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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

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. 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

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