Qualitative Behavior for a Discretized Conformable Fractional-Order Lotka-Volterra Model With Harvesting Effects

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Messaoud Berkal, Juan F. Navarro, M. B. Almatrafi


The predator-prey model is a widely mathematical structure that explains the dynamics between two interacting populations: predators and prey. The predator-prey interaction represents a fundamental dynamic in nature, influencing the stability and balance of ecosystems worldwide. The purpose of this article is to provide insight into the complex interactions and feedback mechanisms between predators and prey in ecological systems via mathematical tools such as stability and bifurcation. We investigate a fractional-order Lotka-Volterra model with a harvesting effect using stability and bifurcation theory. The equilibrium points and local stability of the purposed model are presented in this article. The bifurcation analysis, which is a potent approach used to analyse the qualitative behavior of the predator-prey system as the parameter values are varied, is also explored. In particular, a Neimark-Sacker bifurcation and a period-doubling bifurcation are theoretically and numerically examined. Furthermore, we illustrate some 2D figures to show the phase portriat and bifurcations of this model at various points.

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  1. A.J. Lotka, Elements of Physical Biology, Williams & Wilkins, 1925.
  2. V. Volterra, Variazioni e fluttuazioni del numero di individui in specie animali conviventi, Mem. Acad. Lincei. 2 (1926), 31–113.
  3. M.J. Uddin, S.Md.S. Rana, S. I¸sık, F. Kangalgil, On the Qualitative Study of a Discrete Fractional Order Prey–predator Model With the Effects of Harvesting on Predator Population, Chaos Solitons Fractals. 175 (2023), 113932. https://doi.org/10.1016/j.chaos.2023.113932.
  4. J. Lee, H. Baek, Dynamics of a Beddington-Deangelis Type Predator-Prey System With Constant Rate Harvesting, Electron. J. Qual. Theory Differ. Equ. 2017 (2017), 1–20. https://doi.org/10.14232/ejqtde.2017.1.1.
  5. X. Liu, Q. Huang, Comparison and Analysis of Two Forms of Harvesting Functions in the Two-Prey and OnePredator Model, J. Inequal. Appl. 2019 (2019), 307. https://doi.org/10.1186/s13660-019-2260-y.
  6. B. Sahoo, B. Das, S. Samanta, Dynamics of Harvested-Predator-Prey Model: Role of Alternative Resources, Model. Earth Syst. Environ. 2 (2016), 140. https://doi.org/10.1007/s40808-016-0191-x.
  7. M. Sen, P.D.N. Srinivasu, M. Banerjee, Global Dynamics of an Additional Food Provided Predator–prey SystemWith Constant Harvest in Predators, Appl. Math. Comput. 250 (2015), 193–211. https://doi.org/10.1016/j.amc.2014.10.085.
  8. Q. Lin, Dynamic Behaviors of a Commensal Symbiosis Model With Non-Monotonic Functional Response and Non-Selective Harvesting in a Partial Closure, Commun. Math. Biol. Neurosci. 2018 (2018), 4. https://doi.org/10.28919/cmbn/3652.
  9. A. Das, M. Pal, Theoretical Analysis of an Imprecise Prey-Predator Model with Harvesting and Optimal Control, J. Optim. 2019 (2019), 9512879. https://doi.org/10.1155/2019/9512879.
  10. N. Mohdeb, A Dynamic Analysis of a Prey–predator Population Model With a Nonlinear Harvesting Rate, Arab J. Math. Sci. (2023). https://doi.org/10.1108/ajms-03-2022-0052.
  11. M.B. Almatrafi, M. Berkal, Bifurcation Analysis and Chaos Control for Prey-Predator Model With Allee Effect, Int. J. Anal. Appl. 21 (2023), 131. https://doi.org/10.28924/2291-8639-21-2023-131.
  12. 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. https://doi.org/10.1142/s0218348x23401606.
