Integrability and Non-Existence of Periodic Orbits for a Class of Kolmogorov Systems (original) (raw)

Tatra Mountains Mathematical Publications

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 ( R n - 1 ( x , y ) + P n ( x , y ) + S n + 1 ( x , y ) ) , y ˙ = y ( R n - 1 ( x , y ) + Q n ( x , y ) + S n + 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, Rn−1 , Pn , Qn and Sn +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.