Journal of Mathematical Biology

, Volume 52, Issue 6, pp 745–760 | Cite as

Multiple limit cycles in the standard model of three species competition for three essential resources

  • Steven M. Baer
  • Bingtuan LiEmail author
  • Hal L. Smith


We consider the dynamics of the standard model of 3 species competing for 3 essential (non-substitutable) resources in a chemostat using Liebig's law of the minimum functional response. A subset of these systems which possess cyclic symmetry such that its three single-population equilibria are part of a heteroclinic cycle bounding the two-dimensional carrying simplex is examined. We show that a subcritical Hopf bifurcation from the coexistence equilibrium together with a repelling heteroclinic cycle leads to the existence of at least two limit cycles enclosing the coexistence equilibrium on the carrying simplex- the ``inside'' one is an unstable separatrix and the ``outside'' one is at least semi-stable relative to the carrying simplex. Numerical simulations suggest that there are exactly two limit cycles and that almost every positive solution approaches either the stable limit cycle or the stable coexistence equilibrium, depending on initial conditions. Bifurcation diagrams confirm this picture and show additional features. In an alternative scenario, we show that the subcritical Hopf together with an attracting heteroclinic cycle leads to an unstable periodic orbit separatrix.

Keywords or phrases

Resource competition Coexistence Persistence Hopf bifurcation Carrying simplex Limit cycle Heteroclinic cycle 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Armstrong, R.A., McGehee, R.: Competitive exclusion, Amer. Natur. 115, 151–170 (1980)MathSciNetGoogle Scholar
  2. 2.
    Butler, G.J., Wolkowicz, G.S.K.: A mathematical model of the chemostat with a general class of functions describing nutrient uptake, SIAM J. Appl. Math. 45, 138–151 (1985)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Butler, G.J., Wolkowicz, G.S.K.: Exploitative competition in a chemostat for two complementary, and possible inhibitory, resources, Math. Biosci. 83, 1–48 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Doedel, E.J., Champneys, A.R., Fairgrieve, T.F., Kuznetsov, Y.A., Sandstede, B., Wang, X. Auto 97: Continuation and bifurcation software for ordinary differential equations, Technical report, Concordia University, Montreal, Canada, 1997Google Scholar
  5. 5.
    Chi C-W, Hsu S.B., Wu L-L.: On the asymmetric May-Leonard model of three competing species, SIAM J. Appl. Math. 58, 211–226 (1998)Google Scholar
  6. 6.
    Gause, G.F.: The Struggle for Existence, Williams and Wilkins, Baltimore, Maryland, 1934Google Scholar
  7. 7.
    Grover, J.P.: Resource Competition, Population and Community Biology Series, 19, Chapman & Hall, New York, 1997Google Scholar
  8. 8.
    Guckenheimer, J., Holmes P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer-Verlag, New York, 1983Google Scholar
  9. 9.
    Gyllenberg, M., Yan, P., Wang, Y.: A 3D competitive Lotka-Volterra system with three limit cycles: a falsification of a conjecture of Hofbauer and So, preprintGoogle Scholar
  10. 10.
    Hardin, G.: The competitive exclusion principle, Science 131, 1292–1298 (1960)Google Scholar
  11. 11.
    Hirsch, M.W.: Systems of differential equations which are competitive or cooperative. III: Competing species. Nonlinearity 1, 51–71 (1988)zbMATHMathSciNetGoogle Scholar
  12. 12.
    Hofbauer, J., Sigmund, K.: The Theory of Evolution and Dynamical Systems, London Math. Soc. Student Texts 7, Cambridge University Press, Cambridge, 1988Google Scholar
  13. 13.
    Hofbauer, J., So, J.W.-H.: Multiple limit cycles for three dimesnional Lotka-Volterra Equations, Appl. Math. Lett. 7, 65–70 (1994)zbMATHMathSciNetGoogle Scholar
  14. 14.
    Holm, N.P., Armstrong, D.E.: Role of nutrients limitation and competition in controlling the populations of Asterionella formosa and Microcystis aeruginosa in semicontinuous culture, Limonl. Oceanogr. 26, 622–634 (1981)Google Scholar
  15. 15.
    Hsu, S.B., Hubbell, S., Waltman, P.: A mathematical theory of single-nutrient competition in continuous cultures of micro-organisms, SIAM J. Appl. Math. 32, 366–383 (1977)zbMATHGoogle Scholar
  16. 16.
    Hsu, S.B.: Limiting behavior for competing species, SIAM J. Appl. Math. 34, 760–763 (1978)zbMATHGoogle Scholar
  17. 17.
    Hsu, S.B., Cheng, K.S., Hubbell, S.P.: Exploitative competition of microorganism for two complementary nutrients in continuous culture, SIAM J. Appl. Math. 41, 422–444 (1981)zbMATHGoogle Scholar
