 10.1016/S0025-5564(97)10015-3. [2] E. Beretta, H. Sakakibara and Y. Takeuchi, Analysis of a chemostat model for bacteria and bacteriophage, Vietnam Journal of Mathematics, 30 (2002), 459-472. [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), 45-48. [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), 495-520. [5] B. Bohannan and R. Lenski, Effect of resource enrichment on a chemostat community of bacteria and bacteriophage, Ecology, 78 (1997), 2303-2315. [6] B. Bohannan and R. Lenski, Linking genetic change to community evolution: insights from studies of bacteria and bacteriophage, Ecology Letters, 3 (2000), 362-377. [7] A. Campbell, Conditions for existence of bacteriophages, Evolution, 15 (1961), 153-165. doi: 10.2307/2406076. [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), 369-378. doi: 10.2307/1935611. [9] E. Ellis and M. Delbrück, The growth of bacteriophage, The Journal of General Physiology, 22 (1939), 365-384. doi: 10.1085/jgp.22.3.365. [10] G. Folland, "Real Analysis: Modern Techniques and Their Applications,'' $2^{nd}$ edition, Wiley, New York, 1999. [11] J. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkcialaj Ekvacioj, 21 (1978), 11-41. [12] J. Hale and S. Verduyn Lunel, "Introduction to Functional Differential Equations," Springer-Verlag, New York, 1993. [13] Y. Hino, S. Murakami and T. Naito, "Functional Differential Equations with Infinite Delay," Springer-Verlag, New York, 1991. [14] Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics," Academic Press, San Diego, 1993. [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), 585-602. doi: 10.1086/284364. [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), 3-24. [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), 1043-1086. doi: 10.1007/s00285-011-0439-z. [18] W. Ruess and W. Summers, Linearized stability for abstract differential equations with delay, Journal of Mathematical Analysis and Applications, 198 (1996), 310-336. doi: 10.1006/jmaa.1996.0085. [19] S. Schrag and J. Mittler, Host-parasite coexistence: The role of spatial refuges in stabilizing bacteria-phage interactions, American Naturalist, 148 (1996), 348-377. doi: 10.1086/285929. [20] H. Smith, "An introduction to Delay Differential Equations with Applications to the Life Sciences," Springer-Verlag, New York, 2010. [21] H. Smith and H. Thieme, "Dynamical Systems and Population Persistence," American Mathematical Society, 2010. [22] H. Smith and H. Thieme, Persistence of bacteria and phages in a chemostat, Journal of Mathematical Biology, 64 (2012), 951-979. doi: 10.1007/s00285-011-0434-4. [23] H. Smith and P. Waltman, Perturbation of a globally stable steady state, Proceeding of the American Mathematical Society, 127 (1999), 447-453. doi: 10.1090/S0002-9939-99-04768-1. [24] H. Smith and P. Waltman, "The Theory of the Chemostat: Dynamics of Microbial Competition," Cambridge University Press, 1995. [25] H. Smith and X. Q. Zhao, Robust persistence for semidynamical systems, Nonlinear Analysis Theory Methods Applications, 47 (2001), 6169-6179. doi: 10.1016/S0362-546X(01)00678-2. [26] H. Thieme, Convergence results and a poincaré-bendixson trichotomy for asymptotically autonomous differential equations, Journal of Mathematical Biology, 30 (1992), 755-763. [27] H. Thieme, "Mathematics in Population Biology," Princeton University Press, 2003. [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), 9535-9540. doi: 10.1073/pnas.0504062102.
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