Bacteriophage-resistant and bacteriophage-sensitive bacteria in a chemostat

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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

Zhun Han ^1, and Hal L. Smith ^2,

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

Abstract
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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.

Keywords: delay differential equations., chemostat, persistence, Bacteriophage.

Mathematics Subject Classification: Primary: 92D25; Secondary: 34K6.

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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