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Stability and bionomic analysis of fuzzy parameter based prey–predator harvesting model using UFM

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Abstract

In this paper, a novel concept of fuzzy prey–predator model is introduced by considering the imprecise nature of the biological parameters. We consider the imprecise biological parameters as a form of triangular fuzzy number in nature. These imprecise parameters first transform to the corresponding intervals and then using interval mathematics the related differential equation is converted to two differential equations. Then using utility function method, the converted differential equations is changed to a single differential equation. The possibility of existence of both biological and bionomic equilibrium is presented. We obtain the conditions of local and global stability under impreciseness. We also study the optimal harvesting policy and derive the optimal solution under imprecise biological parameters. Lastly, numerical examples are presented to the support of our proposed approach.

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References

  1. Lotka, A.J.: Elements of Physical Biology. Williams and Wilkins, Baltimore (1925)

    MATH  Google Scholar 

  2. Volterra, V.: Lecons sur la theorie mathematique de la lutte pour la vie. Gauthier-Villars, Paris (1931)

    Google Scholar 

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

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  6. Kar, T.K.: Modelling and analysis of a harvested prey–predator system incorporating a prey refuge. J. Comput. Appl. Math. 185, 19–33 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  7. Shih, S.D., Chow, S.S.: Equivalence of n-point Gauss–Chebyshev rule and 4n-point midpoint rule in computing the period of a Lotka–Volterra system. Adv. Comput. Math. 28, 63–79 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  8. Kim, M.Y.: Global dynamics of approximate solutions to an age-structured epidemic model with diffusion. Adv. Comput. Math. 25, 451–474 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  9. Pei, Y., Liu, S., Li, C.: Complex dynamics of an impulsive control system in which predator species share a common prey. J. Nonlinear Sci. 19, 249–266 (2009)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  11. Wang, J., Wang, K.: Optimal control of harvesting for single population. Appl. Math. Comput. 156, 235–247 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  12. Gao, S., Chen, L., Sun, L.: Optimal pulse fishing policy in stage-structured models with birth pulses. Chaos Solitons Fract. 25, 1209–1219 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  13. Chakraborty, K., Chakraborty, M., Kar, T.K.: Optimal control of harvest and bifurcation of a prey–predator model with stage structure. Appl. Math. Comput. 217, 8778–8792 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  14. Chakraborty, K., Das, S., Kar, T.K.: Optimal control of effort of a stage structured prey–predator fishery model with harvesting. Nonlinear Anal. Real World Appl. 12, 3452–3467 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  15. Liao, X., Chen, Y., Zhou, S.: Traveling wavefrons of a prey–predator diffusion system with stage-structure and harvesting. J. Comput. Appl. Math. 235, 2560–2568 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  16. Qu, Y., Wei, J.: Bifurcation analysis in a time-delay model for prey–predator growth with stage-structure. Nonlinear Dyn. 49, 285–294 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  17. Seo, G., DeAngelis, D.L.: A predator–prey model with a Holling type I functional response including a predator mutual interference. J. Nonlinear Sci. 21, 811–833 (2011)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  19. Lande, R., Engen, S., Sæther, B.E.: Optimal harvesting of fluctuating populations with a risk of extinction. Am. Nat. 145, 728–745 (1995)

    Article  Google Scholar 

  20. Alvarez, L.H.R., Shepp, L.A.: Optimal harvesting of stochastically fluctuating populations. Math. Biosci. 37, 155–177 (1998)

    MATH  MathSciNet  Google Scholar 

  21. Alvarez, L.H.R.: Optimal harvesting under stochastic fluctuations and critical depensation. Math. Biosci. 152, 63–85 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  22. Li, W., Wang, K., Su, H.: Optimal harvesting policy for stochastic logistic population model. Appl. Math. Comput. 218, 157–162 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  24. Wu, R., Zou, X., Wang, K.: Asymptotic properties of a stochastic Lotka–Volterra cooperative system with impulsive perturbations. Nonlinear Dyn. (2014). doi:10.1007/s11071-014-1343-z

  25. Liu, M., Wang, K.: Dynamics of a Leslie–Gower Holling-type II predator–prey system with Lévy jumps. Nonlinear Anal. Theory Methods Appl. 85, 204–213 (2013)

    Article  MATH  Google Scholar 

  26. Ji, C., Jiang, D., Li, X.: Qualitative analysis of a stochastic ratio-dependent predator–prey system. J. Comput. Appl. Math. 235, 1326–1341 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  27. Liu, M., Wang, K.: Population dynamical behavior of Lotka–Volterra cooperative systems with random perturbations. Discrete Contin. Dyn. Syst. 85, 204–213 (2013)

    MATH  Google Scholar 

  28. Liu, M., Wang, K.: Analysis of a stochastic autonomous mutualism model. J. Math. Anal. Appl. 402, 392–403 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  29. Liu, M., Wang, K.: Dynamics of a two-prey one-predator system in random environments. J. Nonlinear Sci. 23, 751–775 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  30. Zadeh, L.A.: Fuzzy sets. Inf. Control 8, 338–353 (1965)

