Skip to main content
SpringerLink
Log in

Convergence in a resource-based competition system

  • Published:
Bulletin of Mathematical Biology Aims and scope Submit manuscript

Abstract

A resource-based competition model of two consumer species and one resource species is formulated in the form of a Lotka-Volterra system. The competition involves both exploitation and interference. By a method of asymptotic estimates, sufficient conditions are derived for the three species system to converge ast→∞ to an equilibrium point with all three species present; a generalization of the result forn≥2 and single resource species is indicated. The strong form of equilibrium perisistence of the three species consumer-resource system is achieved by the ability of each of the consumer species to exploit the resource and interfere with others in such a way which will avoid exclusion by the other.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Literature

  • Armstrong, R. A. and R. McGehee. 1976a. “Coexistence of Competing Species for Shared Resources.”Theor. Pop. Biol. 9, 317–328.

    Article  MATH  MathSciNet  Google Scholar 

  • — and —. 1976b. “Coexistence of Two Competitors on One Resource.”J. theor. Biol. 56, 499–502.

    Article  Google Scholar 

  • — and —. 1980. “Competitive Exclusion.”Am. Nat. 115, 151–170.

    Article  MathSciNet  Google Scholar 

  • Ayala, F. J., M. E. Gilpin and J. G. Ehrenfeld. 1973. “Competition Between Species: Theoretical Models and Experimental Tests.”Theor. Pop. Biol. 4, 331–356.

    Article  MathSciNet  Google Scholar 

  • Brian, M. V. 1956. “Exploitation and Interference in Interspecies Competition.J. Anim. Ecol. 25, 339–347.

    Article  Google Scholar 

  • Butler, G. J. and P. Waltman. 1981. “Bifurcation from a Limit Cycle in a Two Predator-One Prey Ecosystem Modelled on a Chemostat.”J. math. Biol. 12, 295–310.

    Article  MATH  MathSciNet  Google Scholar 

  • — and G. S. K. Wolkowicz. 1985. “A Mathematical Model of the Chemostat with a General Class of Functions Describing Nutrient Uptake.”SIAM J. appl. Math. 45, 138–151.

    Article  MATH  MathSciNet  Google Scholar 

  • —, S. B. Hsu and P. Waltman. 1983. “Coexistence of Competing Predators in a Chemostat.”J. math. Biol. 17, 133–151.

    Article  MATH  MathSciNet  Google Scholar 

  • Case, T. J. and R. G. Casten. 1979. “Global Stability and Multiple Domains of Attraction in Ecological Systems.”Am. Nat. 113, 705–714.

    Article  MathSciNet  Google Scholar 

  • — and M. E. Gilpin. 1974. “Interference Competition and Niche Theory.”Proc. natn. Acad. Sci. U.S.A. 71, 3073–3077.

    Article  Google Scholar 

  • Connell, J. H. 1961. “The Influence of Interspecific Competition and Other Factors on the Distribution of the BarnacleChthamlus Stellatus.”Ecology 42, 710–723.

    Article  Google Scholar 

  • Dayton, P. K. 1971. “Competition, Disturbance and Community Organisation; the Provision and Subsequent Utilization of Space in a Rocky Intertidal Community.”Ecol. Monographys 41, 351–389.

    Article  Google Scholar 

  • Gause, G. F. 1934. “The Struggle for Existence.” Baltimore: Williams and Wilkins.

    Google Scholar 

  • Gilpin, M. E. and F. J. Ayala. 1974. “Global Models of Growth and Competition.”Proc. natn. Acad. Sci. U.S.A. 70, 3590–3593.

    Article  Google Scholar 

  • Gopalsamy, K. 1980. “Time Lags and Global Stability in two Species Competition.”Bull. math. Biol. 42, 728–737.

    Google Scholar 

  • Hardin, G. 1960. “The Competitive Exclusion Principle.”Science 131, 1292–97.

    Google Scholar 

  • Hsu, S. B. 1981. “On a Resource Based Ecological Competition Model With Interference.”J. math. Biol. 12, 45–52.

