Computational dynamics of a Lotka-Volterra Model with additive Allee effect based on Atangana-Baleanu fractional derivative

Hasan S. Panigoro, Emli Rahmi

Abstract


This paper studies an interaction between one prey and one predator following Lotka-Volterra model with additive Allee effect in predator. The Atangana-Baleanu fractional-order derivative is used for the operator. Since the theoretical ways to investigate the model using this operator are limited, the dynamical behaviors are identified numerically. By simulations, the influence of the order of the derivative on the dynamical behaviors is given. The numerical results show that the order of the derivative may impact the convergence rate, the occurrence of Hopf bifurcation, and the evolution of the diameter of the limit-cycle.

Keywords


Lotka-Volterra; Allee Effect; Atangana-Baleanu; Numerical Solution

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References


A. J. Lotka, “Elements of Physical Biology,” Nature, vol. 116, no. 2917, pp. 461–461, 1925.

D. Savitri and H. S. Panigoro, “Bifurkasi Hopf pada model prey-predator-super predator dengan fungsi respon yang berbeda,” Jambura Journal of Biomathematics, vol. 1, no. 2, pp. 65–70, dec 2020.

H. S. Panigoro and D. Savitri, “Bifurkasi Hopf pada model Lotka-Volterra orde-fraksional dengan efek Allee aditif pada predator,” Jambura Journal of Biomathematics, vol. 1, no. 1, pp. 16–24, 2020.

L. K. Beay and M. Saija, “Dynamics of a stage–structure Rosenzweig–MacArthur model with linear harvesting in prey and cannibalism in predator,” Jambura Journal of Biomathematics, vol. 2, no. 1, pp. 42–50, jun 2021.

H. Deng, F. Chen, Z. Zhu, and Z. Li, “Dynamic behaviors of Lotka–Volterra predator–prey model incorporating predator cannibalism,” Advances in Difference Equations, vol. 2019, no. 1, pp. 1–17, 2019.

Z. Zhu, R. Wu, L. Lai, and X. Yu, “The influence of fear effect to the Lotka–Volterra predator–prey system with predator has other food resource,” Advances in Difference Equations, vol. 2020, no. 1, pp. 1–13, 2020.

S. Maisaroh, R. Resmawan, and E. Rahmi, “Analisis kestabilan model predator-prey dengan infeksi penyakit pada prey dan pemanenan proporsional pada predator,” Jambura Journal of Biomathematics, vol. 1, no. 1, pp. 8–15, jun 2020.

W. C. Allee, Animal aggregations, a study in general sociology. Chicago :: The University of Chicago Press„ 1931.

C. Rebelo and C. Soresina, “Coexistence in seasonally varying predator–prey systems with Allee effect,” Nonlinear Analysis: Real World Applications, vol. 55, p. 103140, 2020.

E. Rahmi, I. Darti, A. Suryanto, Trisilowati, and H. S. Panigoro, “Stability Analysis of a Fractional-Order Leslie-Gower Model with Allee Effect in Predator,” Journal of Physics: Conference Series, vol. 1821, no. 1, 2021.

C. Arancibia–Ibarra and J. Flores, “Dynamics of a Leslie–Gower predator–prey model with Holling type II functional response, Allee effect and a generalist predator,” Mathematics and Computers in Simulation, vol. 188, pp. 1–22, 2021.

E. Rahmi, I. Darti, A. Suryanto, and Trisilowati, “A Modified Leslie–Gower Model Incorporating Beddington–DeAngelis Functional Response, Double Allee Effect and Memory Effect,” Fractal and Fractional, vol. 5, no. 3, p. 84, 2021.

H. S. Panigoro, E. Rahmi, N. Achmad, and S. L. Mahmud, “The Influence of Additive Allee Effect and Periodic Harvesting to the Dynamics of Leslie-Gower Predator-Prey Model,” Jambura Journal of Mathematics, vol. 2, no. 2, pp. 87–96, 2020.

L. Lai, Z. Zhu, and F. Chen, “Stability and Bifurcation in a Predator–Prey Model with the Additive Allee Effect and the Fear Effect,” Mathematics, vol. 8, no. 8, p. 1280, 2020.

Y. Cai, C. Zhao, W. Wang, and J. Wang, “Dynamics of a Leslie–Gower predator–prey model with additive Allee effect,” Applied Mathematical Modelling, vol. 39, no. 7, pp. 2092–2106, 2015.

D. Indrajaya, A. Suryanto, and A. R. Alghofari, “Dynamics of modified Leslie-Gower predator-prey model with Beddington-DeAngelis functional response and additive Allee effect,” International Journal of Ecology and Development, vol. 31, no. 3, pp. 60–71, 2016.

I. Podlubny, Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. San Diego CA: Academic Press, 1999.

M. Caputo, “Linear Models of Dissipation whose Q is almost Frequency Independent–II,” Geophysical Journal International, vol. 13, no. 5, pp. 529–539, 1967.

M. Caputo and M. Fabrizio, “A new definition of fractional derivative without singular kernel,” Progress in Fractional Differentiation and Applications, vol. 1, no. 2, pp. 73–85, 2015.

A. Atangana and D. Baleanu, “New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model,” Thermal Science, vol. 20, no. 2, pp. 763–769, 2016.

H. S. Panigoro, A. Suryanto, W. M. Kusumawinahyu, and I. Darti, “Dynamics of an Eco-Epidemic Predator–Prey Model Involving Fractional Derivatives with Power-Law and Mittag–Leffler Kernel,” Symmetry, vol. 13, no. 5, p. 785, 2021.

——, “A Rosenzweig–MacArthur model with continuous threshold harvesting in predator involving fractional derivatives with power law and mittag–leffler kernel,” Axioms, vol. 9, no. 4, p. 122, 2020.

D. Baleanu, A. Jajarmi, and M. Hajipour, “On the nonlinear dynamical systems within the generalized fractional derivatives with Mittag–Leffler kernel,” Nonlinear Dynamics, vol. 94, no. 1, pp. 397–414, 2018.




DOI: https://doi.org/10.34312/jjbm.v2i2.11886

Copyright (c) 2021 Hasan S. Panigoro, Emli Rahmi

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