  18. 18.
    Huisman, J., Weissing, F.J.: Biodiversity of plankton by species oscillations and chaos, Nature 402, 407–410 (1999)Google Scholar
  19. 19.
    Huisman, J., Weissing, F.J.: Fundamental unpredictability in multispecies competition, Amer. Naturalist 157, 488–494 (2001)CrossRefGoogle Scholar
  20. 20.
    Leon, J.A., Tumpson, D.B.: Competition between two species for two complementary or substitutable resources, J. Theor. Biol. 50, 185–201 (1975)CrossRefGoogle Scholar
  21. 21.
    Li, B.: Analysis of Chemostat-Related Models With Distinct Removal Rates, Ph.D thesis, Arizona State University, 1998Google Scholar
  22. 22.
    Li, B.: Global asymptotic behaviour of the chemostat: general response functions and different removal rates, SIAM J. Appl. Math. 59, 411–422 (1999)zbMATHGoogle Scholar
  23. 23.
    Li, B., Smith, H.L.: How many species can two essential resources support? SIAM J. Appl. Math. 62, 336–66 (2001)zbMATHGoogle Scholar
  24. 24.
    Li, B.: Periodic coexistence in the chemostat with three species competing for three essential resources, Math. Biosciences 174, 27–40 (2001)CrossRefzbMATHGoogle Scholar
  25. 25.
    Lu, Z., Luo, Y.: Two limit cycles in three-dimensional Lotka-Volterra systems, Computer and Mathematics with Applications, 44, 51–66 (2002)Google Scholar
  26. 26.
    Lu, Z., Luo, Y.: Three limit cycles for a three-dimensional Lotka-Volterra competitive system with a heteroclinic cycle, Computer and Mathematics with Applications, 46, 51–66 (2003)Google Scholar
  27. 27.
    May, R.M., Leonard, W.J.: Nonlinear aspects of competition between three species, SIAM J. Appl. Math. 29, 243–253 (1975)zbMATHMathSciNetGoogle Scholar
  28. 28.
    Novick, A., Sziliard, L.: Description of the chemostat. Science. 112, 715–716 (1950)Google Scholar
  29. 29.
    Phillips, O.M.: The equilibrium and stability of simple marine biological systems, 1. Primary nutrient consumers, Amer. Natur. 107, 73–93 (1973)Google Scholar
  30. 30.
    K. O. Rothhaup, Laboratory experiments with a mixotrophic chrysophyte and obligately phagotrophic and phototrophic competitors. Ecology 77, 716–724 (1996)Google Scholar
  31. 31.
    Schuster, P., Sigmund, K., Wolf, R.: On ω-limit for competition between three species, SIAM J. Appl. Math. 37, 49–54 (1979)zbMATHMathSciNetGoogle Scholar
  32. 32.
    Smith, H.L.: Monotone Dynamical Systems, An Introduction to the Theory of Competitive and Cooperative Systems. Mathematical Surveys and Monographs 41, Amer. Math. Soc., Providence, RI, 1995Google Scholar
  33. 33.
    Smith, H.L., Li, B.: Competition for essential resources: A brief review, Fields Institute Communications, 36, 213–227 (2003)Google Scholar
  34. 34.
    Smith, H.L., Waltman, P.: The Theory of the Chemostat, Cambridge University Press, 1995Google Scholar
  35. 35.
    Sommer, U.: Comparison between steady states and non-steady competition: experiments with natural phytoplankton. Limnol. Oceanogr. 30, 335–346 (1985)CrossRefGoogle Scholar
  36. 36.
    Sommer, U.: Nitrate-and silicate-competition among Antarctic phytoplankton, Mar. Biol. 91, 345–351 (1986)Google Scholar
  37. 37.
    Tilman, D.: Tests of resources competition theory using four species of Lake Michigan algae, Ecology 62, 802–815 (1981)Google Scholar
  38. 38.
    Tilman, D.: Plant Strategies and the Dynamics and Structure of Plant Communities, Princeton University Press, Princeton, N.J., 1981Google Scholar
  39. 39.
    Tilman, D.: Resource competition and Community Structure, Princeton University Press, Princeton, N.J., 1982Google Scholar
  40. 40.
    Van Donk, E., Kilham, S.S.: Temperature effects on silicon-and phosphorus-limited growth and competitive interactions among three diatoms, J. Phycol. 26, 40–50 (1990)CrossRefGoogle Scholar
  41. 41.
    Von Liebig, J.: Die organische Chemie in ihrer Anwendung auf Agrikultur und Physiologie. Friedrich Vieweg, Braunschweig, 1840Google Scholar
  42. 42.
    Wolkowicz, G.S.K., Lu, Z.: Global dynamics of a mathematical model of competition in the chemostat: general response function and differential death rates, SIAM J. Appl. Math. 52, 222–233 (1992)zbMATHMathSciNetGoogle Scholar
  43. 43.
    Zeeman, M.L.: Hopf Bifurcation in competitive three-dimensional Lotka-Volterra systems, Dynamics and Stability of Systems, 8, 189–217 (1993)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsArizona State UniversityTempeUSA
  2. 2.Department of MathematicsUniversity of LouisvilleLouisvilleUSA

Personalised recommendations