Advertisement

The Behavior of Logistic Equation with Alley Effect in Fuzzy Environment: Fuzzy Differential Equation Approach

  • Soheil Salahshour
  • Ali Ahmadian
  • Animesh Mahata
  • Sankar Prasad Mondal
  • Shariful Alam
Original Paper
  • 69 Downloads

Abstract

In this paper a fuzzy logistic equation with alley effect is introduced by considering some parameter as fuzzy numbers. Due to presence of the fuzzy number the corresponding differential equation in logistic equation model with alley effect becomes fuzzy differential equation. Considering generalized Hukuhara derivative approach the fuzzy logistic equation converted to system of two crisp differential equations. We obtain the conditions of stability criterion for different cases. Different numerical examples are given to support our work.

Keywords

Logistic equation Allee effect Fuzzy differential equation Fuzzy stability analysis 

References

  1. 1.
    Baidosov, V.A.: Fuzzy differential inclusions. J. Appl. Math. Mech. 54, 8–13 (1990)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Hllermeier, E.: An approach to modeling and simulation of uncertain dynamical systems. Int. J. Uncertain. Fuzziness Knowl. Based Syst. 5, 117–137 (1997)CrossRefGoogle Scholar
  3. 3.
    Oberguggenberger, M., Pittschmann, S.: Differential equations with fuzzy parameters. Math. Comput. Model. Dyn. Syst. 5, 181–202 (1999)CrossRefzbMATHGoogle Scholar
  4. 4.
    Buckley, J.J., Feuring, T.: Fuzzy differential equations. Fuzzy Sets Syst. 110, 43–54 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Dubois, D., Prade, H.: Towards fuzzy differential calculus. Part 3: differentiation. Fuzzy Sets Syst. 8, 225–233 (1982)CrossRefzbMATHGoogle Scholar
  6. 6.
    Puri, M.L., Ralescu, D.A.: Differentials of fuzzy functions. J. Math. Anal. Appl. 91, 552–558 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Goetschel, R., Voxman, W.: Elementary fuzzy calculus. Fuzzy Sets Syst. 18, 31–43 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Friedman, M., Ming, M., Kande, A.: Fuzzy derivatives and fuzzy Cauchy problems using LP metric. Fuzzy Log. Found. Ind. Appl. 8, 57–72 (1996)zbMATHGoogle Scholar
  9. 9.
    Seikkala, S.: On the fuzzy initial value problem. Fuzzy Sets Syst. 24, 319–330 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bede, B., Gal, S.G.: Almost periodic fuzzy-number-valued functions. Fuzzy Sets Syst. 147, 385–403 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Yue, Z., Guangyuan, W.: Time domain methods for the solutions of N order fuzzy differential equations. Fuzzy Sets Syst. 94, 77–92 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Chalco-Cano, Y., Roman-Flores, H., Jimnez-Gamero, M.D.: Fuzzy differential equation with-derivative. In: The Proceedings of the Joint 2009 International Fuzzy Systems Association World Congress and 2009 European Society of Fuzzy Logic and Technology Conference, Lisbon, Portugal, 20–24 July 2009Google Scholar
  13. 13.
    Stefanini, L., Bede, B.: Generalized Hukuhara differentiability of interval-valued functions and interval differential equations. Nonlinear Anal. Theory Methods Appl. 71, 1311–1328 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Bede, B., Stefanini, L.: Generalized differentiability of fuzzy-valued functions. Fuzzy Sets Syst. 230, 119–141 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Mazandarani, M., Najariyan, M.: Differentiability of type-2 fuzzy number-valued functions. Commun. Nonlinear Sci. Numer. Simul. 19, 710–725 (2014)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Barros, L.C., Pedro, F.S.: Fuzzy differential equations with interactive derivative. Fuzzy Sets Syst. 309, 64–80 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Mazandarani, M., Pariz, N., Kamyad, A.V.: Granular differentiability of fuzzy-number-valued functions. IEEE Trans. Fuzzy Syst. 26(1), 310–323 (2018)CrossRefGoogle Scholar
  18. 18.
    Barros, L.C., Bassanezi, R.C., Tonelli, P.A.: Fuzzy modelling in population dynamics. Ecol. Model. 128, 27–33 (2000)CrossRefGoogle Scholar
  19. 19.
    Mondal, S.P., Paul, S., Mahata, A., Bhattacharya, P., Roy, T.K.: Classical modelling of HIV virus infected population in imprecise environments. Turk. J. Fuzzy Sets Syst. 6(1), 017–055 (2015)Google Scholar
  20. 20.
    Paul, S., Mondal, S.P., Bhattacharya, P.: Discussion on Fuzzy quota harvesting model in fuzzy environment: fuzzy differential equation approach, modeling earth systems and environment model. Earth Syst. Environ. 2, 70 (2016)CrossRefGoogle Scholar
  21. 21.
