Journal of Mathematical Biology

, Volume 22, Issue 2, pp 123–135 | Cite as

Oscillatory coexistence in a food chain model with competing predators

  • James P. Keener


A food chain model with two predators feeding on a single prey in a chemostat is studied. Using a multiparameter bifurcation analysis, we find parameters values for which there is stable oscillatory coexistence of the predators. It is also shown how these coexistent states provide a transition between two possible states of competitive exclusion. It is shown that the competitive exclusion principle need not hold if one or more of the predators has oscillatory behavior in the absence of other predators.

Key words

Oscillatory coexistence Chemostat Multiparameter bifurcation 


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  1. 1.
    Butler, G. J.: Coexistence in predator-prey systems, In: Modeling and Differential Equations in Biology, T. Burton (eds.), Marcel Dekker: New York 1980Google Scholar
  2. 2.
    Butler, G. J., Waltman, P.: Bifurcation from a limit cycle in a two prey predator ecosystem modeled on a chemostat. J. Math. Biol. 12, 295–310 (1981)Google Scholar
  3. 3.
    Butler, G. J., Hsu, S. B., Waltman, P.: Coexistence of competing predators in a chemostat. J. Math. Biol. 17, 131–151 (1983)Google Scholar
  4. 4.
    Carillo Calvet, H.: The Method of Averaging and Stability under Persistent disturbances with Applications to Phase Locking, preprint, 1983Google Scholar
  5. 5.
    Drake, J. F., Tsuchiya, H. M.: Predation of Escherichia coli by Colpoda Stenii. Appl. Envr. Microbiol. 31, 870–874 (1976)Google Scholar
  6. 6.
    Fredrickson, A. G., Stephanopoulos, G.: Microbial competition. Science 213, 972–979 (1981)Google Scholar
  7. 7.
    Hansen, S. R., Hubbell, S. P.: Single-nutrient microbial competition: Agreement between experimental and theoretical forecast outcomes. Science 207, 1491–1492 (1980)Google Scholar
  8. 8.
    Herbert, D., Elsworth, R., Telling, R. C.: The continuous culture of bacteria: a theoretical and experimental study. J. Gen. Microbiol. 14, 601–622 (1956)Google Scholar
  9. 9.
    Hsu, S. B., Hubbell, S. P., Waltman, P.: A mathematical theory for single-nutrient competition in continuous cultures of microorganisms. SIAM J. Appl. Math. 32, 366–383 (1977)Google Scholar
  10. 10.
    Hsu, S. B., Hubbell, S. P., Waltman, P.: A contribution to the theory of competing predators. Ecol. Mongr. 48, 337–349 (1978)Google Scholar
  11. 11.
    Jannash, H. W., Mateles, R. T.: Experimental bacterial ecology studied in continuous culture. Adv. Microb. Physiol. 11, 165–212 (1974)Google Scholar
  12. 12.
    Jost, J. L., Drake, S. F., Fredrickson, A. G., Tsuchiya, M.: Interaction of tetrahymena pyriformis, Escherichia coli, azotobacter vinelandii and glucose in a minimal medium. J. Bacteriol. 113, 834–840 (1976)Google Scholar
  13. 13.
    Keener, J. P.: Oscillatory coexistence in the chemostat: A codimension two unfolding. SIAM J. Appl. Math. 43, 1005–1018 (1983)Google Scholar
  14. 14.
    Koch, A. L.: Competitive coexistence of two predators utilizing the same prey under constant environmental conditions. J. Theor. Biol. 44, 378–386 (1974)Google Scholar
  15. 15.
    Marsden, J. E., McCracken, M.: The Hopf bifurcation and its Applications. New York: Springer 1976Google Scholar
  16. 16.
    McGehee, R., Armstrong, R. A.: Some mathematical problems concerning the ecological principle of competitive exclusion. J. Differ. Equations 23, 30–52 (1977)Google Scholar
  17. 17.
    Monod, J.: Recherches sur la Croissance des Cultures Bacteriennes. Paris: Hermann 1942Google Scholar
  18. 18.
    Pike, E. B., Cuids, C. R.: The microbial ecology of activated sludge process, In: G. Sykes and F. A. Skinner (eds) Microbial aspects of Pollution. New York: Academic Press 1971Google Scholar
  19. 19.
    Sell, G.: What is a dynamical system?, In: Studies in Ordinary Differential Equations, J. Hale (ed.), MAA Studies in Mathematics No. 14, 1977Google Scholar
  20. 20.
    Smith, H. L.: The interaction of steady state and Hopf bifurcation in a two-predator-one-prey competition model. SIAM J. Appl. Math. 42, 27–43 (1982)Google Scholar
  21. 21.
    Stewart, F. M., Levin, B. R.: Partitioning of resources and the outcome of interspecific competition; a model and some general consideration. Am. Nat. 107, 171–198 (1973)Google Scholar
  22. 22.
    Tsuchiya, H. M., Drake, S. F., Jost, J. L., Fredrickson, A. G.: Predator-prey interactions of dictyostelium discordeum and Escherichia coli in continuous culture. J. Bacteriol. 110, 1147–1153 (1972)Google Scholar
  23. 23.
    Veldcamp, H.: Ecological studies with the chemostat. Adv. Microb. Ecol. 1, 59–95 (1977)Google Scholar
  24. 24.
    Waltman, P., Hubbell, S. P., Hsu, S. B.: Theoretical and experimental investigations of microbial competition in continuous culture, In: T. Burton (eds), Modeling and differential equations in Biology. New York: Marcel Dekker 1980Google Scholar

Copyright information

© Springer-Verlag 1985

Authors and Affiliations

  • James P. Keener
    • 1
  1. 1.Department of MathematicsUniversity of UtahSalt Lake CityUSA

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