Integrability and Non-Existence of Periodic Orbits for a Class of Kolmogorov Systems

In this article, we study the integrability and the non-existence of periodic orbits for the planar Kolmogorov differential systems of the form x˙=x(Rn-1(x,y)+Pn(x,y)+Sn+1(x,y)),y˙=y(Rn-1(x,y)+Qn(x,y)+Sn+1(x,y)), \matrix{ {\dot x = x\left( {{R_{n - 1}}\left( {x,y} \right) + {P_n}\left( {x,y} \right) + {S_{n + 1}}\left( {x,y} \right)} \right),} \hfill \cr {\dot y = y\left( {{R_{n - 1}}\left( {x,y} \right) + {Q_n}\left( {x,y} \right) + {S_{n + 1}}\left( {x,y} \right)} \right),} \hfill \cr } where n is a positive integer, R_n−1, P_n, Q_n and S_n+1 are homogeneous polynomials of degree n − 1, n, n and n + 1, respectively. Applications of Kolmogorov systems can be found particularly in modeling population dynamics in biology and ecology.