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Stability results for delayed-recruitment models in population dynamics

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This paper relates the stability properties of a class of delay-difference equations to those of an associated scalar difference equation. Simple but powerful conditions for testing global stability are presented which are independent of the length of the time delay involved. For models which do not have globally stable equilibria, estimates of stability regions are obtained. Some well known baleen whale models are used to illustrate the results.

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  1. Allen, K. R.: Analysis of stock-recruitment relations in Antarctic fin whales. Cons. Int. pour l'Explor. Mer-Rapp. Proc.-Verb. 164, 132–137 (1963)

  2. Beddington, J. R.: On the dynamics of Sei whales under exploitation. Rep. Int. Whal. Commn. 28, 169–172 (1978a)

  3. Beddington, J. R.: On the risks associated with different harvesting strategies in the harvesting of baleen whales. Rep. Int. Whal. Commn. 28, 165–167 (1978b)

  4. Beddington, J. R., May, R. M.: Harvesting natural populations in a randomly fluctuating environment. Science 197, 463–465 (1977)

  5. Beverton, R. J. H., Holt, S. J.: On the dynamics of exploited fish populations. Fish. Invest. London 19, 1–533 (1957)

  6. Clark, C. W.: A delayed-recruitment model of population dynamics, with an application to baleen whale populations. J. Math. Biol. 3, 381 -391 (1976)

  7. Goh, B. S.: Management and analysis of biological populations. Amsterdam: Elsevier 1980

  8. Horwood, J. W., Knights, B. J., Overy, R. W.: Harvesting of whale populations subject to stochastic variability. Rep. Int. Whal. Commn. 29, 219–224 (1979)

  9. International Whaling Commission: Twenty-Sixth Annual Report of the International Commission on Whaling. London (1976)

  10. International Whaling Commission: Twenty-Eighth Annual Report of the International Commission on Whaling. London (1978)

  11. La Salle, J. P.: The stability of dynamical systems. SIAM, Philadelphia (1976)

  12. Levin, S. A., May, R. M.: A note on difference-delay equations. Theoret. Population Biol. 9, 178–187 (1976)

  13. May, R. M.: Biological populations with nonoverlapping generations: Stable points, stable cycles and chaos. Science 186, 645–647 (1974)

  14. Ricker, W. E.: Stock and recruitment. J. Fish. Res. Bd. Can. 11, 559–623 (1954)

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Fisher, M.E., Goh, B.S. Stability results for delayed-recruitment models in population dynamics. J. Math. Biology 19, 147–156 (1984). https://doi.org/10.1007/BF00275937

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Key words

  • Stability
  • delay
  • difference equations
  • whale models