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Continuous model for the rock–scissors–paper game between bacteriocin producing bacteria

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Abstract

In this work, important aspects of bacteriocin producing bacteria and their interplay are elucidated. Various attempts to model the resistant, producer and sensitive Escherichia coli strains in the so-called rock–scissors–paper (RSP) game had been made in the literature. The question arose whether there is a continuous model with a cyclic structure and admitting an oscillatory dynamics as observed in various experiments. The May–Leonard system admits a Hopf bifurcation, which is, however, degenerate and hence inadequate. The traditional differential equation model of the RSP-game cannot be applied either to the bacteriocin system because it involves positive interaction terms. In this paper, a plausible competitive Lotka–Volterra system model of the RSP game is presented and the dynamics generated by that model is analyzed. For the first time, a continuous, spatially homogeneous model that describes the competitive interaction between bacteriocin-producing, resistant and sensitive bacteria is established. The interaction terms have negative coefficients. In some experiments, for example, in mice cultures, migration seemed to be essential for the reinfection in the RSP cycle. Often statistical and spatial effects such as migration and mutation are regarded to be essential for periodicity. Our model gives rise to oscillatory dynamics in the RSP game without such effects. Here, a normal form description of the limit cycle and conditions for its stability are derived. The toxicity of the bacteriocin is used as a bifurcation parameter. Exact parameter ranges are obtained for which a stable (robust) limit cycle and a stable heteroclinic cycle exist in the three-species game. These parameters are in good accordance with the observed relations for the E. coli strains. The roles of growth rate and growth yield of the three strains are discussed. Numerical calculations show that the sensitive, which might be regarded as the weakest, can have the longest sojourn times.

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References

  1. Ambroziĉ J., Ostroverŝnik A., Starĉiĉ M., Kuhar I., Grabnar M., Zgur-Bertok D. (1998). Escherichia coli ColV plasmid pRK100: genetic organization, stability and conjugal transfer. Microbiology 144: 343–352

    Article  Google Scholar 

  2. Axelrod R., Hamilton W.D. (1981). The evolution of cooperation. Science 211(4489): 1390–1396

    Article  MathSciNet  Google Scholar 

  3. Bockelman B., Deng B., Green E., Hines G., Lippitt L., Sherman J. (2004). Chaotic coexistence in a top-predator mediated competitive exclusive web. J. Dyn. Differ. Equ. 16(4): 1061–1092

    Article  MATH  MathSciNet  Google Scholar 

  4. Butler G., Freedman H.I., Waltman P. (1986). Uniformly persistent systems. Proc. Am. Math. Soc. 96(3): 425–430

    Article  MATH  MathSciNet  Google Scholar 

  5. Chao L., Levin B.R. (1981). Structured habitats and the evolution of anticompetitor toxins in bacteria. Proc. Natl. Acad. Sci. USA 78(10): 6324–6328

    Article  Google Scholar 

  6. Chi S.-B., Hsu C.-W., WuL-I (1998). On the asymmetric May–Leonard model of three competing species. SIAM J. Appl. Math. 58(1): 211–226

    Article  MATH  MathSciNet  Google Scholar 

  7. Coste J., Peyraud J., Coullet P. (1979). Asymptotic behaviors in the dynamics of competing species. SIAM J. Appl. Math. 36: 516–543

    Article  MATH  MathSciNet  Google Scholar 

  8. Czárán T.L., Hoeckstra R.F., Pagie L. (2002). Chemical warfare between microbes promotes biodiversity. Proc. Natl. Acad. Sci. USA 99(2): 786–790

    Article  Google Scholar 

  9. Denison R.F., Kiers E.T. (2004). Lifestyle alternatives for rhizobia: mutualism, parasitism and forgoing symbiosis. FEMS Microbiol. Lett. 237(4): 309–38

    Google Scholar 

  10. Diekmann O., Gyllenberg M., Metz J.A. (2003). Steady-state analysis of structured population models. Theor. Popul. Biol. 63(4): 309–38

    Article  MATH  Google Scholar 

  11. Dykes G.A., Hastings J.W. (1997). Selection and fitness in bacteriocin-producing bacteria. Proc. R. Soc. Lond. B Biol. Sci. 264: 683–687

    Article  Google Scholar 

  12. Feng Z., Smith L.S., McKenzie F.E., Levin S.A. (2004). Coupling ecology and evolution: malaria and the S-gene across time scales. Math. Biosci. 189: 1–19

