Complex Dynamics of a Predator-Prey System With Gompertz Growth and Herd Behavior

Main Article Content

Rizwan Ahmed, M. B. Almatrafi


The complex dynamics of a predator-prey system in discrete time are studied. In this system, we consider the prey’s Gompertz growth and the square-root functional response. The existence of fixed points and stability are examined. Using the center manifold and bifurcation theory, we found that the system undergoes transcritical bifurcation, period-doubling bifurcation, and Neimark-Sacker bifurcation. In addition, numerical examples are presented to illustrate the consistency of the analytical findings. The bifurcation diagrams show that the positive fixed point is stable if the death rate of the predator is greater than a threshold value. Biologically, it means that to prevent the predator population from growing uncontrollably and stability of the positive fixed point, the predator’s death rate should be greater than the threshold value.

Article Details


  1. A.J. Lotka, Elements of Physical Biology, Science Progress in the Twentieth Century (1919-1933), Vol. 21, pp. 341–343, Williams & Wilkins Company, Philadelphia, (1926).
  2. V. Volterra, Fluctuations in the abundance of a species considered mathematically1 , Nature. 118 (1926), 558-560.
  3. C.S. Holling, Some Characteristics of Simple Types of Predation and Parasitism, Canadian Entomol. 91 (1959), 385–398.
  4. P.H. Crowley, E.K. Martin, Functional Responses and Interference within and between Year Classes of a Dragonfly Population, J. North Amer. Benthol. Soc. 8 (1989), 211–221.
  5. J.R. Beddington, Mutual Interference Between Parasites or Predators and its Effect on Searching Efficiency, J. Animal Ecol. 44 (1975), 331–340.
  6. D.L. DeAngelis, R.A. Goldstein, R.V. O’Neill, A Model for Tropic Interaction, Ecology. 56 (1975), 881–892.
  7. M.F. Elettreby, A. Khawagi, T. Nabil, Dynamics of a Discrete Prey-Predator Model with Mixed Functional Response, Int. J. Bifurcation Chaos. 29 (2019), 1950199.
  8. S.M.S. Rana, U. Kulsum, Bifurcation Analysis and Chaos Control in a Discrete-Time Predator-Prey System of Leslie Type with Simplified Holling Type IV Functional Response, Discr. Dyn. Nat. Soc. 2017 (2017), 9705985.
  9. X. Chen, X. Zhang, Dynamics of the Predator-Prey Model With the Sigmoid Functional Response, Stud. Appl. Math. 147 (2021), 300-318.
  10. C. Arancibia-Ibarra, P. Aguirre, J. Flores, P. van Heijster, Bifurcation Analysis of a Predator-Prey Model With Predator Intraspecific Interactions and Ratio-Dependent Functional Response, Appl. Math. Comput. 402 (2021), 126152.
  11. S.M.S. Rana, Bifurcation and Complex Dynamics of a Discrete-Time Predator-Prey System With Simplified Monod-Haldane Functional Response, Adv. Differ. Equ. 2015 (2015), 345.
  12. H. Baek, Complex Dynamics of a Discrete-Time Predator-Prey System with Ivlev Functional Response, Math. Probl. Eng. 2018 (2018), 8635937.
  13. V. Ajraldi, M. Pittavino, E. Venturino, Modeling Herd Behavior in Population Systems, Nonlinear Anal.: Real World Appl. 12 (2011), 2319–2338.
  14. P.A. Braza, Predator-Prey Dynamics With Square Root Functional Responses, Nonlinear Anal.: Real World Appl. 13 (2012), 1837–1843.
  15. S.M. Salman, A.M. Yousef, A.A. Elsadany, Stability, Bifurcation Analysis and Chaos Control of a Discrete PredatorPrey System With Square Root Functional Response, Chaos Solitons Fractals. 93 (2016), 20–31.
