Bifurcation Analysis and Chaos Control for Prey-Predator Model With Allee Effect

Main Article Content

M. B. Almatrafi, Messaoud Berkal


The main purpose of this work is to discuss the dynamics of a predator-prey dynamical system with Allee effect. The conformable fractional derivative is applied to convert the fractional derivatives which appear in the governing model into ordinary derivatives. We use the piecewise-constant approximation method to discritize the considered model. We also investigate the occurrence of positive equilibrium points. Moreover, we analyse the stability of the equilibrium point using some stability theorems. This work also explores a Neimark-Sacker bifurcation and a period-doubling bifurcation using the theory of bifurcations. The distance between the obtained equilibrium point and some closed curves is examined for various values for the considered bifurcation parameter. The chaos control is nicely analysed using the hybrid control approach. Furthermore, we present the maximum Lyapunov exponents for different values for the bifurcation parameter. Numerical simulations are utilized to ensure that the obtained theoretical results are correct. The used techniques can be applied to deal with predator-prey models in various versions.

Article Details


  1. A.Q. Khan, S.A.H. Bukhari, M.B. Almatrafi, Global Dynamics, Neimark-Sacker Bifurcation and Hybrid Control in a Leslie’s Prey-Predator Model, Alexandria Eng. J. 61 (2022), 11391–11404.
  2. A.Q. Khan, F. Nazir, M.B. Almatrafi, Bifurcation Analysis of a Discrete Phytoplankton-Zooplankton Model With Linear Predational Response Function and Toxic Substance Distribution, Int. J. Biomath. 16 (2022), 2250095.
  3. A.Q. Khan, M. Tasneem, M.B. Almatrafi, Discrete-Time COVID-19 Epidemic Model With Bifurcation and Control, Math. Biosci. Eng. 19 (2021), 1944–1969.
  4. R. Ahmed, M.B. Almatrafi, Complex Dynamics of a Predator-Prey System With Gompertz Growth and Herd Behavior, Int. J. Anal. Appl. 21 (2023), 100.
  5. M.B. Almatrafi, Construction of Closed Form Soliton Solutions to the Space-Time Fractional Symmetric Regularized Long Wave Equation Using Two Reliable Methods, Fractals. 31 (2023), 2340160.
  6. M. B. Almatrafi, Solitary Wave Solutions to a Fractional Model Using the Improved Modified Extended TanhFunction Method, Fractal Fract. 7 (2023).
  7. P.H. Leslie, Some Further Notes on the Use of Matrices in Population Mathematics, Biometrika. 35 (1948), 213– 245.
  8. P.H. Leslie, A Stochastic Model for Studying the Properties of Certain Biological Systems by Numerical Methods, Biometrika. 45 (1958), 16–031.
  9. S. Pal, N. Pal, S. Samanta, J. Chattopadhyay, Effect of Hunting Cooperation and Fear in a Predator-Prey Model, Ecol. Complex. 39 (2019), 100770.
  10. S. Kumar, H. Kharbanda, Chaotic Behavior of Predator-Prey Model With Group Defense and Non-Linear Harvesting in Prey, Chaos Solitons Fractals. 119 (2019), 19–28.
  11. Y. Zhou, W. Sun, Y. Song, Z. Zheng, J. Lu, S. Chen, Hopf Bifurcation Analysis of a Predator–prey Model With Holling-II Type Functional Response and a Prey Refuge, Nonlinear Dyn. 97 (2019), 1439–1450.
  12. S. Akhtar, R. Ahmed, M. Batool, N.A. Shah, J.D. Chung, Stability, Bifurcation and Chaos Control of a Discretized Leslie Prey-Predator Model, Chaos Solitons Fractals. 152 (2021), 111345.
  13. H. Deng, F. Chen, Z. Zhu, Z. Li, Dynamic Behaviors of Lotka–volterra Predator–prey Model Incorporating Predator Cannibalism, Adv. Differ. Equ. 2019 (2019), 359.
  14. W.C. Allee, Animal Aggregations: A Study in General Sociology, University of Chicago Press, Chicago, 1931.