    Article  MATH  MathSciNet  Google Scholar 

  • — 1978. “Limiting Behaviour for Competing Species.”SIAM J. appl. Math. 34, 760–763.

    Article  MATH  MathSciNet  Google Scholar 

  • — and S. P. Hubbell. 1979. “Two Predators for Two Prey Species: An Analysis of MacArthur's Model.Mathl Biosci. 47, 143–171.

    Article  MATH  MathSciNet  Google Scholar 

  • —, K. S. Cheng and S. P. Hubbell. 1981. “Exploitative Competition of Microorganisms for Two Complementary Nutrients in Continuous Cultures.”SIAM J. appl. Math. 41, 422–444.

    Article  MATH  MathSciNet  Google Scholar 

  • —, S. P. Hubbell and P. Waltman. 1977. “A Mathematical Theory of Single-Nutrient Competitions in Continuous Cultures of Microorganisms.”SIAM J. Math. 32, 366–383.

    Article  MATH  MathSciNet  Google Scholar 

  • — and —. 1978. “A Contribution to the Theory of Competing Predators.”Ecol. Monographs 48, 337–349.

    Article  Google Scholar 

  • Hutchinson, E. G. 1978. “An Introduction to Population Ecology.” New Haven: Yale University Press.

    MATH  Google Scholar 

  • Kaplan, J. L. and J. A. Yorke. 1977. “Competitive Exclusion and Nonequilibrium Coexistence.”Am. Nat. 111, 1030–1036.

    Article  Google Scholar 

  • Keener, J. P. 1983. “Oscillatory Coexistence in the Chemostat: A Codimension Two Unfolding.”SIAM J. appl. Math. 43, 1005–1018.

    Article  MATH  MathSciNet  Google Scholar 

  • Leon, J. A. and D. B. Tumpson. 1975. “Competition Between Two Species for Two Complementary or Two Substituble Resources.”J. theor. Biol. 50, 185–201.

    Article  Google Scholar 

  • Levin, S. A. 1970. “Community Equilibria and Stability, and an Extension of the Competitive Exclusion Principle.”Am. Nat. 104, 413–423.

    Article  Google Scholar 

  • Levins, R. 1968. Evolution in Changing Environments.” Princeton: Princeton University Press.

    Google Scholar 

  • Leung, A. 1976. “Limiting Behaviour for Several Interacting Populations.”Mathl Biosci. 29, 85–98.

    Article  MATH  Google Scholar 

  • Lotka, A. J. 1925. “Elements of Mathematical Biology.” Baltimore: Williams and Wilkins.

    Google Scholar 

  • MacArthur, R. and R. Levins. 1964. “Competition, Habitat Selection and Character Displacement in a Patchy Environment.”Proc. Natn. Acad. Sci. U.S.A. 51, 1207–1210.

    Article  Google Scholar 

  • MacArthur, R. 1970. “Species Packing and Competitive Equilibrium for Many Species.”Theor. Pop. Biol. 1, 1–11.

    Article  Google Scholar 

  • MacArthur, R. H. 1972.Geographical Ecology. New York: Harper and Row.

    Google Scholar 

  • McGehee, R. and R. A. Armstrong. 1977. “Some Mathematical Problems Concerning the Ecological Principle of Competitive Exclusion.”J. diff. Equns. 23, 30–52.

    Article  MATH  MathSciNet  Google Scholar 

  • Miller, R. S. 1976. “Pattern and Process in Competition.Adv. Ecol. Res. 4, 1–47.

    Article  Google Scholar 

  • Nitecki, Z. 1978. “A Periodic Attractor Determined by One Function.”J. diff. Eqns. 29, 214–234.

    Article  MATH  MathSciNet  Google Scholar 

  • Park, T. 1962. “Beetles, Competition and Populations.”Science 138, 1369–1375.

    Google Scholar 

  • — 1954. “Experimental Studies of Interspecific Competition: II. Temperature, Humidity and Competition Two Species of Tribolium.”Physiol. Zool. 27, 177–238.

