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Bulletin of Mathematical Biology

, Volume 48, Issue 5–6, pp 493–508 | Cite as

Predator-prey systems with group defence: The paradox of enrichment revisited

  • H. I. Freedman
  • G. S. K. Wolkowicz
Article

Abstract

The main concern of this paper is with survival or extinction of predators in models of predator-prey systems exhibiting group defence of the prey. It is shown that if there is no mutual interference among predators, enrichment could result in their extinction. However, if there is mutual interference, the predator population survives (at least deterministically).

Keywords

Periodic Orbit Homoclinic Orbit Positive Equilibrium Predator Population Mutual Interference 
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.

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Literature

  1. Andrews, J. F. 1968. “A Mathematical Model for the Continuous Culture of Microorganisms Utilizing Inhibitory Substrates”.Biotechnol. Bioengng 10, 707–723.CrossRefGoogle Scholar
  2. Aris, R. and A. E. Humphrey. 1977. “Dynamics of a Chemostat in which Two Organisms Compete for a Common Substrate”.Biotechnol. Bioengng 10, 1375–1386.CrossRefGoogle Scholar
  3. Beddington, J. R. 1975. “Mutual Interference Between Parasites of Predators and Its Effect on Searching Efficiency.”J. Anim. Ecol. 44, 331–340.CrossRefGoogle Scholar
  4. Boon, B. and H. Landelout. 1962. “Kinetics of Nitrite Oxidation by Nitrobacter Winogradski.”Biochem. J. 85, 440–447.Google Scholar
  5. Bush, A. W. and A. E. Cook. 1976. “The Effect of Time Delay and Growth Rate Inhibition in the Bacterial Treatment of Wastewater.”J. theor. Biol. 63, 385–395.CrossRefGoogle Scholar
  6. Butler, G. J., H. I. Freedman and P. E. Waltman. 1986. “Uniformly Persistent Systems.”Proc. Am. math. Soc. 96, 425–430.zbMATHMathSciNetCrossRefGoogle Scholar
  7. Erbe, L. H. and H. I. Freedman. 1985. “Modeling Persistence and Mutual Interference among Subpopulations of Ecological Communities.”Bull. math. Biol. 47, 295–304.zbMATHMathSciNetCrossRefGoogle Scholar
  8. Freedman, H. I. 1976. “Graphical Stability, Enrichment, and Pest Control by a Natural Enemy.”Mathl Biosci. 31, 207–225.zbMATHCrossRefGoogle Scholar
  9. —. 1979. “Stability Analysis of a Predator-Prey System with Mutual Interference and Density-dependent Death Rates.”Bull. math. Biol. 41, 167–178.CrossRefGoogle Scholar
  10. —. 1980.Deterministic Mathematical Models in Population Ecology. New York: Marcel Dekker.Google Scholar
  11. — and V. S. H. Rao. 1983. “The Trade-off Between Mutual Interference and Time Lags in Predator-Prey Systems.”Bull. math. Biol. 45, 991–1004.zbMATHMathSciNetGoogle Scholar
  12. Gilpin, M. E. 1972. “Enriched Predator-Prey Systems: Theoretical Stability.”Science 177, 902–904.Google Scholar
  13. Hassell, M. P. 1971. “Mutual Interference between Searching Insect Parasites.”J. Anim. Ecol. 40, 473–486.CrossRefGoogle Scholar
  14. Holling, C. S. 1965. “The Functional Response of Predators to Prey Density and its Role in Mimicry and Population Regulation.”Mem. ent. Soc. Can. 45, 3–60.Google Scholar
  15. Holmes, J. C. and W. M. Bethel. 1972. “Modification of Intermediate Host Behaviour by Parasites.”Zool. J. Linn. Soc., Suppl. 1 51, 123–149.Google Scholar
  16. Huffaker, C. B., K. P. Shea, S. G. Herman. 1963. “Experimental Studies on Predator: Complex Dispersion and Levels of Food in an Acarine Predator-Prey Interaction.”Hilgardia 34, 305–329.Google Scholar
  17. Luckinbill, L. S. 1973. “Coexistence in Laboratory Populations ofParamecium Aurelia and Its PredatorDidinium Nasutum.”Ecology 54, 1320–1327.CrossRefGoogle Scholar
  18. McAllister, C. D., R. J. Lebrasseur and T. R. Parsons. 1972. “Stability of Enriched Aquatic Ecosystems.”Science 175, 562–564.Google Scholar
  19. May, R. M. 1972. “Limit Cycles in Predator-Prey Communities.”Science 177, 900–902.Google Scholar
  20. Riebesell, J. F. 1974. “Paradox of Enrichment in Competitive Systems.”Ecology 55, 183–187.CrossRefGoogle Scholar
  21. Rogers, D. J. and M. P. Hassell. 1974. “General Models for Insect Parasite and Predator Searching Behaviour: Interference.”J. Anim. Ecol. 43, 239–253.CrossRefGoogle Scholar
  22. Rosenzweig, M. L. 1971. “Raradox of Enrichment: Destabilization of Exploitation Ecosystems in Ecological Time.”Science 171, 385–387.Google Scholar
  23. —. 1972a. “Reply to McAllisteret al.Science 175, 564–565.Google Scholar
  24. —. 1972b. “Reply to Gilpin.”Science 177, 904.Google Scholar
  25. — and W. M. Schaffer. 1978. “Homage to the Red Queen II. Coevolutionary Response to Enrichment of Exploitation Ecosystems.”Theor. Pop. Biol. 14, 158–163.zbMATHMathSciNetCrossRefGoogle Scholar
  26. Schaffer, W. M. and M. L. Rosenzweig. 1978. “Homage to the Red Queen I. Coevolution of Predators and their Victims.”Theor. Pop. Biol. 14 135–157.zbMATHMathSciNetCrossRefGoogle Scholar
  27. Tener, J. S. 1965.Muskoxen. Ottawa: Queen's Printer.Google Scholar
  28. Yang, R. D. and A. E. Humphrey. 1975. “Dynamics and Steady State Studies of Phenol Biodegeneration in Pure and Mixed Cultures.”Biotechnol. Bioengng 17, 1211–1235.CrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 1986

Authors and Affiliations

  • H. I. Freedman
    • 1
  • G. S. K. Wolkowicz
    • 2
  1. 1.Department of MathematicsUniversity of AlbertaEdmontonCanada
  2. 2.Department of Mathematics and Computer ScienceEmory UniversityAtlantaU.S.A.

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