  13. M.B. Almatrafi, Solitary Wave Solutions to a Fractional Model Using the Improved Modified Extended TanhFunction Method, Fractal Fract. 7 (2023), 252. https://doi.org/10.3390/fractalfract7030252.
  14. M. Berkal, J.F. Navarro, Qualitative Behavior of a Two-dimensional Discrete-time Prey-predator Model, Comput. Math. Methods. 3 (2021), e1193. https://doi.org/10.1002/cmm4.1193.
  15. A.Q. Khan, M.B. Almatrafi, Two-Dimensional Discrete-Time Laser Model With Chaos and Bifurcations, AIMS Math. 8 (2023), 6804–6828. https://doi.org/10.3934/math.2023346.
  16. 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. https://doi.org/10.1016/j.aej.2022.04.042.
  17. 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. https://doi.org/10.1142/s1793524522500954.
  18. 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. https://doi.org/10.3934/mbe.2022092.
  19. 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. https://doi.org/10.28924/2291-8639-21-2023-100.
  20. S. Kartal, F. Gurcan, Discretization of Conformable Fractional Differential Equations by a Piecewise Constant Approximation, Int. J. Comput. Math. 96 (2019), 1849–1860. https://doi.org/10.1080/00207160.2018.1536782.
  21. R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, A New Definition of Fractional Derivative, J. Comput. Appl. Math. 264 (2014), 65–70. https://doi.org/10.1016/j.cam.2014.01.002.
  22. T. Abdeljawad, On Conformable Fractional Calculus, J. Comput. Appl. Math. 279 (2015), 57–66. https://doi.org/10.1016/j.cam.2014.10.016.
  23. Y. Liu, X. Li, Dynamics of a Discrete Predator-Prey Model With Holling-II Functional Response, Int. J. Biomath. 14 (2021), 2150068. https://doi.org/10.1142/S1793524521500686.
  24. M. Berkal, M.B. Almatrafi, Bifurcation and Stability of Two-Dimensional Activator-Inhibitor ModelWith FractionalOrder Derivative, Fractal Fract. 7 (2023), 344. https://doi.org/10.3390/fractalfract7050344.
  25. M. Berkal, J.F. Navarro, Qualitative Study of a Second Order Difference Equation, Turk. J. Math. 47 (2023), 516–527. https://doi.org/10.55730/1300-0098.3375.
  26. Q. Din, M.A. Zulfiqar, Qualitative Behavior of a Discrete Predator-Prey System Under Fear Effects, Z. Naturforsch. A. 77 (2022), 1023–1043. https://doi.org/10.1515/zna-2022-0129.
  27. Q. Din, Controlling Chaos in a Discrete-Time Prey-Predator Model With Allee Effects, Int. J. Dyn. Control. 6 (2018), 858–872. https://doi.org/10.1007/s40435-017-0347-1.
  28. Q. Din, Neimark-Sacker Bifurcation and Chaos Control in Hassell-Varley Model, J. Differ. Equ. Appl. 23 (2017), 741–762. https://doi.org/10.1080/10236198.2016.1277213.
  29. 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. https://doi.org/10.11948/20190426.
  30. S.Md.S. Rana, Md.J. Uddin, Dynamics of a Discrete-Time Chaotic Lü System, Pan-Amer. J. Math. 1 (2022), 7. https://doi.org/10.28919/cpr-pajm/1-7.
  31. G. Chen, X. Dong, From Chaos to Order: Methodologies, Perspectives and Applications, World Scientific, 1998.
  32. S.N. Elaydi, An Introduction to Difference Equations, Springer, New York, 1996.
  33. S. Lynch, Dynamical Systems With Applications Using Mathematica, Springer, 2007.
  34. S. Vinoth, R. Sivasamy, K. Sathiyanathan, B. Unyong, R. Vadivel, N. Gunasekaran, A Novel Discrete-Time LeslieGower Model with the Impact of Allee Effect in Predator Population, Complexity. 2022 (2022), 6931354. https://doi.org/10.1155/2022/6931354.
  35. M. Imran, M.B. Almatrafi, R. Ahmed, Stability and Bifurcation Analysis of a Discrete Predator-prey System of Ricker Type With Harvesting Effect, Commun. Math. Biol. Neurosci. 2024 (2024), 11. https://doi.org/10.28919/cmbn/8313.