Nonlinear Dynamics

, Volume 80, Issue 1–2, pp 777–789 | Cite as

Fractional-order delayed predator–prey systems with Holling type-II functional response

  • F. A. Rihan
  • S. Lakshmanan
  • A. H. Hashish
  • R. Rakkiyappan
  • E. Ahmed
Original Paper


In this paper, a fractional dynamical system of predator–prey with Holling type-II functional response and time delay is studied. Local and global stability of existence steady states and Hopf bifurcation with respect to the delay is investigated, with fractional-order \(0< \alpha \le 1\). It is found that Hopf bifurcation occurs when the delay passes through a sequence of critical values. Unconditionally, stable implicit scheme for the numerical simulations of the fractional-order delay differential model is introduced. The numerical simulations show the effectiveness of the numerical method and confirm the theoretical results. The presence of fractional order in the delayed differential model improves the stability of the solutions and enrich the dynamics of the model.


Fractional calculus Predator–prey Hopf bifurcation Stability Time delay 



This work was supported by NRF Grant Project (UAE University). Dr. R. Rakkiyappan was supported by DST SERB Project # SB/FTP/MS-045/2013. The authors thank Prof. J. A. Tenreiro Machado and referees for their valuable comments.


  1. 1.
    Ahmed, E., Hashish, A., Rihan, F.A.: On fractional order cancer model. J. Fract. Calc. Appl. 3(2), 1–6 (2012)Google Scholar
  2. 2.
    Anguelov, R., Lubuma, J.M.S.: Nonstandard finite difference method by nonlocal approximation. Math. Comput. Simul. 61, 465–475 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Assaleh, K., Ahmad, W.M.: Modeling of speech signals using fractional calculus. In: 9th International Symposium on Signal Processing and Its Applications (ISSPA 2007) (2007)Google Scholar
  4. 4.
    Baleanu, D., Diethelm, K., Scalas, E., Trujillo, J.J.: Fractional Calculus Models and Numerical Methods. World Scientific, Singapore (2012)zbMATHGoogle Scholar
  5. 5.
    Caponetto, R., Dongola, G., Fortuna, L.: Fractional Order Systems: Modeling and Control Applications. World Scientific, London (2010)Google Scholar
  6. 6.
    Chen, W.C.: Nonlinear dynamics and chaos in a fractional-order financial system. Chaos Solitons Fract. 36(5), 1305–1314 (2008)CrossRefGoogle Scholar
  7. 7.
    Cole, K.S.: Electric conductance of biological systems. In: Cold Spring Harbor Symposium on Quantitative Biology, pp. 107–116 (1993)Google Scholar
  8. 8.
    Das, S., Gupta, P.: A mathematical model on fractional Lotka–Volterra equations. J. Theor. Biol. 277, 1–6 (2001)Google Scholar
  9. 9.
    Debnath, L.: Recent applications of fractional calculus to science and engineering. Int. J. Math. Math. Sci. 54, 3413–3442 (2003)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Deng, W., Li, C., Lu, J.: Stability analysis of linear fractional differential system with multiple time delays. Nonlinear Dyn. 48, 409–416 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Diethelm, K.: An algorithm for the numerical solution of differential equations of fractional order. Electron. Trans. Numer. Anal. 5, 1–6 (1997)zbMATHMathSciNetGoogle Scholar
  12. 12.
    Diethelm, K., Ford, N., Freed, A.: A predictor–corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn. 29, 3–22 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Edelman, M.: Fractional maps as maps with power-law memory. In: Afraimovich, A., Luo, A.C.J., Fu, X. (eds.) Nonlinear Dynamics and Complexity, pp. 79–120. Springer, New York (2014)CrossRefGoogle Scholar