    Article  MATH  MathSciNet  Google Scholar 

  31. Bassanezi, R.C., Barros, L.C., Tonelli, A.: Attractors and asymptotic stability for fuzzy dynamical systems. Fuzzy Sets Syst. 113, 473–483 (2000)

    Article  MATH  Google Scholar 

  32. Barros, L.C., Bassanezi, R.C., Tonelli, P.A.: Fuzzy modelling in population dynamics. Ecol. Model. 128, 27–33 (2000)

    Article  Google Scholar 

  33. Peixoto, M., Barros, L.C., Bassanezi, R.C.: Predator–prey fuzzy model. Ecol. Model. 214, 39–44 (2008)

    Article  Google Scholar 

  34. Mizukoshi, M.T., Barros, L.C., Bassanezi, R.C.: Stability of fuzzy dynamic systems. Int. J. Uncertainty Fuzziness Knowl. Based Syst. 17, 69–84 (2009)

  35. Guo, M., Xu, X., Li, R.: Impulsive functional differential inclusions and fuzzy population models. Fuzzy Sets Syst. 138, 601–615 (2003)

    Article  MATH  Google Scholar 

  36. Pal, D., Mahapatra, G.S., Samanta, G.P.: A proportional harvesting dynamical model with fuzzy intrinsic growth rate and harvesting quantity. Pacific-Asian J. Math. 6, 199–213 (2012)

    Google Scholar 

  37. Pal, D., Mahapatra, G.S., Samanta, G.P.: Optimal harvesting of prey–predator system with interval biological parameters: a bioeconomic model. Math. Biosci. 241, 181–187 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  38. Pal, D., Mahapatra, G.S.: A bioeconomic modeling of two-prey and one-predator fishery model with optimal harvesting policy through hybridization approach. Appl. Math. Comput. 242, 748–763 (2014)

    Article  MathSciNet  Google Scholar 

  39. Pal, D., Mahapatra, G.S., Samanta, G.P.: Bifurcation analysis of predator–prey model with time delay and harvesting efforts using interval parameter. Int. J. Dyn. Control (2014). doi:10.1007/s40435-014-0083-8

  40. Sharma, S., Samanta, G.P.: Optimal harvesting of a two species competition model with imprecise biological parameters. Nonlinear Dyn. (2014). doi:10.1007/s11071-014-1354-9

    MathSciNet  Google Scholar 

  41. Kar, T.K., Chaudhuri, K.S.: Harvesting in a two-prey one predator fishery: a bioeconomic model. ANZIAM J. 45, 443–456 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  42. Moore, R.E.: Interval Analysis. Prentice-Hall Inc., Englewood Cliffs, NJ (1966)

    MATH  Google Scholar 

  43. Puri, M., Ralescu, D.: Differentials of fuzzy functions. J. Math. Anal. Appl. 91, 552–558 (1983)

  44. Seikkala, S.: On the fuzzy initial value problem. Fuzzy Sets Syst. 24, 309–330 (1987)

    Article  MathSciNet  Google Scholar 

  45. Murray, J.D.: Mathematical Biology. Springer, Berlin (1993)

  46. Kar, T.K., Misra, S.: Influence of prey reverse in a prey–predator fishery. Nonlinear Anal. 65, 1725–1735 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  47. Lv, Y., Yuan, R., Pai, Y.: A prey–predator model with harvesting for fishery resource with reserve area. Appl. Math. Model. 37, 3048–3062 (2013)

  48. Arrow, K.J., Kurz, M.: Public Investment. The Rate of Return and Optimal Fiscal Policy. John Hopkins, Baltimore (1970)

  49. Pontryagin, L.S., Boltyanski, V.S., Gamkrelidze, R.V., Mishchenco, E.F.: The Mathematical Theory of Optimal Processes. Wiley, New York (1962)

    MATH  Google Scholar 

  50. Chaudhuri, K.S.: A bioeconomic model of harvesting a multispecies fishery. Ecol. Model. 32, 267–279 (1986)

    Article  Google Scholar 

Download references

Acknowledgments

The authors would like to express their gratitude to the Editor and Referees for their encouragement and constructive comments in revising the paper.

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Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah, 711103, India

    D. Pal & G. P. Samanta

  2. Department of Mathematics, National Institute of Technology Puducherry, Karaikal, 609605, India

    G. S. Mahapatra

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  1. D. Pal
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  3. G. P. Samanta
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Correspondence to G. S. Mahapatra.

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Pal, D., Mahapatra, G.S. & Samanta, G.P. Stability and bionomic analysis of fuzzy parameter based prey–predator harvesting model using UFM. Nonlinear Dyn 79, 1939–1955 (2015). https://doi.org/10.1007/s11071-014-1784-4

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  • DOI: https://doi.org/10.1007/s11071-014-1784-4

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