  16. M. Berkal, M.B. Almatrafi, Bifurcation and Stability of Two-Dimensional Activator–Inhibitor Model with FractionalOrder Derivative, Fractal Fract. 7 (2023), 344.
  17. M.B. Almatrafi, Solitary Wave Solutions to a Fractional Model Using the Improved Modified Extended TanhFunction Method, Fractal Fract. 7 (2023), 252.
  18. A.Q. Khan, M.B. Almatrafi, Two-Dimensional Discrete-Time Laser Model With Chaos and Bifurcations, AIMS Math. 8 (2023), 6804–6828.
  19. 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.
  20. M.B. Almatrafi, Abundant Traveling Wave and Numerical Solutions for Novikov-Veselov System With Their Stability and Accuracy, Appl. Anal. 102 (2022), 2389–2402.
  21. M.G. Mortuja, M.K. Chaube, S. Kumar, Dynamic Analysis of a Predator-Prey System With Nonlinear Prey Harvesting and Square Root Functional Response, Chaos Solitons Fractals. 148 (2021), 111071.
  22. P. Chakraborty, U. Ghosh, S. Sarkar, Stability and Bifurcation Analysis of a Discrete Prey-predator Model With Square-Root Functional Response and Optimal Harvesting, J. Biol. Syst. 28 (2020), 91-110.
  23. P. Panja, Combine Effects of Square Root Functional Response and Prey Refuge on Predator-Prey Dynamics, Int. J. Model. Simul. 41 (2020), 426-433.
  24. B. Gompertz, On the Nature of the Function Expressive of the Law of Human Mortality, and on a New Mode of Determining the Value of Life Contingencies, Phil. Trans. R. Soc. Lond. 115 (1825), 513–583.
  25. P.A. Naik, Z. Eskandari, M. Yavuz, J. Zu, Complex Dynamics of a Discrete-Time Bazykin-Berezovskaya PreyPredator Model With a Strong Allee Effect, J. Comput. Appl. Math. 413 (2022), 114401.
  26. S.M.S. Rana, Dynamics and Chaos Control in a Discrete-Time Ratio-Dependent Holling-Tanner Model, J. Egypt Math. Soc. 27 (2019), 48.
  27. M. Zhao, C. Li, J. Wang, Complex Dynamic Behaviors of a Discrete-Time Predator-Prey System, J. Appl. Anal. Comput. 7 (2017), 478–500.
  28. K.S.N. Al-Basyouni, A.Q. Khan, Bifurcation Analysis of a Discrete-Time Chemostat Model, Math. Probl. Eng. 2023 (2023), 7518261.
  29. B. Hong, C. Zhang, Neimark-Sacker Bifurcation of a Discrete-Time Predator-Prey Model with Prey Refuge Effect, Mathematics. 11 (2023), 1399.
  30. Q. Din, Complex Dynamical Behavior and Control of a Discrete Ecological Model, J. Vib. Control. (2022).
  31. A. Tassaddiq, M.S. Shabbir, Q. Din, H. Naaz, Discretization, Bifurcation, and Control for a Class of Predator-Prey Interactions, Fractal Fract. 6 (2022), 31.
  32. A.C.J. Luo, Regularity and Complexity in Dynamical Systems, Springer, New York, 2012.
  33. Y.A. Kuznetsov, Elements of Applied Bifurcation Theory, Springer, New York, 2004.
  34. S. Wiggins, M. Golubitsky, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer, New York, 2003.
  35. 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.
  36. R. Ahmed, A. Ahmad, N. Ali, Stability Analysis and Neimark-Sacker Bifurcation of a Nonstandard Finite Difference Scheme for Lotka-Volterra Prey-Predator Model, Commun. Math. Biol. Neurosci. 2022 (2022), 61.
  37. A. Al Khabyah, R. Ahmed, M.S. Akram, S. Akhtar, Stability, Bifurcation, and Chaos Control in a Discrete PredatorPrey Model With Strong Allee Effect, AIMS Math. 8 (2023), 8060–8081.
  38. M.S. Shabbir, Q. Din, R. Alabdan, A. Tassaddiq, K. Ahmad, Dynamical Complexity in a Class of Novel DiscreteTime Predator-Prey Interaction With Cannibalism, IEEE Access. 8 (2020), 100226–100240.
  39. A.A. Elsadany, Q. Din, S.M. Salman, Qualitative Properties and Bifurcations of Discrete-Time Bazykin-berezovskaya Predator-prey Model, Int. J. Biomath. 13 (2020), 2050040.