Integrability of Lotka-Volterra differential equation via integrating factors

Abstract

We consider the Lotka-Volterra differential equation:y(a_2*x +b_2*y+c_2)*dx=x*(a_1*x +b_1*y+c_1)*d_y in which the coefficients a_1, b_1, c_1, a_2, b_2, c_2 and variables x, y are assumed to be real. This equation introduced by Lotka and Volterra appears in ecology where it models two species in competition. It has been widely used in chemistry, applied mathematics and in a large variety of physical topics such as laser physics, plasma physics, neural networks, hydrodynamics etc. In this paper, depending on the coefficients of the equation, we study its integrability by using different methods including the existence of integrating factors of the form µ=h_(n)^(-1) where h_(n) (x,y) is a polynomial of degree n, n €{1,2,3}.