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A bioeconomic model of a ratio-dependent predator-prey system and optimal harvesting

  • T. K. Kar
  • Swarnakamal Misra
  • B. Mukhopadhyay
Article

Abstract

This paper deals with the problem of a ratio-dependent prey-predator model with combined harvesting. The existence of steady states and their stability are studied using eigenvalue analysis. Boundedness of the exploited system is examined. We derive conditions for persistence and global stability of the system. The possibility of existence of bionomic equilibria has been considered. The problem of optimal harvest policy is then solved by using Pontryagin’s maximal principle.

AMS Mathematics Subject Classification

92D25 34K20 49K15 

Key words and phrases

Harvesting ratio-dependent permanence bionomic-equilibrium optimal harvesting 

References

  1. 1.
    Arrow, K. J. and Kurz, M.,Public Investment, The Rate of Return and Optimal Fiscal Policy, John Hopkins, 1970.Google Scholar
  2. 2.
    Beretta, E. and Kuang, Y.,Global qualitative analysis of a ratio dependent prey-predator systems, J. Math. Biol.36 (1998), 389–406.MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Beretta, E. and Kuang, Y.,Global analysis in some delayed ratio-dependent prey-predator system, Nonlinear Analysis — TMA,32(3) (1998), 381–408.MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Berryman, A. A.,The origins and evolution of predator-prey theory, Ecology,73 (1992), 1530–1535.CrossRefGoogle Scholar
  5. 5.
    Bhattacharya, D. K. and Begum, S.,Bionomic equilibrium of two species system, Math. Biosci.135(2) (1996) 111–127.MATHCrossRefGoogle Scholar
  6. 6.
    Birkhoff, G. and Rota, G. C.,Ordinary Differential Equations, Ginn, 1982.Google Scholar
  7. 7.
    Brauer, F. and Soudack, A. C.,Stability regions and transition phenomena for harvested predator-prey systems, J. Math. Biol.7 (1979), 319–337.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Brauer, F. and Soudack,, A. C.,Stability regions in predator-prey systems with constant rate prey harvesting, J. Math. Biol.8 (1979), 55–71.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Brauer, F. and Soudack, A. C.,Constant rate stocking of predator-prey systems, J. Math. Biol.11 (1981), 1–14.MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Chaudhuri, K. S.,A bioeconomic model of harvesting, a multispecies fishery, Ecol. Model,32 (1986), 267–279.CrossRefGoogle Scholar
  11. 11.
    Chaudhuri, K. S. and Saha Ray, S.,On the combined harvesting of a preypredator system, J. Biol. Syst.4(1996), 376–389.CrossRefGoogle Scholar
  12. 12.
    Clark, C. W.,Mathematical Bioeconomics, the optimal Management of Renewable Resources, John Wiley, New York, 1990.MATHGoogle Scholar
  13. 13.
    Clark, C. W.,Bioeconomic Modeling and Fisheries Management, John Wiley, New York, 1985.Google Scholar
  14. 14.
    Dai, G. and Tang, M.,Coexistence region and global dynamics of a harvested predatorprey system, SI AM J. Appl. Math.13 (1998), 193–210.CrossRefMathSciNetGoogle Scholar
  15. 15.
    Gamito, S.,Growth models and their use in ecological modeling: an application to a fish population, Ecol. Model.113 (1998), 83–94.CrossRefGoogle Scholar
  16. 16.
    Jerry, M., and Raissi, N.,A policy of fisheries management based on continuous fishing effort, J. Biol. Syst.9 (2001), 247–254.CrossRefGoogle Scholar
  17. 17.
    Kar, T.K. and Chaudhuri, K. S.,On non-selective harvesting of two competing fish species in the presence of toxicity, Ecol. Model.161 (2003), 125–137.CrossRefGoogle Scholar
  18. 18.
    Kar, T. K.,Selective harvesting in a prey-predator fishery with time delay, Math. Comp. Model38 (2003), 449–458.MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Leung, A. W.,Optimal harvesting coefficient control of steady state prey predator diffusive Volterra-Lotka systems, Appl. Math. Optim.31 (1985), 219.CrossRefGoogle Scholar
  20. 20.
    Mesterton-Gibbons, M.,On the optimal policy for combined harvesting of independent species, Natural Resource Modeling,2 (1987), 107–132.Google Scholar
  21. 21.
    Mesterton-Gibbons, M.,On the optimal policy for combined harvesting of predator-prey, Nat. Res. Model3 (1988), 63–90.MathSciNetGoogle Scholar
  22. 22.
    Pontryagin, L. S., Boltyonskii, V. G., R. V. Gamkrelidre and E. F. Mishchenko,The Mathematical Theory of Optimal processes, Wiley, New York, 1962.MATHGoogle Scholar
  23. 23.
    Pradhan, T., and Chaudhuri, K. S.,Bioeconomic modelling of a single-species fishery with Gompertz law of growth, J. Biol. Syst.,6(4) (1988), 393–409.CrossRefGoogle Scholar
  24. 24.
    Pradhan, T., and Chaudhuri, K. S.,Bioeconomic harvesting of a schooling fish species: a dynamic reaction model, Korean J. Comput, & Appl. Math,6(1) (1999), 127–142.MATHMathSciNetGoogle Scholar
  25. 25.
    Ragozin, D. L. and Brown G. Jr.,Harvest policies and non-market valuations in a predator-prey system, J. Environ-Econ. Managm.12 (1985), 155–168.CrossRefGoogle Scholar
  26. 26.
    Rao, S. H. and Rao, R. S.,Stability analysis of a ratio prey-predator models, Nonlinear Analysis-Real World Application6 (2005), 245–262.MATHCrossRefGoogle Scholar
  27. 27.
    Samanta, G. P., Manna, D. and Maiti, A.,Bioeconomic modeling of a three-species fishery with switching effect, J. Appl. Math. Comp.12 (2003), 219–232.MATHMathSciNetCrossRefGoogle Scholar
  28. 28.
    Wilen, J. E.,Bioeconomics of renewable resource use in A.V. Kneese and J.L. Sweeney (eds.) Handbook of natural resource and energy economics, Vol. 1, North Holland, Amsterdam, 1985, 61–124.Google Scholar
  29. 29.
    Zheng, X., Chen, L. and Newmann, A. V.,The stage structured predator-prey model and optimal harvesting policy, Math. Biosci.,168(2) (2000), 201–210.CrossRefMathSciNetGoogle Scholar

Copyright information

© Korean Society for Computational & Applied Mathematics and Korean SIGCAM 2006

Authors and Affiliations

  • T. K. Kar
    • 1
  • Swarnakamal Misra
    • 2
  • B. Mukhopadhyay
    • 1
  1. 1.Department of MathematicsBengal Engineering and Science UniversityShibpur, HowrahIndia
  2. 2.Department of MathematicsDhakuria Ram Chandra High SchoolDhakuria, KolkataIndia

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