Nonlinear Dynamics

, Volume 95, Issue 2, pp 1457–1470 | Cite as

Fractal analysis and control of the fractional Lotka–Volterra model

  • Yupin Wang
  • Shutang LiuEmail author
Original Paper


This paper reports the investigation of a fractional Lotka–Volterra model from the fractal viewpoint. A Julia set of a discrete version of this model is introduced and its state feedback control is realized. Coupled terms are designed to realize the synchronization of two Julia sets with different parameters. Numerical simulations are presented to further verify the correctness and effectiveness of the main theoretical results.


Fractional Lotka–Volterra system Discretization Julia set Feedback control Synchronization 

Mathematics Subject Classification

26A33 34K35 37P40 92D25 



This research is supported by the National Natural Science Foundation of China (No. 61533011). The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have led to the present improved version of the original manuscript. The first author is also indebted to Dr. H. Li for numerous helpful discussions.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

Human and animals rights

This article does not contain any studies with human participants or animals performed by any of the authors.


  1. 1.
    Lotka, A.: Elem. Phys. Biol, Williams and Wilkins, Baltimore (1925)Google Scholar
  2. 2.
    Volterra, V.: Variazioni e fluttuazioni del numero di individui in specie animali conviventi. Mem. R. Acad. Lincei Ser. VI 2, 31–113 (1926)zbMATHGoogle Scholar
  3. 3.
    Takeuchi, Y.: Global Dynamical Properties of Lotka-Volterra Systems. World Scientific, Singapore (1996)CrossRefzbMATHGoogle Scholar
  4. 4.
    Ahmed, E., El-Sayed, A., El-Saka, H.: Equilibrium points, stability and numerical solutions of fractional-order predator-prey and rabies models. J. Math. Anal. Appl. 325(7), 542–553 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    El-Saka, H., Ahmed, E., Shehata, M., El-Sayed, A.: On stability, persistence, and Hopf bifurcation in fractional order dynamical systems. Nonlinear Dyn. 56, 121–126 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Das, S., Gupta, P.: Rajeev, : A fractional predator-prey model and its solution. Int. J. Nonlinear Sci. Numer. Simul. 10(7), 873–876 (2009)CrossRefGoogle Scholar
  7. 7.
    Das, S., Gupta, P.: A mathematical model on fractional Lotka-Volterra equations. J. Theor. Biol. 277(1), 1–6 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Agrawal, S., Srivastava, M., Das, S.: Synchronization between fractional-order Ravinovich-Fabrikant and Lotka-Volterra systems. Nonlinear Dyn. 69(4), 2277–2288 (2012)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Elsadany, A., Matouk, A.: Dynamical behaviors of fractional-order Lotka-Volterra predator-prey model and its discretization. J. Appl. Math. Comput. 49, 269–283 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Matouk, A., Elsadany, A.: Dynamical analysis, stabilization and discretization of a chaotic fractional-order GLV mode. Nonlinear Dyn. 85(3), 1597–1612 (2016)CrossRefzbMATHGoogle Scholar
  11. 11.
    Nosrati, K., Shafiee, M.: Dynamic analysis of fractional-order singular Holling type-II predator-prey system. Appl. Math. Comput. 313, 159–179 (2017)MathSciNetGoogle Scholar
  12. 12.
    Li, C., Sprott, J., Mei, Y.: An infinite 2-D lattice of strange attractors. Nonlinear Dyn. 89(4), 2629–2639 (2017)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Řadulescu, A., Pignatelli, A.: Symbolic template iterations of complex quadratic maps. Nonlinear Dyn. 84(4), 2025–2042 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Jackson, E., Kodoeorgiou, A.: Entrainment and migration controls of two-dimensional maps. Phys. D Nonlinear Phenom. 54(3), 253–265 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Gilpin, M., Hanski, I.: Metapopulation Dynamics: Empirical and Theoretical Investigations. Academic Press, London (1991)Google Scholar
  16. 16.
    Sun, W., Zhang, Y., Zhang, X.: Fractal analysis and control in the predator-prey model. Int. J. Comput. Math. 94(4), 737–746 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Zhang, M., Zhang, Y.: Fractal analysis and control of the competition model. Int. J. Biomath. 9(3), 1650045 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and Some of their Applications. Academic Press, San Diego (1999)zbMATHGoogle Scholar
  19. 19.
    Petráš, I.: Fractional-Order Nonlinear Systems: Modeling. Analysis and Simulation. Springer, Berlin (2011)CrossRefzbMATHGoogle Scholar
  20. 20.
    Wang, Y., Sun, S.: Solvability to infinite-point boundary value problems for singular fractional differential equations on the half-line. J. Appl. Math. Comput. 57, 359–373 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Arfken, G., Weber, H.: Mathematical Methods for Physicists, 6th edn. Academic Press, San Diego (2005)zbMATHGoogle Scholar
  22. 22.
    Zhu, S., Cai, C., Spanos, P.: A nonlinear and fractional derivative viscoelastic model for rail pads in the dynamic analysis of coupled vehicle-slab track systems. J. Sound Vib. 335, 304–320 (2015)CrossRefGoogle Scholar
  23. 23.
    Spanos, P., Evangelatos, G.: Response of a non-linear system with restoring forces governed by fractional derivatives–time domain simulation and statistical linearization solution. Soil Dyn. Earthq. Eng. 30(9), 811–821 (2010)CrossRefGoogle Scholar
  24. 24.
    Falconer, K.: Fractal Geometry: Mathematical Foundations and Applications. Wiley, Chichester (2003)CrossRefzbMATHGoogle Scholar
  25. 25.
    Liu, S., Wang, P.: Fractal Control Theory. Springer, Singapore (2018)CrossRefGoogle Scholar
  26. 26.
    Zhang, Y., Liu, S.: Gradient control and synchronization of Julia sets. Chin. Phys. B 17(2), 543–549 (2008)CrossRefGoogle Scholar
  27. 27.
    Sun, W., Zhang, Y.: Control and synchronization of Julia sets in the forced Brusselator model. Int. J. Bifurc. Chaos 25(9), 1550113 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Elaydi, S.: An Introduction to Difference Equations. Springer, New York (2005)zbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Institute of Marine Science and Technology, Shandong UniversityJinanPeople’s Republic of China
  2. 2.School of Control Science and Engineering, Shandong UniversityJinanPeople’s Republic of China

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