Article Contents
Article Contents

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

• 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.
Mathematics Subject Classification: Primary: 92D25; Secondary: 34K60.

 Citation:

•  [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.