Jump to ContentJump to Main Navigation
Show Summary Details
More options …

International Journal of Nonlinear Sciences and Numerical Simulation

Editor-in-Chief: Birnir, Björn

Editorial Board: Armbruster, Dieter / Chen, Xi / Bessaih, Hakima / Chou, Tom / Grauer, Rainer / Marzocchella, Antonio / Rangarajan, Govindan / Trivisa, Konstantina / Weikard, Rudi

IMPACT FACTOR 2017: 1.162

CiteScore 2017: 1.41

SCImago Journal Rank (SJR) 2017: 0.382
Source Normalized Impact per Paper (SNIP) 2017: 0.636

Mathematical Citation Quotient (MCQ) 2017: 0.12

See all formats and pricing
More options …
Volume 19, Issue 7-8


Asymptotic Behavior of the Fractional Order three Species Prey–Predator Model

M. Sambath / P. Ramesh / K. Balachandran
Published Online: 2018-08-24 | DOI: https://doi.org/10.1515/ijnsns-2017-0273


In this work, we introduce fractional order predator–prey model with infected predator. First, we prove different mathematical results like existence, uniqueness, non-negativity and boundedness of the solutions of fractional order dynamical system. Further, we investigate the local and global stability of all feasible equilibrium points of the system. Numerical results are illustrated as several examples.

Keywords: boundedness; existence and uniqueness; fractional dynamical system; stability; prey–predator model

MSC 2010: 65L05; 92B05; 26A33


  • [1]

    W. M. Abd-Elhameed and Y. H. Youssri, Spectral Tau algorithm for certain coupled system of fractional differential equations via generalized Fibonacci polynomial sequence, Iran. J. Sci. Technol.: Trans. Mech. Eng. 3 (2017), 1–12.

  • [2]

    M. S. Osman, Multiwave solutions of time-fractional (2 + 1)-dimensional Nizhnik–Novikov–Veselov equations, Pramana J. Phys. 88 (2017), 1–9.Web of Science

  • [3]

    M. Hassell, The Dynamics of Arthropod Predator-Prey System, Princeton University Press, Princeton, 1978.Google Scholar

  • [4]

    Z. Ma, The research of predator-prey models incorporating prey refuges. Ph.D. Thesis, Lanzhou University, Lanzhou, 2010.

  • [5]

    T. Kar, Stability analysis of a prey-predator model incorporating a prey refuge, Commun. Nonlinear Sci. Numer. Simul. 10 (2005), 681–691.Crossref

  • [6]

    Y. Huang, F. Chen and Z. Li, Stability analysis of a prey-predator model with Holling type III response function incorporating a prey refuge, Appl. Math. Comput. 182 (2006), 672–683.

  • [7]

    J. Tripathi, S. Abbas and M. Thakur, Dynamical analysis of a prey-predator model with Beddington-DeAngelis type function response incorporating a prey refuge, Nonlinear Dyn. 80 (2015), 177–196.Web of ScienceCrossref

  • [8]

    C. Bianca, C. Dogba and L. Guerrini, A thermostatted kinetic framework with particle refuge for the modeling of tumors hiding, Appl. Math. Inf. Sci. 8 (2014), 469–473.Web of ScienceCrossref

  • [9]

    C. Bianca, Modeling complex systems with particles refuge by thermostatted kinetic theory methods, Abstr. Appl. Anal. Hindawi Publishing Corporation, (2013), 1–13.Google Scholar

  • [10]

    E. Ahmed and A. Elgazzar, On fractional order differential equations model for nonlocal epidemics, J. Phys. A: Math. Theor. 379 (2007), 607–614.

  • [11]

    A. Elsadany and A. Matouk, Dynamical behaviors of fractional-order LotkaVolterra predator-prey model and its discretization, J. Appl. Math. Comput. 49 (2015), 269–283.Crossref

  • [12]

    M. Caputo, Linear models of dissipation whose Q is almost frequency independent-II. Geophys, J. Royal Astron. Soc. Can. 13 (1967), 529–539.Crossref

  • [13]

    F. Ben Adda, Geometric interpretation of the fractional derivative, J. Fractional Calculus Appl. 11 (1997), 21–52.

  • [14]

    I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation, Fractional Calculus Appl. Anal. 5 (2002), 367–386.

  • [15]

    A. S. Hegazi, E. Ahmed and A. E. Matouk, The effect of fractional order on synchronization of two fractional order chaotic and hyperchaotic systems, J. Fractional Calculus Appl. 1 (2011), 1–15.

