On the class of two dimensional Kolmogorov systems

Abstract

In this paper we charaterize the integrability and the non-existence of limit cycles of Kolmogorov systems of the form

\[

\left\{

\begin{array}{l}

x^{\prime }=x\left( P\left( x,y\right) +R\left( x,y\right) \ln \left\vert

\frac{A\left( x,y\right) }{B\left( x,y\right) }\right\vert \right) , \\

y^{\prime }=y\left( Q\left( x,y\right) +R\left( x,y\right) \ln \left\vert

\frac{A\left( x,y\right) }{B\left( x,y\right) }\right\vert \right) ,

\end{array}

\right.

\]

where $A\left(x,y\right)$, $B\left(x,y\right)$, $P\left( x,y\right)$, $Q\left(x,y\right)$, $R\left(x,y\right)$ are homogeneous polynomials of degree $a$, $a$, $n$, $n$, $m$ respectively. Concrete example exhibiting the applicability of our result is introduced.