On the non existence of periodic orbits for a class of two dimensional Kolmogorov systems

Abstract

Abstract In this paper we characterize the integrability and the non-existence of limit cycles of Kolmogorov systems of the form x ′ = x ( B 1 ( x , y ) ln | A 3 ( x , y ) A 4 ( x , y ) | + B 3 ( x , y ) ln | A 1 ( x , y ) A 2 ( x , y ) | ) , y ′ = y ( B 2 ( x , y ) ln | A 5 ( x , y ) A 6 ( x , y ) | + B 3 ( x , y ) ln | A 1 ( x , y ) A 2 ( x , y ) | ) \matrix{{x' = x\left( {{B_1}\left( {x,y} \right)\ln \left| {{{{A_3}\left( {x,y} \right)} \over {{A_4}\left( {x,y} \right)}}} \right| + {B_3}\left( {x,y} \right)\ln \left| {{{{A_1}\left( {x,y} \right)} \over {{A_2}\left( {x,y} \right)}}} \right|} \right),} \hfill \cr {y' = y\left( {{B_2}\left( {x,y} \right)\ln \left| {{{{A_5}\left( {x,y} \right)} \over {{A_6}\left( {x,y} \right)}}} \right| + {B_3}\left( {x,y} \right)\ln \left| {{{{A_1}\left( {x,y} \right)} \over {{A_2}\left( {x,y} \right)}}} \right|} \right)} \hfill \cr } where A 1 (x, y), A 2 (x, y), A 3 (x, y), A 4 (x, y), A 5 (x, y), A 6 (x, y), B 1 (x, y), B 2 (x, y), B 3 (x, y) are homogeneous polynomials of degree a, a, b, b, c, c, n, n, m respectively. Concrete example exhibiting the applicability of our result is introduced.