Arabian Journal of Mathematics

, Volume 8, Issue 1, pp 1–14 | Cite as

Bifurcation analysis and chaos control in discrete-time system of three competing species

  • Irfan AliEmail author
  • Umer Saeed
  • Qamar Din
Open Access


In this paper, we investigate the complex dynamics of three-dimensional Ricker-type discrete-time competition model. We discuss the existence and uniqueness, and find parametric conditions for local asymptotic stability of positive fixed point of this model. It is also proved that the system undergoes Neimark–Sacker (NS) and period-doubling bifurcation (PDB) at certain parametric values for positive fixed point with the help of an explicit criterion for NS and PDB. The system shows chaotic dynamics at increasing values of bifurcation parameter. To control the chaos, we apply the hybrid control methodology. Finally, numerical simulations are provided to illustrate the theoretical discussions. These results of numerical simulations show chaotic long-term behavior over a broad range of parameters. The computation of the maximum Lyapunov exponents confirms the presence of chaotic behavior in the model.

Mathematics Subject Classification

37C25 37H20 37C75 



Maximum Lyapunov exponents


Period-doubling bifurcation


Neimark–Sacker bifurcation



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© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.School of Natural Sciences (SNS)National University of Sciences and Technology (NUST)IslamabadPakistan
  2. 2.NUST Institute of Civil Engineering (NICE)National University of Sciences and Technology (NUST)IslamabadPakistan
  3. 3.Department of MathematicsThe University of Poonch RawalakotRawalakotPakistan

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