Advertisement

Nonlinear Dynamics

, Volume 72, Issue 3, pp 591–603 | Cite as

Conservation of a resource based fishery through optimal taxation

  • U. K. Pahari
  • T. K. Kar
Original Paper

Abstract

In the present paper, we study a dynamic reaction model in which (i) the predator is provided with an alternative food in addition to the prey species, (ii) the predator is harvested, and (iii) a tax is imposed to regulate the system. The existence of possible steady states along with their local as well as global stability is discussed for both the exploited and unexploited systems. Boundedness of the system is also discussed. It is seen that the system undergoes a Hopf bifurcation by the addition of alternative prey and the criteria for the Hopf-bifurcation is also discussed. Optimal tax policy is discussed using Pontryagin’s maximal principle. Finally, some numerical simulations are given to show the consistency with theoretical analysis.

Keywords

Alternative prey Dynamic reaction Harvesting Global stability Bifurcation Taxation 

Notes

Acknowledgements

Research of U.K. Pahari was supported by the UGC, India (Grant No. F. PSW-180/09-10(ERO)).

References

  1. 1.
    Birkhoff, G., Rota, G.C.: Ordinary Differential Equations. Blaisdell, Waltham (1982) Google Scholar
  2. 2.
    Clark, C.W.: Mathematical Bioeconomics: The Optimal Management of Renewable Resources, 2nd edn. Wiley, New York (1990) MATHGoogle Scholar
  3. 3.
    Conway, M.I., Smoller, J.A.: Global analysis of a system of predator–prey equations. SIAM J. Appl. Math. 46, 630–642 (1986) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Dhar, J., Sharma, A.K., Tegar, S.: The role of delay in digestion of plankton by fish population: a fishery model. J. Nonlinear Sci. Appl. 1(1), 13–19 (2008) MathSciNetMATHGoogle Scholar
  5. 5.
    Dia, G., Tang, M.: Coexistence region and global dynamics of harvested prey predator system. SIAM J. Appl. Math. 58, 193–210 (1998) MathSciNetCrossRefGoogle Scholar
  6. 6.
    Dubey, B., Sinha, P., Chandra, P.A.: A model of an inshore-offshore fishery. J. Biol. Syst. 11(1), 27–41 (2003) MATHCrossRefGoogle Scholar
  7. 7.
    Freedman, H.I.: Deterministic Mathematical Models in Population Ecology. Decker, New York (1980) MATHGoogle Scholar
  8. 8.
    Harwood, J.D., Obryeki, J.J.: The role of alternative prey in sustaining predator population. In: Hoddle, M.S. (ed.) Proceedings of 2nd International Symposium on Biological Control of Arthropods, Davos, Switzerland, vol. 2, pp. 453–462 (2005) Google Scholar
  9. 9.
    Holling, C.S.: The functional response of predator to prey density and its role in mimicry and population regulation. Mem. Ent. Sec. Can. 45, 1–60 (1965) CrossRefGoogle Scholar
  10. 10.
    Holt, R.D., Lawton, J.H.: The ecological consequences of shard natural enemies. Annu. Rev. Ecol. Syst. 25, 495–520 (1994) CrossRefGoogle Scholar
  11. 11.
    Kapur, J.N.: Mathematical Models in Biology and Medicine. Affiliated East-West Press, New Delhi (1985) MATHGoogle Scholar
  12. 12.
    Kapur, J.N.: Mathematical Modeling. Wiley, Easter (1985) Google Scholar
  13. 13.
    Kar, T.K., Chaudhuri, K.S.: Regulation of a prey–predator fishery by taxation: a dynamic reaction model. J. Biol. Syst. 11(2), 173–187 (2003) MATHCrossRefGoogle Scholar
  14. 14.
    Kar, T.K., Pahari, U.K., Chaudhari, K.S.: Management of a single species fishery with stage-structure. Int. J. Math. Educ. Sci. Technol. 35(3), 403–414 (2004) CrossRefGoogle Scholar
  15. 15.
    Kar, T.K., Matsuda, H.: Controllability of a harvested prey–predator system with time delay. J. Biol. Syst. 14(2), 1–12 (2006) CrossRefGoogle Scholar
  16. 16.
    Kar, T.K., Pahari, U.K.: Modelling and analysis of a prey–predator systems with stage-structure and harvesting. Nonlinear Anal., Real World Appl. 8, 601–609 (2007) MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Marsden, J.E., McCracken, M.: The Hopf Bifurcation and Its Applications. Appl. Math. Sci., vol. 19. Springer, Berlin (1976) MATHCrossRefGoogle Scholar
  18. 18.
    Myerscough, M.R., Gray, B.F., Hogarth, W.L., Norbury, J.: An analysis of an ordinary differential equations model for a two species predator–prey system with harvesting and stocking. J. Math. Biol. 30, 389–411 (1992) MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Colinvaux, P.A.: Ecology. Wiley, New York (1986) Google Scholar
  20. 20.
    Wootton, J.T.: The nature and consequences of indirect effects in ecological communities. Annu. Rev. Ecol. Syst. 25, 443–466 (1994) CrossRefGoogle Scholar
  21. 21.
    Xiao, D., Ruan, S.: Bogdanov–Takens bifurcations in predator–prey systems with constant rate harvesting. Fields Inst. Commun. 21, 493–506 (1999) MathSciNetGoogle Scholar
  22. 22.
    Zhang, Z., Zhang, X., Chen, L.: The effect of pulsed harvesting policy on the inshore-offshore fishery model with the impulsive diffusion. Nonlinear Dyn. 63, 537–545 (2011) CrossRefGoogle Scholar
  23. 23.
    Zhang, Y., Zhang, Q.: Dynamic behavior in a delayed stage-structured population model with stochastic fluctuation and harvesting. Nonlinear Dyn. 66, 231–245 (2011) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Department of MathematicsSree Chaitanya CollegeHabraIndia
  2. 2.Department of MathematicsBengal Engineering and Science UniversityHowrahIndia

Personalised recommendations