  14. 14.
    El-Sayed, A.: Nonlinear functional differential equations of arbitrary orders. Nonlinear Anal.: Theory Methods Appl. 33(2), 181–186 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    El-Sayed, A., El-Mesiry, A., El-Saka, H.: On the fractional-order logistic equation. Appl. Math. Lett. 20(7), 817–823 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Ferdri, Y.: Some applications of fractional order calculus to design digital filters for biomedical signal processing. J. Mech. Med. Biol. 12(2), 13 (2012)Google Scholar
  17. 17.
    Freedman, H.: Deterministic Mathematical Models in Population Ecology. Marcel Dekker, New York (1980)zbMATHGoogle Scholar
  18. 18.
    Grahovac, N.M., Zigic, M.M.: Modelling of the hamstring muscle group by use of fractional derivatives. Comput. Math. Appl. 59, 1695–1700 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Hilfer, R., Ed.: Applications of Fractional Calculus in Physics. World Scientific, River Edge (2000)Google Scholar
  20. 20.
    Javidi, M., Nyamoradi, N.: Dynamic analysis of a fractional order prey–predator interaction with harvesting. Appl. Math. Model. 37, 8946–8956 (2013)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Laskin, N., Zaslavsky, G.M.: Nonlinear fractional dynamics on a lattice with long-range interactions. Phys. A 368, 38–54 (2006)CrossRefGoogle Scholar
  22. 22.
    Li, C., Zhang, F.: A survey on the stability of fractional differential equations. Eur. Phys. J. Spec. Top. 193, 27–47 (2011)CrossRefGoogle Scholar
  23. 23.
    Li, L., Wang, Z.J.: Global stability of periodic solutions for a discrete predator–prey system with functional response. Nonlinear Dyn. 72, 507–516 (2013)CrossRefzbMATHGoogle Scholar
  24. 24.
    Lin, W.: Global existence theory and chaos control of fractional differential equations. J. Math. Anal. Appl. 332, 709–726 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Lotka, A.: Elements of Physical Biology. Williams and Wilkins, Baltimore (1925)zbMATHGoogle Scholar
  26. 26.
    Luo, A.C., (Eds.), V.A.: Long-Range Interaction, Stochasticity and Fractional Dynamics. New York, Springer (2010)Google Scholar
  27. 27.
    Machado, J.A.T.: Analysis and design of fractional order digital control systems. Syst. Anal. Model. Simul. 27, 107–122 (1997)Google Scholar
  28. 28.
    Machado, J.A.T.: Fractional-order derivative approximations in discrete-time control systems. Syst. Anal. Model. Simul. 34, 419–434 (1999)Google Scholar
  29. 29.
    Machado, J.A.T.: Entropy analysis of integer and fractional dynamical systems. Nonlinear Dyn. 62(1–2), 371–378 (2010)Google Scholar
  30. 30.
    Machado, J.A.T., Galhano, A.M.S.F.: Fractional order inductive phenomena based on the skin effect. Non-linear Dyn. 68(1–2), 107–115 (2012)Google Scholar
  31. 31.
    Meng, X., Jiao, J., Chen, L.: The dynamics of an age structured predator–prey model with disturbing pulse and time delays. Nonlinear Anal.: Real World Appl. 9, 547561 (2008)MathSciNetGoogle Scholar
  32. 32.
    Muth, E.: Transform Methods with Applications to Engineering and Operations Research. Prentice-Hall, New Jersey (1977)zbMATHGoogle Scholar
  33. 33.
    Petras, I.: Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation. HEP/Springer, London (2011)Google Scholar
  34. 34.
    Podlubny, I.: Fractional Differential Equations. Academic Press, London (1999)zbMATHGoogle Scholar
  35. 35.
    Rihan, F.A.: Computational methods for delay parabolic and time fractional partial differential equations. Num. Meth. Partial Differ. Eqn. 26(6), 1556–1571 (2010)zbMATHMathSciNetGoogle Scholar
