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

For two-dimensional Kolmogorov system, where $R\left( x,y\right)$, $S\left( x,y\right)$, $P\left( x,y\right)$, $Q\left( x,y\right)$, $M\left( x,y\right)$, and $N\left( x,y\right) $ are homogeneous polynomials of degrees $m$, $a$, $n$, $n$, $b$, and $b$, respectively, we obtain an explicit expression of the first integral and prove the non-existence of periodic orbits and of limit cycles. We adduce an example of applicability of our result.