Competitive Exclusion Through Discrete Time Models

  • Azmy S. AcklehEmail author
  • Paul L. Salceanu
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 102)


In biology, the principle of competitive exclusion, largely attributed to the Russian biologist G. F. Gause, states that two species competing for common resources (food, territory etc.) cannot coexist, and that one of the species drives the other to extinction. We make a survey of discrete-time mathematical models that address this issue and point out the main mathematical methods used to prove the occurrence of competitive exclusion in these models. We also offer examples of models in which competitive exclusion fails to take place, or at least it is not the only outcome. Finally, we present an extension of the competitive exclusion results in [1, 5] to a more general model.


Interspecific Competition Competitive Exclusion Competition Model Interior Equilibrium Strong Nonlinearity 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



The work of A.S. Ackleh is partially supported by the National Science Foundation under grant # DMS-1312963. The work of P.L. Salceanu is partially supported by the Louisiana Board of Regents under grant # LEQSF(2012-15)-RD-A-29.


  1. 1.
    Ackleh, A.S., Dib, Y., Jang, S.: A discrete-time Beverton-Holt competition model. In: Allen, L.J.S., Aulback, B., Elaydi, S., Sacker, R. (eds.) Difference Equations and Discrete Dynamical Systems, pp. 1–10. World Scientific, Singapore (2005)CrossRefGoogle Scholar
  2. 2.
    Ackleh, A.S., Sacker, R.J., Salceanu, P.L.: On a discrete selection-mutation model. Manuscript under review.Google Scholar
  3. 3.
    Ackleh, A.S., Zhang, P.: Competitive exclusion in a discrete stage-structured two species model. Math. Model. Nat. Phenom. 6, 156–175 (2009)CrossRefMathSciNetGoogle Scholar
  4. 4.
    AlSharawi, Z., Rhouma, M.: Coexistence and extinction in a competitive exclusion Leslie/Gower model with harvesting and stocking. J. Diff. Equ. Appl. 15, 1031–1053 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Chow, Y., Hsieh, J.: On multi-dimensional discrete-time Beverton-Holt competition models. J. Diff. Equ. Appl. 19, 491–506 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Costantino, R.F., Desharnais, R.A., Cushing, J.M., Dennis, B.: Chaotic dynamics in an insect population. Science 275, 389–391 (1997)CrossRefzbMATHGoogle Scholar
  7. 7.
    Costantino, R.F., Cushing, J.M., Dennis, B., Desharnais, R.A., Henson, S.M.: Resonant population cycles in temporally fluctuating habitats. Bull. Math. Biol. 60, 247–273 (1998)zbMATHGoogle Scholar
  8. 8.
    Cushing, J.M.: An Introduction to Structured Population Dynamics. SIAM, Philadelphia (1998)CrossRefzbMATHGoogle Scholar
  9. 9.
    Cushing, J.M.: The LPA model. Fields Inst. Commun. 43, 29–55 (2004)MathSciNetGoogle Scholar
  10. 10.
    Cushing, J.M., Costantino, R.F., Dennis, B., Desharnais, R.A., Henson, S.M.: Chaos in Ecology. Academic Press/Elsevier, San Diego (2003)Google Scholar
  11. 11.
    Cushing, J.M., Levarge, S., Chitnis, N., Henson, S.M.: Some discrete competition models and the competitive exclusion principle. J. Diff. Equ. Appl. 10, 1139–1151 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Cushing, J.M., Henson, S.M., Blackburn, C.C.: Multiple mixed-type attractors in a competition model. J. Biol. Dyn. 4, 347–362 (2007)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Cushing, J.M., Henson, S.M., Roeger, L.-I.: Coexistence of competing juvenile-adult structured populations. J. Biol. Dyn. 1, 201–231 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Desharnais, R.A., Costantino, R.F., Cushing, J.M., Henson, S.M., Dennis, B.: Chaos and population control of insect outbreaks. Ecol. Lett. 4, 229–235 (2001)CrossRefGoogle Scholar
  15. 15.
    Dowson, P.S.: Developmental rate and competitive ability in Tribolium. III. Competition in unfavorable environments. J. Stored Prod. Res. 3, 193–198 (1967)CrossRefGoogle Scholar
