Journal of Nonlinear Science

, Volume 27, Issue 3, pp 817–846 | Cite as

The Impact of Prophage on the Equilibria and Stability of Phage and Host



In this paper, we present a bacteriophage model that includes prophage, that is, phage genomes that are incorporated into the host cell genome. The general model is described by an 18-dimensional system of ordinary differential equations. This study focuses on asymptotic behaviour of the model, and thus the system is reduced to a simple six-dimensional model, involving uninfected host cells, infected host cells and phage. We use dynamical system theory to explore the dynamic behaviour of the model, studying in particular the impact of prophage on the equilibria and stability of phage and host. We employ bifurcation and stability theory, centre manifold and normal form theory to show that the system has multiple equilibrium solutions which undergo a series of bifurcations, finally leading to oscillating motions. Numerical simulations are presented to illustrate and confirm the analytical predictions. The results of this study indicate that in some parameter regimes, the host cell population may drive the phage to extinction through diversification, that is, if multiple types of host emerge; this prediction holds even if the phage population is likewise diverse. This parameter regime is restricted, however, if infecting phage are able to recombine with prophage sequences in the host cell genome.


SIV model Bacteriophage Equilibria Stability Hopf bifurcation 

Mathematics Subject Classfication

92D30 37L10 37N25 



This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) (Nos. R2686A02 and 238389-RGPIN).


  1. Bouchard, J.D., Moineau, S.: Homologous recombination between a lactococcal bacteriophage and the chromosome of its host strain. Virology 270, 65–75 (2000)CrossRefGoogle Scholar
  2. Casjens, S.: Prophages and bacterial genomics: what have we learned so far? Mol. Microbiol. 49, 277–300 (2003)CrossRefGoogle Scholar
  3. Chepyzov, V.V., Vishik, M.I.: Attractors for Equations of Mathematical Physics. American Mathematical Society, Providence (2002)Google Scholar
  4. De Paepe, M., Taddei, F.: Viruses life history: towards a mechanistic basis of a trade-off between survival and reproduction among phages. PLoS Biol. e193 4, 1248–1256 (2006)Google Scholar
  5. Durmaz, E., Klaenhammer, T.R.: Genetic analysis of chromosomal regions of Lactobacillus lactis acquired by recombinant lytic phages. Appl. Environ. Microbiol. 66, 895–903 (2000)CrossRefGoogle Scholar
  6. Hendrix, R.W.: Bacteriophages: evolution of the majority. Theor. Popul. Biol. 61, 471–480 (2002)CrossRefGoogle Scholar
  7. Hubbarde, J.E., Wild, G., Wahl, L.M.: Fixation probabilities when generation times are variable: the burst-death model. Genetics 176, 1703–1712 (2007)CrossRefGoogle Scholar
  8. Koskella, B., Brockhurst, M.A.: Bacteria-phage coevolution as a driver of ecological and evolutionary processes in microbial communities. FEMS Microbiol. Rev. 38(5), 916–931 (2014)CrossRefGoogle Scholar
  9. Labrie, S.J., Moineau, S.: Abortive infection mechanisms and prophage sequences significantly influence the genetic makeup of emerging lytic lactococcal phages. J. Bacteriol. 189, 1482–1487 (2007)CrossRefGoogle Scholar
  10. LaSalle, J.P.: The Stability of Dynamics Systems. SIAM, Philadelphia (1976)CrossRefMATHGoogle Scholar
  11. Manuel, M., Surovtsev, I.V., Kato, S., Paintdakhi, A., Beltran, B., Ebmeier, S.E., Jacobs-Wagner, C.: A constant size extension drives bacterial cell size homeostasis. Cell 159, 1433–1446 (2014)CrossRefGoogle Scholar
  12. Meyer, J.R., Dobias, D.T., Weitz, J.S., Barrick, J.E., Quick, R.T., Lenski, R.E.: Repeatability and contingency in the evolution of a key innovation in phage lambda. Science 335(6067), 428–432 (2012)CrossRefGoogle Scholar
  13. Moineau, S., Pandian, S., Klaenhammer, T.R.: Evolution of a lytic bacteriophage via DNA acquisition from the Lactococcus lactis chromosome. Appl. Environ. Microbiol. 60, 1832–1841 (1994)Google Scholar
  14. Stewart, F.M., Levin, B.R.: The population biology of bacterial viruses: why be temperate. Theor. Popul. Biol. 26, 93–117 (1984)MathSciNetCrossRefMATHGoogle Scholar
  15. Suresh, S., Ow, D.S.W., Lee, S.Y., Lee, M.M., Oh, S.K.W., Karimi, I.A., Lee, D.Y.: Characterizing Escherichia coli DH5a growth and metabolism in a complex medium using genome-scale flux analysis. Biotechnol. Bioeng. 102, 923–934 (2009)CrossRefGoogle Scholar
  16. Wang, P., Robert, L., Pelletier, J., Dang, W.L., Taddei, F., Wright, A., Jun, S.: Robust growth of Escherichia coli. Curr. Biol. 20, 1099–1103 (2010)CrossRefGoogle Scholar
  17. Weitz, J.S., Hartman, H., Levin, S.A.: Coevolutionary arms races between bacteria and bacteriophage. PNAS 102, 9535–9540 (2005)CrossRefGoogle Scholar
  18. Yu, P.: Computation of normal forms via a perturbation technique. J. Sound Vib. 211, 19–38 (1998)MathSciNetCrossRefMATHGoogle Scholar
  19. Yu, P., Huseyin, K.: A perturbation analysis of interactive static and dynamic bifurcations. IEEE Trans. Autom. Control 33, 28–41 (1988)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of Applied MathematicsThe University of Western OntarioLondonCanada

Personalised recommendations