Painlevé analysis and exact solutions of a predator–prey system with diffusion

Abstract

A system of equations for description of the predator-prey relations is considered. The model corresponds to the modified Lotka-Volterra system with logistic growth of the prey and with both predator and prey dispersing by diffusion. The Painlevé analysis of the system of equations is studied. Exact traveling wave solutions are found by means of the Q-function method.

Analytic Travelling Wave Solutions of Nonlinear Coupled Equations of Fractional Order

“…Solving the result system of algebraic equations in Step 3 and using solution (8), finally we can compose the family of exact travelling wave solutions for Eq. (9).…”

“…The method of Q-functions is a very powerful approach for obtaining exact solutions of nonlinear ordinary and partial differential equations arising in mathematical physics [6] and also the advantage of this method is discussed in the papers [7,8]. More recently, Kudryashov and Zakharchenko [9] obtained exact traveling wave solutions of a predator-prey system with diffusion by means of the Q-function method. In this paper, the Qfunction method is employed to obtain some exact solutions of the nonlinear coupled equations of fractional order [4][5][6][7].…”

“…More recently, Kudryashov and Zakharchenko [9] obtained exact traveling wave solutions of a predator-prey system with diffusion by means of the Q-function method. In this paper, the Qfunction method is employed to obtain some exact solutions of the nonlinear coupled equations of fractional order [4][5][6][7].…”

Some Explicit Solutions of Nonlinear Evolution Equations

“…Apart from physical importance, the closed-form solutions of nonlinear partial differential equations assist the numerical solvers to compare the correctness of their results and help them in the stability analysis. * In this paper, we employ the system technique to obtain exact solutions for nonlinear partial differential equations that contain exponential function based on a suitable choice of parameters through the Painlevé test [19,20,21,22]. The aim of this paper is to obtain more exact explicit solutions and to analyze the motions of exact solutions as the values of parameters and proper coefficients about the equal width wave equation and the (2+1)-dimensional Maccari's system.…”

Painlevé analysis and exact solutions of the nonlinear diffusion equation with a polynomial source

“…It is well known that the Painlevé test for nonlinear differential equations is a powerful approach for testing integrable differential equation. Besides, we are going to use the logistic function method [11][12][13][14][15][16][17] to find exact solutions of Equation (1). Note that some exact solutions of Equation (1) are given in [18].…”