A comparison between Caputo and Caputo-Fabrizio fractional derivatives for modelling Lotka-Volterra differential equations

In this paper, we apply the concept of the fractional calculus to study three-dimensional LotkaVolterra differential equations. Our goal is to compare the results of this system with respect to Caputo and Caputo-Fabrizio fractional derivatives. According to the existence of non-singular kernel in the definition of Caputo-Fabrizio operator, we analyze the stability of the system and try to improve a numerical method based on a corrected Adams-Bashforth method. Numerical results show that the behaviors of the Lotka-Volterra system depend on the fractional derivative order as well as the differential operators.

[1]  Badr Saad T. Alkahtani,et al.  Controlling the wave movement on the surface of shallow water with the Caputo–Fabrizio derivative with fractional order , 2016 .

[2]  L. Greller,et al.  Explosive route to chaos through a fractal torus in a generalized lotka-volterra model , 1988 .

[3]  Alireza Nazemi,et al.  Nonlinear fractional optimal control problems with neural network and dynamic optimization schemes , 2017, Nonlinear Dynamics.

[4]  Chang Phang,et al.  New predictor‐corrector scheme for solving nonlinear differential equations with Caputo‐Fabrizio operator , 2018, Mathematical Methods in the Applied Sciences.

[5]  G. Zaman,et al.  Mathematical analysis of HIV/AIDS infection model with Caputo-Fabrizio fractional derivative , 2018 .

[6]  M. Caputo,et al.  A new Definition of Fractional Derivative without Singular Kernel , 2015 .

[7]  S Das,et al.  A mathematical model on fractional Lotka-Volterra equations. , 2011, Journal of theoretical biology.

[8]  Shouming Zhong,et al.  Stability Analysis of a Fractional-Order Linear System Described by the Caputo–Fabrizio Derivative , 2019, Mathematics.

[9]  Sara Dadras,et al.  Control of a fractional-order economical system via sliding mode , 2010 .

[10]  Călin-Adrian Comes,et al.  Banking System: Three level Lotka-Volterra Model , 2012 .

[11]  S. Arabia,et al.  Properties of a New Fractional Derivative without Singular Kernel , 2015 .

[12]  Marcel Ausloos,et al.  Fractional Dynamics of Network Growth Constrained by Aging Node Interactions , 2016, PloS one.

[13]  Ahmed M. A. El-Sayed,et al.  On the fractional-order logistic equation , 2007, Appl. Math. Lett..

[14]  Dos Santos,et al.  Non-Gaussian Distributions to Random Walk in the Context of Memory Kernels , 2018 .

[15]  Michele Caputo,et al.  Applications of New Time and Spatial Fractional Derivatives with Exponential Kernels , 2016 .

[16]  N. Azimi-Tafreshi,et al.  Memory effects on epidemic evolution: The susceptible-infected-recovered epidemic model , 2017, Physical review. E.

[17]  D. Matignon.  Stability results for fractional differential equations with applications to control processing , 1996 .

[18]  Tamer Nabil,et al.  Fractional-Order Model of Two-Prey One-Predator System , 2017 .

[19]  E. Ahmed,et al.  Equilibrium points, stability and numerical solutions of fractional-order predator–prey and rabies models , 2007 .

[20]  Alireza Nazemi,et al.  Solving fractional optimal control problems with fixed or free final states by Haar wavelet collocation method , 2016, IMA J. Math. Control. Inf..

[21]  Gabriel A. D. Lopes,et al.  A fast sampling method for estimating the domain of attraction , 2016 .

[22]  Dumitru Baleanu,et al.  A numerical framework for the approximate solution of fractional tumor-obesity model , 2019, Int. J. Model. Simul. Sci. Comput..

[23]  Hasan Bulut,et al.  Cancer treatment model with the Caputo-Fabrizio fractional derivative , 2018 .