On the dynamics of a class of Kolmogorov systems

We characterize the integrability and the non-existence of periodic orbits for the 2-dimensional Kolmogorov systems of the formẋ=x(Pn(x,y)+Rm(x,y)),ẏ=y(Qn(x,y)+Rm(x,y)),where n and m   are positive integers and Pn,QnPn,Qn and RmRm are homogeneous polynomials of degree n,nn,n and m, respectively.