    Google Scholar 

  • Rescigno, A. and I. W. Richardson. 1965. “On the Competitive Exclusion Principle.”Bull. math. Biophys. 27 (special issue) 85–89.

    MathSciNet  Google Scholar 

  • Schoener, T. W. 1973. “Population Growth Regulated by Intraspecific Competition for Energy or Time; Some Simple Representations.”Theor. Pop. Biol. 4, 56–84.

    Article  MATH  Google Scholar 

  • — 1974. “Competition and the Form of Habitat Shift.”Theor. Pop. Biol. 6, 265–307.

    Article  MATH  Google Scholar 

  • — 1976. “Alternatives to Lotka-Volterra Competition: Models of Intermediate Complexity.”Theor. Pop. Biol. 10, 309–333.

    Article  MATH  MathSciNet  Google Scholar 

  • Shigasada, N., K. Kawasaki and E. Teramoto. 1985. “The Effects of Interference Competition on Stability Structure and Invasion of a Multispecies System.J. math. Biol. 21, 97–113.

    Article  Google Scholar 

  • Smith, H. L. 1982. “The Interaction of a Steady State and Hopf Bifurcation in a Two Predator-One Prey Competition Models.”SIAM J. appl. Math. 42, 27–43.

    Article  MATH  MathSciNet  Google Scholar 

  • Smouse, P. E. 1980. “Mathematical Models for Continuous Culture Growth Dynamics of Mixed Populations Subsisting on a Heterogeneous Resource Base: I. Simple Competition.”Theor. Pop. Biol. 17, 16–36.

    Article  MATH  MathSciNet  Google Scholar 

  • — 1981. “Mathematical Models for Continuos Culture Growth Dynamics of Mixed Populations Subsisting on a Heterogeneous Base: II. Predation and Trophic Structure.”Theor. Pop. Biol. 20, 127–149.

    Article  MATH  MathSciNet  Google Scholar 

  • Stewart, F. M. and B. R. Levin. 1973. “Partitioning of Resources and the Outcome of Interspecific Competition: a Model and General Considerations.”Am. Nat. 107, 171–198.

    Article  Google Scholar 

  • Verhulst, P. F. 1838. Notice Sur la que la Population Pursuit Dans Son Accroissement.Correspond. math. Phys. 10, 113–121.

    Google Scholar 

  • Volterra, V. 1926. “Variations and Fluctuations of the Number of Individuals of Animal Species Living Together.”Animal Ecology, R. N. Chapman (Ed.). New York: McGraw Hill.

    Google Scholar 

  • Waltman, P. E., S. P. Hubbell and S. B. Hsu. 1979. “Theoretical and Experimental Investigation of Microbial Competition in Continuous Cultures.”Proc. Conference on Mathematical Modelling, Carbondale, Il.

  • — 1983.Competition Models in Population Biology. Philadelphia: SIAM.

    Google Scholar 

  • Wangersky, P. J. 1978. “Lotka-Volterra Population Models.”A. Rev. Ecol. Syst. 9, 189–218.

    Article  Google Scholar 

  • Winsor, C. P. 1934. “Mathematical Analysis of the Growth of Mixed Populations.”Cold Spring Harb. Symp. Quant. Biol. 2, 181–187.

    Google Scholar 

  • Zicarelli, J. 1975. “Mathematical Analysis of a Population Model with Several Predators on a Single Prey.” Ph.D. thesis, University of Minnesota, Minneapolis.

Download references

Author information

Author notes

  1. K. Gopalsamy

    Present address: The Flinders University of South Australia, 5042, Bedford Park, S.A., Australia

Authors and Affiliations

  1. Department of Mathematics, University of Alberta, T6G 2G1, Edmonton, Alberta, Canada

    K. Gopalsamy

Authors
  1. K. Gopalsamy
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gopalsamy, K. Convergence in a resource-based competition system. Bltn Mathcal Biology 48, 681–699 (1986). https://doi.org/10.1007/BF02462330

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02462330

Keywords

Search

Navigation