On the dynamics of a class of Kolmogorov systems

In this paper we charaterize the integrability and the non-existence of limit cycles of Kolmogorov systems of the form
\begin{equation*} \left\{ \begin{array}{l} x^{\prime }=x\left( P\left( x,y\right) +\sqrt{R\left( x,y\right) }\right) , \\ y^{\prime }=y\left( Q\left( x,y\right) +\sqrt{R\left( x,y\right) }\right) , \end{array} \right. \end{equation*}
where $P\left( x,y\right) ,$ $Q\left( x,y\right) ,$ $R\left( x,y\right) ,$ homogeneous polynomials of degree $n,$ $n,$ $m,$ respectively.