THERMAL SCIENCE

International Scientific Journal

Jagdev Singh

,

Adem Kilicman

,

Devendra Kumar

,

Ram Swroop

,

Fadzilah Md Ali

NUMERICAL STUDY FOR FRACTIONAL MODEL OF NON-LINEAR PREDATOR-PREY BIOLOGICAL POPULATION DYNAMICAL SYSTEM

ABSTRACT
The key objective of the present paper is to propose a numerical scheme based on the homotopy analysis transform technique to analyze a time fractional nonlinear predator-prey population model. The population model are coupled fractional order nonlinear partial differential equations often employed to narrate the dynamics of biological systems in which two species interact, first is a predator and the second is a prey. The proposed scheme provides the series solution with a great freedom and flexibility by choosing appropriate parameters. The convergence of the results is free from small or large parameters. Three examples are discussed to demonstrate the correctness and efficiency of the used computational approach.
KEYWORDS
Fractional nonlinear predator-prey population model, Biological systems, Homotopy analysis transform technique
PAPER SUBMITTED: 2019-07-25
PAPER REVISED: 2019-08-24
PAPER ACCEPTED: 2019-08-27
PUBLISHED ONLINE: 2019-10-06
DOI REFERENCE: https://doi.org/10.2298/TSCI190725366S
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2019, VOLUME 23, ISSUE Supplement 6, PAGES [S2017 - S2025]
REFERENCES
  1. Caputo, M., Elasticita e Dissipazione, Zani-Chelli, Bologna, 1969.
  2. Kilbas, A. A., Srivastava, H. M., Trujillo, J. J., Theory and Applications of Fractional Differential Equations, Elsevier Science B.V., Amsterdam, 204, 2006.
  3. Miller, K.S., Ross, B., An Introduction to the fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993.
  4. Rivero, M., Trujillo, J., Vazquez, L., Velasco, M., Fractional dynamics of populations, Appl. Math. Comput., 218 (2011), pp. 1089-1095.
  5. S. G. Samko, A.A. Kilbas, O. L. Marichev, Fractional integrals and derivatives: Theory and Applications, Gordon and Breach Science (1993).
  6. Kumar, D., Singh, J., Baleanu, D, Numerical computation of a fractional model of differentialdifference equation, Journal of Computational and Nonlinear Dynamics, 11 (2016),doi: 10.1115/1.4033899.
  7. Srivastava, H.M., Kumar, D., Singh, J., An efficient analytical technique for fractional model of vibration equation, Appl. Math.Model.,45 (2017), pp. 192-204.
  8. Singh, J., Kumar, D., Nieto, J.J., A reliable algorithm for local fractional Tricomi equation arising in fractal transonic flow, Entropy, 18 (2016), 6, doi: 10.3390/e18060206.
  9. Hristov, J., Transient Heat Diffusion with a Non-Singular Fading Memory: From the Cattaneo Constitutive Equation with Jeffrey's kernel to the Caputo-Fabrizio time-fractional derivative, Thermal Science, 20 (2016), 2, pp.765-770.
  10. Yang, X.J., Srivastava, H.M., Machado, J.A.T., A new fractional derivative without singular kernel: Application to the modelling of the steady heat flow, Thermal Science, 20, (2016), pp. 753-756.
  11. Atangana A., Baleanu D., New fractional derivatives with nonlocal and non-singular kernel, Theory and application to heat transfer model, Thermal Science, 20, (2016), 2, pp. 763-769.
  12. Khan, M., Gondal, M. A., Hussain, I., CarimiVanani, S., A new comparative study between homotopy analysis transform method and homotopy perturbation transform method on semi in finite domain, Math. Comput. Model., 55, (2012), pp. 1143-1150.
  13. Kumar, D., Singh, J.,Baleanu, D., A fractional model of convective radial fins with temperature-dependent thermal conductivity, Romanian Reports in Physics, 69, (2017),1, 103.
  14. Singh, J., Rashidi, M.M., Sushila, Kumar, D., A hybrid computational approach for Jeffery-Hamel flow in non-parallel walls, Neural Comput. Appl., (2017), DOI: 10.1007/s00521-017-3198-y.
  15. Kumar, D., Agarwal, R.P., Singh, J., A modified numerical scheme and convergence analysis for fractional model of Lienard's equation, J. Comput. Appl. Math., 339, (2018), pp. 405-413.
  16. Liao, S.J., Beyond Perturbation: Introduction to the homotopy analysis method, Chaoman and Hall/CRC Press, Boca Raton, 2003.
  17. Liao, S.J., On the homotopy analysis method for nonlinear problems, Appl. Math. Comput., 147 (2004), pp. 499-513.
  18. Liao, S.J., An approximate solution technique not depending on small parameters: a special example, Int. J. Non-Linear Mech., 30, (1995), 3, pp. 371-380.
  19. Lotka, A., Element of physical biology, Williams and Wilkins, Baltimore, 1925.
  20. Volterra, V., Variazioni e fluttuazionidelnumero di individui in specie animaliconvivent, Mem. Accd. Linc., 2, (1926), pp. 31-113.
  21. Verhulst, P.F., Notice sur la loi que la population suit dans son accrocisse-ment, Corr. Math. Phys., 10, (1838), pp. 113-121.
  22. Freedman, H.I., Deterministic mathematical model in population ecology, Marcel Dekker, New York, 57, (1980), pp. 95-252.
  23. Gourely, S.A., Britton, N.F., A predator-prey reaction-diffusion system with nonlocal effects, J. Math. Bio., 34, (1996), pp. 297-333.
  24. Devaney, R., Hirsch, M., Smale, S., Differential equations, dynamical systems and an introduction to Chaos (2nd Ed.), USA: Academic Press, 2004.
  25. Petrovskii, S., Malchow, H., Li, B.L., An exact solution of a diffusive predator-prey system, Proc. Royal Soc. London A, 161, (2005), pp. 1029-1053.
  26. Das, S., Gupta, P.K., Rajeev, A fractional predator prey model and its solution, Int. J. Nonlin. Sci. Numer. Simul., 10, (2009), pp. 873-876.
  27. Yanqin, L., Baogui, X., Numerical solutions of a fractional predator-prey system, Adv. Differ. Equ., 2011, (2011), 11 pages.
PDF VERSION [DOWNLOAD]
Numerical study for fractional model of non-linear predator-prey biological population dynamical system

© 2024 Society of Thermal Engineers of Serbia. Published by the Vinča Institute of Nuclear Sciences, National Institute of the Republic of Serbia, Belgrade, Serbia. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International licence