Journal of Mathematical Biology

, Volume 7, Issue 4, pp 319–337

Stability regions and transition phenomena for harvested predator-prey systems

  • F. Brauer
  • A. C. Soudack
Article

Summary

We analyze the global behaviour of a predator-prey system under constant-rate predator harvesting, showing how to classify the possibilities and determine the region of asymptotic stability by a combination of relatively elementary theoretical methods and computer simulations.

Key words

Predator-prey systems Harvesting 

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References

  1. Albrecht, F., Gatzke, H., Haddad, A., Wax, N.: The dynamics of two interacting populations. J. Math. Analysis and Appl. 46, 658–670 (1974)Google Scholar
  2. Brauer, F.: Destabilization of predator-prey systems under enrichment. Int. J. Control 23, 541–552 (1976)Google Scholar
  3. Brauer, F.: Boundedness of solutions of predator-prey systems. Theor. Pop. Biol. to appear, 1979Google Scholar
  4. Brauer, F., Soudack, A. C., Jarosch, H. S.: Stabilization, and destabilization of predator-prey systems under harvesting and nutrient enrichment. Int. J. Control 23, 553–573 (1976)Google Scholar
  5. Bulmer, M. G.: The theory of prey-predator oscillations. Theor. Pop. Biol. 9, 137–150 (1976)Google Scholar
  6. Coddington, E. A., Levinson, N.: Theory of Ordinary Differential Equations. New York: McGraw-Hill, 1955Google Scholar
  7. Conley, C. C.: Isolated Invariant Sets and the Morse Index. Conference Board on Mathematical Sciences, 1978Google Scholar
  8. Holling, C. S.: The functional response of predators to prey density and its rate in mimicry and population regulation. Mem. Ent. Soc. Canada 45, 1–73 (1965)Google Scholar
  9. Kolmogorov, A. N.: Sulla teoria di Volterra della lotta per l'esistenza. Giorn. Inst. Ital. Attuari 7, 74–80 (1936)Google Scholar
  10. Kopell, N., Howard, L. N.: Bifurcations and trajectories joining critical points. Adv. in Math. 18, 306–358 (1975)Google Scholar
  11. Ludwig, D., Jones, D. S., Holling, C. S.: Qualitative analyses of insect outbreak systems: The spruce budworm and forest. J. Animal Ecology 47, 315–332 (1978)Google Scholar
  12. May, R. M.: Stability and Complexity in Model Ecosystems. Princeton: Princeton Univ. Press, 1973Google Scholar
  13. Thom, R.: Structural Stability and Morphogenesis. Reading, Mass.: W. A. Benjamin, 1975Google Scholar

Copyright information

© Springer-Verlag 1979

Authors and Affiliations

  • F. Brauer
    • 1
  • A. C. Soudack
    • 2
  1. 1.Department of MathematicsUniversity of WisconsinMadisonUSA
  2. 2.Department of Electrical EngineeringUniversity of British ColumbiaVancouverCanada

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