    Paul, S., Mondal, S.P., Bhattacharya, P.: Discussion on proportional harvesting model in fuzzy environment: fuzzy differential equation approach. Int. J. Appl. Comput, Math (2016)Google Scholar
  22. 22.
    Paul, S., Jana, D., Mondal, S.P., Bhattacharya, P.: Optimal harvesting of two species mutualism model with interval parameters. J. Intell. Fuzzy Syst. 33(4), 1991–2005 (2017)CrossRefzbMATHGoogle Scholar
  23. 23.
    Wazwaz, A.M.: A reliable modification of Adomian decomposition method. Appl. Math. Comput. 102(1), 77–86 (1999)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Deniz, S., Bildik, N.: Comparison of Adomian decomposition method and Taylor matrix method in solving different kinds of partial differential equations. Int. J. Model. Optim. 4(4), 292 (2014)CrossRefGoogle Scholar
  25. 25.
    Bildik, N., Deniz, S.: The use of Sumudu decomposition method for solving predator–prey systems. Math. Sci. Lett. 5(3), 285–289 (2016)CrossRefGoogle Scholar
  26. 26.
    Allee, W.C.: Animal Aggretions: A Study in General Sociology. University of Chicago Press, Chicago (1931)CrossRefGoogle Scholar
  27. 27.
    Merdan, H., Duman, O., Akın, O., Celik, C.: Allee effects on population dynamics in continuous (overlapping) case. Chaos Solitons Fractals 39, 1994–2001 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Celik, C., Merdan, H., Duman, O., Akın, O.: Allee effects on population dynamics with delay. Chaos Solitons Fractals 37, 65–74 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Fowler, M.S., Ruxton, G.D.: Population dynamic consequences of Allee effects. J. Theor. Biol. 215, 39–46 (2002)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Gao, S., Chen, L.: The effect of seasonal harvesting on a single-species discrete population model with stage structure and birth pulses. Chaos Solitons Fractals 24, 1013–1023 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Hadjiavgousti, D., Ichtiaroglou, S.: Existence of stable localized structures in population dynamics through the Allee effect. Chaos Solitons Fractals 21, 119–131 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    López-Ruiz, R., Fournier-Prunaret, D.: Indirect Allee effect, bistability and chaotic oscillations in a predator–prey discrete model of logistic type. Chaos Solitons Fractals 24, 85–101 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Pederson, S., Sambandham, M.: Numerical solution of hybrid fuzzy differential equation IVP by characterization theorem. Inf. Sci. 179, 319–328 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Bede, B.: Note on “Numerical solution of fuzzy differential equations by predictor corrector method”. Inf. Sci. 178, 1917–1922 (2008)CrossRefzbMATHGoogle Scholar
  35. 35.
    Nieto, J.J., Khastan, A., Ivaz, K.: Numerical solution of fuzzy differential equations under generalized differentiability. Nonlinear Anal. Hybrid Syst. 3, 700–707 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Allahviranloo, T., Salahshour, S.: Euler method for solving hybrid fuzzy differential equation. Soft. Comput. 15, 1247–1253 (2011)CrossRefzbMATHGoogle Scholar
  37. 37.
    Veiseh, H., Lotfi, T., Allahviranloo, T.: On general conditions for nestedness of the solution set of fuzzy-interval linear systems. Fuzzy Sets Syst. 331, 105–115 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Ezadi, S., Allahviranloo, T.: Numerical solution of linear regression based on Z-numbers by improved neural network. Intell. Autom. Soft Comput. (2017).  https://doi.org/10.1080/10798587.2017.1328812 Google Scholar
  39. 39.
    Tapaswini, S., Chakraverty, S., Allahviranloo, T.: A new approach to nth order fuzzy differential equations. Comput. Math. Model. 28, 278–300 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Gouyandeh, Z., Allahviranloo, T., Abbasbandy, S., Armand, A.: A fuzzy solution of heat equation under generalized Hukuhara differentiability by fuzzy Fourier transform. Fuzzy Sets Syst. 309, 81–97 (2017)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer (India) Private Ltd., part of Springer Nature 2018

Authors and Affiliations

  • Soheil Salahshour
    • 1
  • Ali Ahmadian
    • 2
  • Animesh Mahata
    • 3
    • 4
  • Sankar Prasad Mondal
    • 5
  • Shariful Alam
    • 4
  1. 1.Young Researchers and Elite Club, Mobarakeh BranchIslamic Azad UniversityMobarakehIran
  2. 2.Institute for Mathematical ResearchUniversity Putra Malaysia (UPM)SerdangMalaysia
  3. 3.Department of MathematicsNetaji Subhash Engineering CollegeKolkataIndia
  4. 4.Department of MathematicsIndian Institute of Engineering Science and Technology, ShibpurHowrahIndia
  5. 5.Department of MathematicsMidnapore College (Autonomous), MidnaporeWest MidnaporeIndia

Personalised recommendations