Skip to main content

Advertisement

SpringerLink
Log in

Optimal Management of a Renewable Resource Subjected to Strong Allee effect in a Periodically Fluctuating Environment

  • Original Research
  • Published:
Differential Equations and Dynamical Systems Aims and scope Submit manuscript

Abstract

In this article we present the qualitative properties of solutions of a periodic differential equation that describes dynamics of a renewable resource subjected to strong Allee effect (multiplicative Allee effect) with periodic harvesting in a periodically fluctuating environment. We also present the bio-economic aspects of the considered renewable resource studied from sole owner perspective. An optimal harvesting problem with periodic dynamic constraints has been solved using Pontryagin’s maximum principle. It is interesting to find that the considered optimal control problem admits multiple periodic singular solutions. Techniques are developed to identify the optimal singular solution and the associated optimal singular periodic effort function. Optimal, suboptimal harvest policies and their corresponding approach paths are constructed. It is found that the optimal harvest policies are periodic after some time.

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.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. Aguirre, P., González-Olivares, E., Sáez, E.: Three limit cycles in a Leslie–Gower predator–prey model with additive Allee effect. SIAM J. Appl. Math. 69(5), 1244–1262 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  2. Aguirre, P., González-Olivares, E., Torres, S.: Stochastic predatorprey model with Allee effect on prey. Nonlinear Anal. Real World Appl. 14, 768–779 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  3. Allee, W.C.: Animal Aggregations. University of Chicago Press, Chicago (1931)

    Google Scholar 

  4. Allee, W.C.: Cooperation Among Animals. Henry Schuman, New York (1951)

    Google Scholar 

  5. Anderson, L.G., Seijo, J.C.: Bio-economics of Fisheries Management. Wiley-Blackwell, Singapore (2010)

    Google Scholar 

  6. Berec, L., Angulo, E., Counchamp, F.: Multiple Allee effects and population management. Trends Ecol. Evol. 22(4), 185–191 (2006)

    Article  Google Scholar 

  7. Bowden, D.A.: Seasonality of recruitment in Antarctic sessile marine bentho. Marine Ecol. Prog. Ser. 97, 10–18 (2005)

    Google Scholar 

  8. Castilho, C., Srinivasu, P.D.N.: Bio-economics of a renewable resource in a seasonally varying environment. Math. Biosci. 205, 1–18 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  9. Clark, C.W.: Mathematical Bioeconomics: The Optimal Management of Renewable Resources, 2nd edn. Wiley, New York (2005)

    Google Scholar 

  10. Correigh, M.G.: Habitat selection reduces extinction of populations subject to Allee effects. Theor. Popul. Biol. 64, 1–10 (2003)

    Article  MATH  Google Scholar 

  11. Courchamp, F., Berec, L., Gascoigne, J.: Allee Effects in Ecology and Conservation. Oxford University Press, New York (2008)

    Book  Google Scholar 

  12. Cushing, J.M.: Oscillatory population growth in periodic environments. Theor. Popul. Biol. 30, 289–308 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  13. David, S.B., Berec, L.: Single-species models of the Allee effect: extinction boundaries, sex ratios and mate encounters. J. Theor. Biol. 218, 375–394 (2002)

    Article  Google Scholar 

  14. David, S.B., Maurice, W.S., Berec, L.: How predator functional responses and Allee effects in prey affect the paradox of enrichment and population collapses. Theor. Popul. Biol. 72, 136–147 (2007)

    Article  MATH  Google Scholar 

  15. Drake, J.M., Kramer, A.M.: Allee Effects. Nat. Educ. Knowl. 3(10), 2 (2011)

    Google Scholar 

  16. Bayramoğlu Rizaner, F., Rogovchenko, S.P.: Dynamics of a single species under periodic habitat fluctuations and Allee effect. Nonlinear Anal. Real World Appl. 13, 141–157 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  17. Graef, J.R., Padhi, S., Pati, S.: Periodic solutions of some models with strong Allee effects. Nonlinear Anal. Real World Appl. 13, 569–581 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  18. Hilker, F.M., Langlais, M., Petrovskii, S.V., Malchow, H.: A diffusive SI model with Allee effect and application to FIV. Math. Biosci. 206, 61–80 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  19. King, A.C., Billingham, J., Otto, S.R.: Differential Equations Linear, Nonlinear, Ordinary, Partial. Cambridge University Press, Cambridge (2003)

    Book  MATH  Google Scholar 

  20. Lidicker Jr, W.Z.: The Allee effect: its history and future importance. Open Ecol. J. 3, 71–82 (2010)

    Article  Google Scholar 

  21. Lin, Z.S., Li, B.L.: The maximum sustainable yield of Allee dynamic system. Ecol. Model. 154, 1–7 (2002)

    Article  Google Scholar 

  22. Sen, M., Banerjee, M., Morozov, A.: Bifurcation analysis of a ratio-dependent prey predator model with the Allee effect. Ecol. Complex. 11, 12–27 (2012)

    Article  Google Scholar 

  23. Padhi, S., Srinivasu, P.D.N., Kiran Kumar, G.: Periodic solutions for an equation governing dynamics of a renewable resource subjected to Allee effects. Nonlinear Anal. Real World Appl. 11, 2610–2618 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  24. Patric, C.T., Ludek, B., Andrew, M.L.: Exploiting Allee effects for managing biological invasions. Ecol. Lett. 14, 615–624 (2011)

    Article  Google Scholar 

  25. Pontryagin, L.S., Bol’tyankii, V.G., Gamkrelidge, R.V., Mishchenko, E.F.: The Mathematical Thoery of Optimal Processes. Pergamon press, London (1964)

    Google Scholar 

  26. Rojas-Palma, A., González-Olivares, E.: Optimal harvesting in a predator–prey model with Allee effect and sigmoid functional response. Nonlinear Anal. Real World Appl. 12(3), 1864–1874 (2012)

    Google Scholar 

  27. Zhou, S.R., Wang, G.: Allee-like effects in metapopulation dynamics. Math. Biosci. 189, 103–113 (2004)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

The authors gratefully acknowledge three anonymous reviewers for their valuable suggestions. First author acknowledges the support received from National Board for Higher Mathematics, (DAE), Govt. of India under sponsored research scheme, vide Grant no. 2/48(8)/2009-R&D II/437. Second author acknowledges the support received from Council of Scientific and Industrial Research, sanction no. 09/002(0481)/2007-EMR-1 and NBHM Award No. 2/40(47)/2011/R&D-II.

Author information

Authors and Affiliations

  1. Department of Mathematics, Andhra University, Visakhapatnam , 530 003, India

    P. D. N. Srinivasu

  2. Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur , 208 016, India

    G. Kiran Kumar

Authors
  1. P. D. N. Srinivasu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. G. Kiran Kumar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to P. D. N. Srinivasu.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Srinivasu, P.D.N., Kiran Kumar, G. Optimal Management of a Renewable Resource Subjected to Strong Allee effect in a Periodically Fluctuating Environment. Differ Equ Dyn Syst 22, 397–412 (2014). https://doi.org/10.1007/s12591-013-0185-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12591-013-0185-3

Keywords

Search

Navigation