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Chaotic behaviour of fractional predator-prey dynamical system

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  • Kumar, Sunil
  • Kumar, Ranbir
  • Cattani, Carlo
  • Samet, Bessem

Abstract

In this endeavour, Bernstein wavelet and Euler methods are used to solve a nonlinear fractional predator-prey biological model of two species. The theoretical results with their corresponding biological consequence due to Bernstein wavelet are considered and discussed. A test problem of predator-prey model with two different cases are examined to determined the capability of our proposed methods. We showed that the obtained solutions are the most powerful and, wherever it is possible the comparison, in a very good coincidence with the other numerical solution. Few numerical simulations are finding for predator and prey populations and new chaotic behaviours of predator-prey population model are also obtained by using the Euler method. Moreover, a comparison have been done between the capability of the Bernstein wavelet and the Euler approach. The numerical simulations and behaviours of Rabies model are depicted through graphically which is a special case of predator-prey model.

Suggested Citation

  • Kumar, Sunil & Kumar, Ranbir & Cattani, Carlo & Samet, Bessem, 2020. "Chaotic behaviour of fractional predator-prey dynamical system," Chaos, Solitons & Fractals, Elsevier, vol. 135(C).
    • Handle: RePEc:eee:chsofr:v:135:y:2020:i:c:s0960077920302125
    DOI: 10.1016/j.chaos.2020.109811
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    1. Wang, Jiao & Xu, Tian-Zhou & Wang, Gang-Wei, 2018. "Numerical algorithm for time-fractional Sawada-Kotera equation and Ito equation with Bernstein polynomials," Applied Mathematics and Computation, Elsevier, vol. 338(C), pages 1-11.
    2. Guo, Xiaoxia & Zhu, Chunjuan & Ruan, Dehao, 2019. "Dynamic behaviors of a predator–prey model perturbed by a complex type of noises," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 523(C), pages 1024-1037.
    3. Hajipour, Mojtaba & Jajarmi, Amin & Malek, Alaeddin & Baleanu, Dumitru, 2018. "Positivity-preserving sixth-order implicit finite difference weighted essentially non-oscillatory scheme for the nonlinear heat equation," Applied Mathematics and Computation, Elsevier, vol. 325(C), pages 146-158.
    4. Erfanian, Majid & Mansoori, Amin, 2019. "Solving the nonlinear integro-differential equation in complex plane with rationalized Haar wavelet," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 165(C), pages 223-237.
    5. Mirzaee, Farshid & Samadyar, Nasrin, 2019. "Numerical solution based on two-dimensional orthonormal Bernstein polynomials for solving some classes of two-dimensional nonlinear integral equations of fractional order," Applied Mathematics and Computation, Elsevier, vol. 344, pages 191-203.
    6. Bhatter, Sanjay & Mathur, Amit & Kumar, Devendra & Singh, Jagdev, 2020. "A new analysis of fractional Drinfeld–Sokolov–Wilson model with exponential memory," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 537(C).
    7. Thierry, Hugo & Sheeren, David & Marilleau, Nicolas & Corson, Nathalie & Amalric, Marion & Monteil, Claude, 2015. "From the Lotka–Volterra model to a spatialised population-driven individual-based model," Ecological Modelling, Elsevier, vol. 306(C), pages 287-293.
    8. Fernandez, Arran & Özarslan, Mehmet Ali & Baleanu, Dumitru, 2019. "On fractional calculus with general analytic kernels," Applied Mathematics and Computation, Elsevier, vol. 354(C), pages 248-265.
    9. Ghalib, M. Mansha & Zafar, Azhar A. & Riaz, M. Bilal & Hammouch, Z. & Shabbir, Khurram, 2020. "Analytical approach for the steady MHD conjugate viscous fluid flow in a porous medium with nonsingular fractional derivative," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 554(C).
    10. Ghanbari, Behzad & Kumar, Sunil & Kumar, Ranbir, 2020. "A study of behaviour for immune and tumor cells in immunogenetic tumour model with non-singular fractional derivative," Chaos, Solitons & Fractals, Elsevier, vol. 133(C).
    11. Yousefi, S. & Razzaghi, M., 2005. "Legendre wavelets method for the nonlinear Volterra–Fredholm integral equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 70(1), pages 1-8.
    12. Peng, Yahong & Zhang, Guoying, 2020. "Dynamics analysis of a predator–prey model with herd behavior and nonlocal prey competition," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 170(C), pages 366-378.
    13. Asgari, M. & Ezzati, R., 2017. "Using operational matrix of two-dimensional Bernstein polynomials for solving two-dimensional integral equations of fractional order," Applied Mathematics and Computation, Elsevier, vol. 307(C), pages 290-298.
    14. Heydari, Mohammad Hossein & Avazzadeh, Zakieh, 2018. "Legendre wavelets optimization method for variable-order fractional Poisson equation," Chaos, Solitons & Fractals, Elsevier, vol. 112(C), pages 180-190.
    15. Li, Haihong & Cong, Fuzhong, 2019. "Dynamics of a stochastic Holling–Tanner predator–prey model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 531(C).
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