    Article  MATH  MathSciNet  Google Scholar 

  13. Frank S.A. (1996). Models of parasite virulence. Q. Rev. Biol. 71(1): 37–78

    Article  Google Scholar 

  14. Frean M., Abraham E.R. (2001). Rock–scissors–paper and the survival of the weakest. Proc. R. Soc. Lond. B Biol. Sci. 268: 1323–1327

    Article  Google Scholar 

  15. Gardner A., West S.A., Buckling A. (2004). Bacteriocins, spite and virulence. Proc. R. Soc. Lond. B Biol. Sci. 271: 1529–1535

    Article  Google Scholar 

  16. Govaerts, W., Kuznetsov, Y.A., De Feo, O., Dhooge, A., Govorukhin, V., Khoshsiar Ghaziani, R., Meijer, H.G.E., Mestrom, W., Riet, A., Sautois, B.: MATCONT. http://www.matcont.ugent.be/

  17. Grohmann E., Muth G., Espinosa M. (2003). Conjugative plasmid transfer in Gram-positive bacteria. Microbiol. Mol. Biol. Rev. 67(2): 277–301

    Article  Google Scholar 

  18. Guckenheimer J.A., Holms P.J. (1983). Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer, New York

    MATH  Google Scholar 

  19. Hauert C., De Monte S., Hofbauer J., Sigmund K. (2002). Volunteering as red queen mechanism for cooperation in public goods games. Science 296: 1129–1132

    Article  Google Scholar 

  20. Hirsch M.W. (1988). Systems of differential equations which are competitive or cooperative: III. Competing species. Nonlinearity 1: 051–07

    Article  MathSciNet  Google Scholar 

  21. Hofbauer J., So J.W.H. (1994). Multiple limit cycles for three-dimensional Lotka–Volterra equations. Appl. Math. Lett. 7(6): 65–70

    Article  MATH  MathSciNet  Google Scholar 

  22. Hofbauer J., Sigmund K. (1998). Evolutionary Games and Population Dynamics. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  23. Hofbauer J., Sigmund K. (1988). The Theory of Evolution and Dynamical Systems. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  24. Horn R.A., Johnson C.R. (1985). Matrix Analysis. University Press, Cambridge

    MATH  Google Scholar 

  25. Huisman J., Weissing F.J. (2001). Biological conditions for oscillations and chaos generated by multispecies competition. Ecology 82(10): 2682–2695

    Article  Google Scholar 

  26. Hurwitz A. (1895). On the conditions under which an equation has only roots with negative real parts. Math. Ann. 46: 273–284

    Article  MathSciNet  Google Scholar 

  27. Ifti M., Bergersen B. (2004). Crossover behaviour of 3-species systems with mutations or migrations. Eur. Phys. J. B 37: 101–107

    Article  Google Scholar 

  28. Iwasa Y., Nakamaru M., Levin S.A (1998). Allelopathy of bacteria in a lattice population: competition between colicin-sensitive and colicin-producing strains. Evol. Ecol. 12: 785–802

    Article  Google Scholar 

  29. Kerr B., Riley M.A., Feldman M.W., Bohannan B.J.M. (2002). Local dispersal promotes biodiversity in a real-life game of rock–paper–scissors. Nature 418: 171–174

    Article  Google Scholar 

  30. Kirkup B.C., Riley M.A. (2004). Antibiotic-mediated antagonism leads to a bacterial game of rock-paper-scissors in vivo. Nature 428: 412–414

    Article  Google Scholar 

  31. Kuznetsov Y.A. (2004). Elements of Applied Bifurcation Theory, 3rd edn. Springer, Berlin

    MATH  Google Scholar 

  32. Levin S.A. (2000). Multiple scales and the maintenance of biodiversity. Ecosystems (N. Y., Print) 3: 498–506

    Article  Google Scholar 

  33. May R.M., Leonard W.J. (1975). Nonlinear aspects of competition between three species. SIAM J. Appl. Math. 29(2): 243–253

    Article  MathSciNet  MATH  Google Scholar 

  34. Maynard-Smith J., Price G.R. (1973). The logic of animal conflict. Nature 246: 15–18

    Article  Google Scholar 

  35. Murray J.D. (2002). Mathematical Biology: I. An Introduction. Springer, New York

    Google Scholar 

  36. Neumann G., Schuster S. (2007). Modeling the Rock–Scissors–Paper Game between Bacteriocin Producing Bacteria by Lotka–Volterra Equations. DCDS-B 8(1): 207–228