  36. 36.
    Rihan, F.A.: Numerical modeling of fractional-order biological systems. Abstr. Appl. Anal. 2013, 11 (2013)Google Scholar
  37. 37.
    Rihan, F.A., Abdelrahman, D.H.: Delay differential model for tumor-immune dynamics with HIV infection of CD4\(^{+}\) T-cells. Int. J. Comput. Math. 90(3), 594–614 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  38. 38.
    Rihan, F.A., Abdelrahman, D.H., Lakshmanan, S.: A time delay model of tumour–immune system interactions: global dynamics, parameter estimation, sensitivity analysis. Appl. Math. Comput. 232, 606–623 (2014)CrossRefMathSciNetGoogle Scholar
  39. 39.
    Rihan, F.A., Baleanu, D., Lakshmanan, S., Rakkiyappan, R.: On fractional SIRC model with salmonella bacterial infection. Abstr. Appl. Anal. 2014, 9 (2014)Google Scholar
  40. 40.
    Rivero, M., Trujillo, J., Vazquez, L., Velasco, M.: Fractional dynamics of populations. Appl. Math. Comput. 218, 1089–1095 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  41. 41.
    Sheng, H., Chen, Y.Q., Qiu, T.S.: Fractional Processes and Fractional-Order Signal Processing. Springer, New York (2012)zbMATHGoogle Scholar
  42. 42.
    Suzuki, T.: A generalized banach contraction principle that characterizes metric completeness. Proc. Am. Math. Soc. 136(5), 1861–1869 (2008)CrossRefzbMATHGoogle Scholar
  43. 43.
    Tang, G., Tang, S., Cheke, R.A.: Global analysis of a holling type II predator–prey model with a constant prey refuge. Nonlinear Dyn. 76, 635–664 (2014)CrossRefMathSciNetGoogle Scholar
  44. 44.
    Tarasov, V.E.: Discrete map with memory from fractional differential equation of arbitrary positive order. J. Math. Phys. 50, 122,703 (2009)CrossRefMathSciNetGoogle Scholar
  45. 45.
    Volterra, V.: Variazioni e fluttuazioni del numero di individui in specie animali conviventiGoogle Scholar
  46. 46.
    Xia, Y., Cao, J., Cheng, S.: Multiple periodic solutions of a delayed stage-structured predator–prey model with non-monotone functional responses. Appl. Math. Model. 31, 1947–1959 (2007)CrossRefzbMATHGoogle Scholar
  47. 47.
    Xu, H.: Analytical approximations for a population growth model with fractional order. Commun. Nonlinear Sci. Numer. Simul. 14, 1978–1983 (2009)CrossRefzbMATHGoogle Scholar
  48. 48.
    Yuste, S.B., Acedo, L., Lindenberg, K.: Subdiffusion-limited A+B \(\rightarrow \) C reaction–subdiffusion process. Phys. Rev. E 69(3), 036,126 (2004)CrossRefGoogle Scholar
  49. 49.
    Zaslavsky, G.M.: Chaos, fractional kinetics, and anomalous transport. Phys. Rep. 371, 461580 (2002)CrossRefMathSciNetGoogle Scholar
  50. 50.
    Zaslavsky, G.M., Edelman, M., Tarasov, V.E.: Dynamics of the chain of forced oscillators with long-range interaction: from synchronization to chaos. Chaos 17(4), 043,124 (2007)CrossRefMathSciNetGoogle Scholar
  51. 51.
    Zhang, J.F.: Bifurcation analysis of a modified Holling–Tanner predator–prey model with time delay. Appl. Math. Model. 36, 1219–1231 (2012)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  • F. A. Rihan
    • 1
  • S. Lakshmanan
    • 1
  • A. H. Hashish
    • 2
  • R. Rakkiyappan
    • 3
  • E. Ahmed
    • 4
  1. 1.Department of Mathematical Sciences, College of ScienceUAE UniversityAl-AinUAE
  2. 2.Department of Physics, College of ScienceUAE UniversityAl-AinUAE
  3. 3.Department of MathematicsBharathiar UniversityCoimbatoreIndia
  4. 4.Department of Mathematics, Faculty of ScienceMansoura UniversityMansouraEgypt

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