A fractional model for population dynamics of two interacting species by using spectral and Hermite wavelets methods

• Published in: Numerical methods for partial differential equations Vol. 37; no. 2; pp. 1652 - 1672
• Main Authors: Kumar, Sunil, Ghosh, Surath, Kumar, Ranbir, Jleli, Mohamed
• Format: Journal Article
• Language: English
• Published: Hoboken, USA John Wiley & Sons, Inc 01.03.2021; Wiley Subscription Services, Inc
• Subjects: Caputo derivative; Chebyshev approximation; Collocation methods; Comparative studies; fractional Lotka‐Volterra model; Hermite wavelets method; Integers; Iterative methods; Mathematical models; Nonlinear equations; Polynomials; Spectra; spectral method; Spectral methods; Volterra integral equations
• ISSN: 0749-159X; 1098-2426
• DOI: 10.1002/num.22602

The Lotka‐Volterra model is a very famous model and frequently used to describe the dynamics of ecological systems in which two species interact, one a predator and one its prey. In this present work, a comparative study is presented for solving Lotka‐Volterra model which has an important role in Biological sciences. The Lotka‐Volterra equations are solved by spectral collocation method (SCM) by using shifted Chebyshev polynomials of first kind. Then, a numerical example with two different cases is presented to demonstrate the accuracy and effectiveness of the numerical schemes. Using spectral method, we get a system of nonlinear algebraic equations which are solved by Newton's iteration method. Also the achieved solutions are compared with another solutions obtained by the Hermite wavelets method (HWM). Further, this work reflects that the integer order Lotka‐Volterra model and the fractional order one have close relationship where integer order model is a special case of fractional order. The behaviors study of predator and prey populations due to presence of fractional derivative are the key features of the present work. Our numerical and graphical results indicate that the proposed methods are simple, powerful and fully compatible for treatment of the Lotka‐Volterra model of fractional order.