  15. B. Dennis, Allee Effects: Population Growth, Critical Density, and the Chance of Extinction, Nat. Resource Model. 3 (1989), 481–538.
  16. M. Sen, M. Banerjee, A. Morozov, Bifurcation Analysis of a Ratio-Dependent Prey-predator Model With the Allee Effect, Ecol. Complex. 11 (2012), 12–27.
  17. L. Cheng, H. Cao, Bifurcation Analysis of a Discrete-Time Ratio-Dependent Predator-prey Model With Allee Effect, Commun. Nonlinear Sci. Numer. Simul. 38 (2016), 288–302.
  18. S. Vinoth, R. Sivasamy, K. Sathiyanathan, B. Unyong, G. Rajchakit, R. Vadivel, N. Gunasekaran, The Dynamics of a Leslie Type Predator–prey Model With Fear and Allee Effect, Adv. Differ. Equ. 2021 (2021), 338.
  19. R. Ahmed, S. Akhtar, U. Farooq, S. Ali, Stability, Bifurcation, and Chaos Control of Predator-Prey System With Additive Allee Effect, Commun. Math. Biol. Neurosci. 2023 (2023), 9.
  20. Y. Li, M. Rafaqat, T.J. Zia, I. Ahmed, C.Y. Jung, Flip and Neimark-Sacker Bifurcations of a Discrete Time PredatorPre Model, IEEE Access. 7 (2019), 123430–123435.
  21. P.A. Stephens, W.J. Sutherland, R.P. Freckleton, What Is the Allee Effect?, Oikos. 87 (1999), 185–190.
  22. W.B. Zhang, Discrete Dynamical Systems, Bifurcations and Chaos in Economics, Elsevier, 2006.
  23. W. Yang, Y. Lin, Y. Dai, Rank One Strange Attractors in Periodically Kicked Lorenz System with Time-Delay, Discr. Dyn. Nat. Soc. 2015 (2015), 915614.
  24. S. Kartal, Flip and Neimark–sacker Bifurcation in a Differential Equation With Piecewise Constant Arguments Model, J. Differ. Equ. Appl. 23 (2017), 763–778.
  25. R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, A New Definition of Fractional Derivative, J. Comput. Appl. Math. 264 (2014), 65–70.
  26. T. Abdeljawad, On Conformable Fractional Calculus, J. Comput. Appl. Math. 279 (2015), 57–66.
  27. Y. Liu, X. Li, Dynamics of a Discrete Predator-Prey Model With Holling-II Functional Response, Int. J. Biomath. 14 (2021), 2150068.
  28. M. Berkal, M.B. Almatrafi, Bifurcation and Stability of Two-Dimensional Activator–Inhibitor Model with FractionalOrder Derivative, Fractal Fract. 7 (2023), 344.
  29. M. Berkal, J.F. Navarro, Qualitative Behavior of a Two-dimensional Discrete-time Prey-predator Model, Comput. Math. Methods. 3 (2021), e1193.
  30. M. Berkal, J.F. Navarro, Qualitative Study of a Second Order Difference Equation, Turk. J. Math. 47 (2023), 516–527.
  31. Q. Din, Bifurcation Analysis and Chaos Control in Discrete-Time Glycolysis Models, J. Math. Chem. 56 (2017), 904–931.
  32. M.O. AL-Kaff, H.A. El-Metwally, E.M.M. Elabbasy, Qualitative Analysis and Phase of Chaos Control of the Predator-Prey Model With Holling Type-III, Sci. Rep. 12 (2022), 20111.
  33. R. Ma, Y. Bai, F. Wang, Dynamical Behavior Analysis of a Two-Dimensional Discrete Predator-Prey Model With Prey Refuge and Fear Factor, J. Appl. Anal. Comput. 10 (2020), 1683–1697.
  34. D. Mukherjee, Dynamics of A Discrete-Time Ecogenetic Predator-Prey Model, Commun. Biomath. Sci. 5 (2023), 161–169.
  35. S.Md.S. Rana, Md.J. Uddin, Dynamics of a Discrete-Time Chaotic Lü System, Pan-Amer. J. Math. 1 (2022), 7.
  36. Q. Lin, Allee Effect Increasing the Final Density of the Species Subject to the Allee Effect in a Lotka-volterra Commensal Symbiosis Model, Adv. Differ. Equ. 2018 (2018), 196.
  37. S. Işık, A study of stability and bifurcation analysis in discrete-time predator–prey system involving the Allee effect, Int. J. Biomath. 12 (2019), 1950011.
  38. F. Kangalgil, Neimark–sacker Bifurcation and Stability Analysis of a Discrete-Time Prey-predator Model With Allee Effect in Prey, Adv. Differ. Equ. 2019 (2019), 92.