  16. 16.
    Edmunds, J.L.: Multiple attractors in a discrete competition model. Theor. Popul. Biol. 72, 379–388 (2007)CrossRefzbMATHGoogle Scholar
  17. 17.
    Edmunds, J.L., Cushing, J.M., Costantino, R.F., Henson, S.M., Dennis, B., Desharnais, R.A.: Park’s Tribolium competition experiments: a non-equilibrium species coexistence hypothesis. J. Anim. Ecol. 72, 703–712 (2003)CrossRefGoogle Scholar
  18. 18.
    Gause, G.F.: The Struggle for Existence. The Williams & Wilkins Company, Baltimore (1934)CrossRefGoogle Scholar
  19. 19.
    Henson, S.M., Costantino, R.F., Cushing, J.M., Desharnais, R.A., Dennis, B., King, A.A.: Lattice effects observed in chaotic dynamics of experimental populations. Science 294, 602–605 (2001)CrossRefGoogle Scholar
  20. 20.
    Henson, S.M., Cushing, J.M., Costantino, R.F., Dennis, B., Desharnais, R.A.: Phase switching in population cycles. Proc. Roy. Soc. Lond. B 265, 2229–2234 (1998)CrossRefGoogle Scholar
  21. 21.
    Hsu, S.B., Smith, H.L., Waltman, P.: Competitive exclusion and coexistence for competitive systems on ordered Banach spaces. Trans. Am. Math. Soc. 348, 4083–4094 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    King, A.A., Desharnais, R.A., Henson, S.M., Costantino, R.F., Cushing, J.M., Dennis, B.: Random perturbations and lattice effects in chaotic population dynamics. Science 297, 2163a (2002)CrossRefGoogle Scholar
  23. 23.
    Leslie, P.H., Gower, J.C.: The properties of a stochastic model for two competing species. Biometrika 45, 316–330 (1958)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Leslie, P.H., Park, T., Mertz, D.B.: The effect of varying the initial numbers on the outcome of competition between two Tribolium species. J. Anim. Ecol. 37, 9–23 (1968)CrossRefGoogle Scholar
  25. 25.
    Luis, R., Elaydi, S., Oliveira, H.: Stability of a Ricker-type competition model and the competitive exclusion principle. J. Biol. Dyn. 5, 636–660 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Park, T.: Experimental studies of interspecies competition. I. Competition between populations of the flour beetles Tribolium confusum Duval and Tribolium castaneum Herbst. Ecol. Monogr. 18, 265–308 (1948)CrossRefGoogle Scholar
  27. 27.
    Park, T.: Experimental studies of interspecies competition. II. Temperature, humidity and competition in two species of Tribolium. Physiol. Zool. 27, 177–238 (1954)Google Scholar
  28. 28.
    Park, T.: Experimental studies of interspecies competition. III. Relation of initial species proportion to the competitive outcome in populations of Tribolium. Physiol. Zool. 30, 22–40 (1957)Google Scholar
  29. 29.
    Park, T., Leslie, P.H., Mertz, D.B.: Genetic strains and competition in populations of Tribolium. Physiol. Zool. 37, 97–162 (1964)Google Scholar
  30. 30.
    Rael, R.C., Vincent, T.L., Cushing, J.M.: Competitive outcomes changed by evolution. J. Biol. Dyn. 5, 227–252 (2011)CrossRefMathSciNetGoogle Scholar
  31. 31.
    Smith, H.L.: Planar competitive and cooperative difference equations. J. Diff. Equ. Appl. 3, 335–357 (1998)CrossRefzbMATHGoogle Scholar
  32. 32.
    Smith, H.L.: A discrete, size-structured model of microbial growth and competition in the chemostat. J. Math. Biol. 34, 734–754 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    Smith, H.L., Zhao, X.-Q.: Competitive exclusion in a discrete-time, size-structured chemostat model. Discrete Contin. Dyn. Syst. Ser. B 1, 183–191 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  34. 34.
    Zhang, P., Ackleh, A.S.: A discrete stage-structured two species competition model with sexual and clonal reproduction. J. Biol. Dyn. 6, 2–16 (2012)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.University of Louisiana at LafayetteLafayetteUSA

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