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Stability regions and transition phenomena for harvested predator-prey systems

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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.

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References

  • 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 

  • Brauer, F.: Destabilization of predator-prey systems under enrichment. Int. J. Control 23, 541–552 (1976)

    Google Scholar 

  • Brauer, F.: Boundedness of solutions of predator-prey systems. Theor. Pop. Biol. to appear, 1979

  • 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 

  • Bulmer, M. G.: The theory of prey-predator oscillations. Theor. Pop. Biol. 9, 137–150 (1976)

    Google Scholar 

  • Coddington, E. A., Levinson, N.: Theory of Ordinary Differential Equations. New York: McGraw-Hill, 1955

    Google Scholar 

  • Conley, C. C.: Isolated Invariant Sets and the Morse Index. Conference Board on Mathematical Sciences, 1978

  • 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 

  • Kolmogorov, A. N.: Sulla teoria di Volterra della lotta per l'esistenza. Giorn. Inst. Ital. Attuari 7, 74–80 (1936)

    Google Scholar 

  • Kopell, N., Howard, L. N.: Bifurcations and trajectories joining critical points. Adv. in Math. 18, 306–358 (1975)

    Google Scholar 

  • 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 

  • May, R. M.: Stability and Complexity in Model Ecosystems. Princeton: Princeton Univ. Press, 1973

    Google Scholar 

  • Thom, R.: Structural Stability and Morphogenesis. Reading, Mass.: W. A. Benjamin, 1975

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Wisconsin, 53706, Madison, Wisconsin, USA

    F. Brauer

  2. Department of Electrical Engineering, University of British Columbia, V6T1W5, Vancouver, B.C., Canada

    A. C. Soudack

Authors
  1. F. Brauer
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  2. A. C. Soudack
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Additional information

Research sponsored in part by the National Research Council of Canada. Grant No. A-3138

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Brauer, F., Soudack, A.C. Stability regions and transition phenomena for harvested predator-prey systems. J. Math. Biol. 7, 319–337 (1979). https://doi.org/10.1007/BF00275152

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  • DOI: https://doi.org/10.1007/BF00275152

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