    Article  MathSciNet  MATH  Google Scholar 

  37. Pfeiffer T., Schuster S., Bonhoeffer S. (2001). Cooperation and competition in the evolution of ATP-producing pathways. Science 292: 504–507

    Google Scholar 

  38. Pfeiffer T., Schuster S. (2005). Game-theoretical approaches to studying the evolution of biochemical systems. Trends Biochem. Sci. 30(1): 20–25

    Article  Google Scholar 

  39. Riley M.A. (1998). Molecular mechanisms of bacteriocin evolution. Annu. Rev. Genet. 32: 255–278

    Article  Google Scholar 

  40. Riley M.A., Wertz J.E. (2002). Bacteriocin diversity: ecological and evolutionary perspectives. Biochimie 84(5–6): 357–364

    Article  Google Scholar 

  41. Riley M.A., Wertz J.E. (2002). Bacteriocins: evolution, ecology and application. Annu. Rev. Microbiol. 56: 117–37

    Article  Google Scholar 

  42. Schreiber S.J. (2000). Criteria for C r robust permanence. J. Differ. Equ. 162: 400–426

    Article  MATH  MathSciNet  Google Scholar 

  43. Semmann D., Krambeck H-J., Millinski M. (2003). Volunteering leads to rock-paper-scissors dynamics in a public goods game. Nature 425: 390–393

    Article  Google Scholar 

  44. Sinervo B., Lively C.M. (1996). The rock–paper–scissors game and the evolution of alternative male strategies. Nature 380: 240–243

    Article  Google Scholar 

  45. Sinervo, B.: Runaway social games, genetic cycles driven by alternative male and female strategies, and the origin of morphs. Genetica 112–113, 417–434 (2001)

    Google Scholar 

  46. Smith, H.: Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems Mathematical Surveys and Monographs, vol. 41, AMS, Providence, RI (1995)

  47. Smith H.L., Waltman P. (1994). The Theory of the Chemostat: Dynamics of Microbial Competition. Cambridge University Press, Cambridge

    Google Scholar 

  48. Szabó G., Szolnoki A., Izsák R. (2004). Rock–scissors–paper game on regular small-world networks. J. Phys. A Math. Gen. 37: 2599–2609

    Article  MATH  Google Scholar 

  49. Tagg J.R., Dajani A.S., Wannamaker L.W. (1975). Bacteriocin of a group B streptococcus: partial purification and characterization. Antimicrob. Agents Chemother. 7(6): 764–772

    Google Scholar 

  50. van den Driessche P., Zeeman M.L. (1998). Three-dimensional competitive Lotka–Volterra systems with no periodic orbits. SIAM J. Appl. Math. 58: 227–234

    Article  MATH  MathSciNet  Google Scholar 

  51. van den Driessche P., Zeeman M.L. (2004). Disease induced oscillations between two competing species. SIAM J. Appl. Dyn. Syst. 3(4): 601–619

    Article  MATH  MathSciNet  Google Scholar 

  52. Vu-Delcarte, C.D.: Normal Forms and Bifurcations of Vector Fields. Perruquettiet, W., Barbot, J.P. (eds.) Chaos in automatic control, Chap. 3. Taylor & Francis, New York (2005)

  53. Wilhelm T., Heinrich R. (1996). Mathematical analysis of the smallest chemical reaction system with Hopf bifurcation. J. Math. Chem. 19: 111–130

    Article  MATH  MathSciNet  Google Scholar 

  54. Xiao D., Li W. (2000). Limit cycles for the competitive three dimensional Lotka–Volterra systems. J. Differ. Equ. 164: 1–15

    Article  MATH  MathSciNet  Google Scholar 

  55. Zeeman M.L. (1993). Hopf bifurcations in competitive three dimensional Lotka–Volterra systems. Dyn. Stab. Syst. 8(3): 189–217

    MATH  MathSciNet  Google Scholar 

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  1. Ernst-Abbé Platz 1-2, 07743, Jena, Germany

    Gunter Neumann & Stefan Schuster

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Neumann, G., Schuster, S. Continuous model for the rock–scissors–paper game between bacteriocin producing bacteria. J. Math. Biol. 54, 815–846 (2007). https://doi.org/10.1007/s00285-006-0065-3

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