The Pure and Applied Mathematics (한국수학교육학회지시리즈B:순수및응용수학)

Korean Society of Mathematical Education (한국수학교육학회)
권호 표지

STABILITY ANALYSIS FOR PREDATOR-PREY SYSTEMS

  • Shim, Seong-A (DEPARTMENT OF MATHEMATICS, SUNGSHIN WOMEN'S UNIVERSITY)
  • Received : 2009.12.31
  • Accepted : 2010.08.19
  • Published : 2010.08.31
Download PDF
⟨ Previous Next ⟩

Abstract

Various types of predator-prey systems are studied in terms of the stabilities of their steady-states. Necessary conditions for the existences of non-negative constant steady-states for those systems are obtained. The linearized stabilities of the non-negative constant steady-states for the predator-prey system with monotone response functions are analyzed. The predator-prey system with non-monotone response functions are also investigated for the linearized stabilities of the positive constant steady-states.

Keywords

Acknowledgement

Supported by : Korea Research Foundation

References

  1. J. Andrews: A Mathematical model for the continuous culture of micro-organisms utilizing inhibitory substrates. Biotechnol. bioengng. 10 (1968), 707-723. https://doi.org/10.1002/bit.260100602
  2. R. Bhattacharyya, B. Mukhopadhyay & M. Bandyopadhyay: Diffusion-driven stability analysis of a prey-predator system with Holling type-IV functional response. Systems Analysis Modelling Simulation 43 (2003), no. 8, 1085-1093. https://doi.org/10.1080/0232929031000150409
  3. A. Bush & A. Cook: The effect of time delay and growth rate inhibition in the bacterial treatment of wastwater. J. Theor. Bio. 63 (1976), 385-395. https://doi.org/10.1016/0022-5193(76)90041-2
  4. H. Freedman: Deterministic Mathematical Models in Population Ecology. Marcel Dekker, New York, 1980.
  5. H. Freedman & G. Wolkowicz: Predator-prey systems with group defence: the paradox of enrichment revisited. Bull. Math. Biol. 48 (1986), 493-508. https://doi.org/10.1007/BF02462320
  6. J. Haldane: Enzymes. Longman, London, 1930.
  7. C. Holling: The components of predation as revealed by a study of small-mammal predation of the European pine sawfly. Can. Entomol. 91 (1959), 293-320. https://doi.org/10.4039/Ent91293-5
  8. C. Holling: The functional response of predators to prey density and its role in mimicry and population regulation. Mem. Entomol. Soc. Can. 45 (1965), 3-60.
  9. Y. Kuang & H. Freedman: Uniqueness of limit cycles in Gause-type models of predatorprey system. Math. Biosci. 88 (1988), 67-84. https://doi.org/10.1016/0025-5564(88)90049-1
  10. R. May: Limit cycles in predator-prey communities. Science 177 (1972), 900-902. https://doi.org/10.1126/science.177.4052.900
  11. J. Riebesell: Paradox of enrichment in competitive systems. Ecology 55 (1974), 183-187. https://doi.org/10.2307/1934634
  12. M. Rosenzweig: Paradox of enrichment: destabilization of exploitation ecosystems in ecological time. Science 171 (1971), 385-387. https://doi.org/10.1126/science.171.3969.385
  13. M. Rosenzweig: Reply to McAllister et al. Science 175 (1972), 564-565.
  14. M. Rosenzweig: Reply to Gilpin. Science 177 (1972), 904. https://doi.org/10.1126/science.177.4052.904
  15. L. Real: Ecological determinants of functional response. Ecology 60 (1972), 481-485.
  16. S. Shim: Hopf Bifurcation properties of Holling Type Predator-prey Systems. J. Korea Soc. Math. Educ. Ser. B: Pure Appl. Math. 15 (2008), no. 3, 329-342
  17. J. Sugie, R. Kohno & R. Miyazaki: On a predator-prey system of Holling type. Proc. Amer. Math. Soc. 125 (1997), 2041-2050. https://doi.org/10.1090/S0002-9939-97-03901-4
  18. J. Tenor: Muskoxen. Queen's Printer, Ottawa (1965).