Skip to main content

Advertisement

SpringerLink
Log in

Continuous versus pulse harvesting for population models in constant and variable environment

  • Published:
Journal of Mathematical Biology Aims and scope Submit manuscript

Abstract

We consider both autonomous and nonautonomous population models subject to either impulsive or continuous harvesting. It is demonstrated in the paper that the impulsive strategy can be as good as the continuous one, but cannot outperform it. We introduce a model, where certain harm to the population is incorporated in each harvesting event, and study it for the logistic and the Gompertz laws of growth. In this case, impulsive harvesting is not only the optimal strategy but is the only possible one.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Clark C.W. (1976). Mathematical Bioeconomics: The Optimal Management of Renewable Resources. Wiley, New York

    MATH  Google Scholar 

  2. Cooke K.L. and Witten M. (1986). One-dimensional linear and logistic harvesting models. Math. Model. 7: 301–340

    Article  MATH  MathSciNet  Google Scholar 

  3. Gao S.J., Chen L.S. and Sun L.H. (2005). Optimal pulse fishing policy in stage-structured models with birth pulses. Chaos Solutions Fractals 25(5): 1209–1219

    Article  MATH  MathSciNet  Google Scholar 

  4. Tang S., Cheke R.A. and Xiao Y. (2006). Optimal impulsive harvesting on non-autonomous Beverton–Holt difference equations. Nonlinear Anal. Ser. A: Theory Methods 65: 2311–2341

    Article  MATH  MathSciNet  Google Scholar 

  5. Xu C., Boyce M.S. and Daley D.J. (2005). Harvesting in seasonal environments. J. Math. Biol. 50: 663–682

    Article  MATH  MathSciNet  Google Scholar 

  6. Zhang X., Shuai Z. and Wang K. (2003). Optimal impulsive harvesting policy for single population. Nonlinear Anal. Real World Appl. 4: 639–651

    Article  MATH  MathSciNet  Google Scholar 

  7. Zhang Y., Xiu Z. and Chen L. (2006). Optimal impulsive harvesting of a single species with Gompertz law of growth. J. Biol. Syst. 14: 303–314

    Article  MATH  Google Scholar 

  8. Fan M. and Wang K. (1998). Optimal harvesting policy for single population with periodic coefficients. Math. Biosci. 152: 165–177

    Article  MATH  MathSciNet  Google Scholar 

  9. Xiao Y., Cheng D. and Qin H. (2006). Optimal impulsive control in periodic ecosystems. Syst. Control Lett. 55: 556–565

    MathSciNet  Google Scholar 

  10. Angelova J. and Dishliev A. (2000). Optimization problems for one-impulsive models from population Dynamics. Nonlinear Anal. 39: 483–497

    Article  MathSciNet  Google Scholar 

  11. Braverman E., Israeli M., Averbuch A. and Vozovoi L. (1998). A fast 3-D Poisson solver of arbitrary order accuracy. J. Comput. Phys. 144: 109–136

    Article  MathSciNet  Google Scholar 

  12. Boyce M.S. and Daley D.J. (1980). Population tracking of fluctuating environments and natural selection for tracking ability. Am. Nat. 115: 480–491

    Article  MathSciNet  Google Scholar 

  13. Chen L., Sun L. and Dong (2007). Optimal harvesting policies for periodic Gompertz systems. Nonlinear Anal. Real World Appl. 8: 572–578

    Article  MathSciNet  Google Scholar 

  14. Berezansky L. and Braverman E. (2004). On impulsive Beverton–Holt difference equations and their applications. J. Differ. Equ. Appl. 10: 851–868

    Article  MATH  MathSciNet  Google Scholar 

  15. Braverman E. (2005). On a discrete model of population dynamics with impulsive harvesting or recruitment. Nonlinear Anal. Ser. A: Theory Methods 63: e751–e759

    Article  Google Scholar 

  16. Braverman E. and Kinzebulatov D. (2006). On linear perturbations of the Ricker model. Math. Biosci. 202: 323–339

    Article  MATH  MathSciNet  Google Scholar 

  17. Schreiber A.J. (2001). Chaos and population disappearances in simple ecological models. J. Math. Biol. 42: 239–260

    Article  MATH  MathSciNet  Google Scholar 

  18. Braverman, E., Mamdani, R.: On optimal impulsive sustainable harvesting, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 14(S2) (suppl.), Adv Dynamical Syst., 112–116 (2007)

  19. Artstein Z. (1993). Chattering limit for a model of harvesting in a rapidly changing environment. Appl. Math. Optim. 28: 133–147

    Article  MATH  MathSciNet  Google Scholar 

  20. Ludwig D. (1980). Harvesting strategies for a randomly fluctuating population. J. Cons. Int. Explor. Mer. 39: 168–174

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Calgary, 2500 University Drive NW, Calgary, AB, T2N 1N4, Canada

    Elena Braverman

  2. University of Calgary, Calgary, AB, Canada

    Reneeta Mamdani

Authors
  1. Elena Braverman
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Reneeta Mamdani
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Elena Braverman.

Additional information

The authors were partially supported by the NSERC Research Grant and the AIF Research Grant.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Braverman, E., Mamdani, R. Continuous versus pulse harvesting for population models in constant and variable environment. J. Math. Biol. 57, 413–434 (2008). https://doi.org/10.1007/s00285-008-0169-z

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00285-008-0169-z

Keywords

Mathematics Subject Classification (2000)

Search

Navigation