  • [16]

    A. E. Matouk, Chaos, feedback control and synchronization of a fractional-order modified autonomous Van der Pol-Duffing circuit, Commun. Nonlinear Sci. Numer. Simul. 16 (2011), 975–986.CrossrefWeb of Science

  • [17]

    A. E. Matouk, Stability conditions, hyperchaos and control in a novel fractional order hyperchaotic system, Phys. Lett. A. 373 (2009), 2166–2173.CrossrefWeb of Science

  • [18]

    A. E. Matouk, Dynamical behaviors, linear feedback control and synchronization of the fractional order Liu system, J. Nonlinear Syst. Appl. 1 (2010), 135–140.

  • [19]

    D. Matignon, Stability results for fractional differential equations with applications to control processing, in: Proceedings of the Computational Engineering in Systems and Application Multiconference, vol. 2, pp. 963–968 (IMACS, IEEE-SMC), Lille, France, 1996.

  • [20]

    M. Odibat and N. T. Shawagfeh, Generalized Taylor’s formula, Appl. Math. Comput. 186 (2007), 286–293.Web of Science

  • [21]

    A. Kilbas, H. Srivastava and J. Trujillo, Theory and Application of Fractional Differential Equations, Elsevier, New York, 2006.

  • [22]

    Y. Li, Y. Chen and I. Podlubny, Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Laffler stability, Computers and Mathematics with Applications. 59 (2010), 1810–1821.Google Scholar

  • [23]

    J. Huo, H. Zhao and L. Zhu, The effect of vaccines on backward bifurcation in a fractional order HIV model, Nonlinear Anal. Real World Anal. 26 (2015), 289–305.Web of ScienceCrossref

  • [24]

    A. E. Matouk, A. A. Elsadany, E. Ahmed and H. N. Agiza, Dynamical behavior of fractional-order Hastings-Powell food chain model and its discretization, Commun. Nonlinear Sci. Numer. Simul. 27 (2015), 153–167.Web of ScienceCrossref

  • [25]

    C. Vargas-De-Leon, Volterra-type Lyapunov functions for fractional-order epidemic systems, Commun. Nonlinear Sci. Numer. Simul. 24 (2014), 1–17.Web of Science

  • [26]

    K. Diethelm and A. Freed, On the solution of nonlinear fractional order differential equations used in the modelling of viscoplasticity, in: F. Keil, W. Mackens, H. Voss and J. Werther (Eds.), Scientific Computing in Chemical Engineering II-Computational Fluid Dynamics, pp. 217–224, Reaction Engineering, and Molecular Properties, Springer, Heidelberg, 1999.Google Scholar

  • [27]

    K. Diethelm and A. Freed, The Frac PECE subroutine for the numerical solution of differential equations of fractional order, in: S. Heinzel and T. Plesser (Eds.), Forschung und wissenschaftliches Rechnen 1998, pp. 57–71, Gesellschaft fur Wisseschaftliche Datenverarbeitung, Gottingen, 1999.Google Scholar

  • [28]

    E. Ahmed, A. M. A. El-Sayed, E. M. El-Mesiry and H. A. A. El-Saka, Numerical solution for the fractional replicator equation, Int. J. Modern Phys. C. 16 (2005), 1–9.

  • [29]

    K. Diethelm, Predictor-corrector strategies for single and multi-term fractional differential equations, in: E. A. Lipitakis (Ed.), Proceedings of the 5th Hellenic-European Conference on Computer Mathematics and Its Applications, pp. 117–122, LEA Press, Athens, 2002.Google Scholar

  • [30]

    M. A. Z. Raja, M. A. Manzar and R. Samar, An efficient computational intelligence approach for solving fractional order Riccati equations using ANN and SQP, Appl. Math. Modell. 39 (2015), 3075–3093.Crossref

  • [31]

    M. A. Z. Raja, R. Samar, M. A. Manzar and S. M. Shah, Design of unsupervised fractional neural network model optimized with interior point algorithm for solving BagleyTorvik equation, Math. Comput. Simul. 132 (2017), 139–158.Crossref

About the article

Received: 2017-12-12

Accepted: 2018-08-10

Published Online: 2018-08-24

Published in Print: 2018-12-19

Citation Information: International Journal of Nonlinear Sciences and Numerical Simulation, Volume 19, Issue 7-8, Pages 721–733, ISSN (Online) 2191-0294, ISSN (Print) 1565-1339, DOI: https://doi.org/10.1515/ijnsns-2017-0273.

Export Citation

© 2018 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Comments (0)

Please log in or register to comment.
Log in