Affiliation:
1. Department of Mathematics, Faculty of Science , Soran University , Soran, Kurdistan region - Iraq
2. Department of Mathematics , University Mohamed El Bachir El Ibrahimi , Bordj Bou Arreridj , El-anasser , Algeria
3. LMIB — School of Mathematical Sciences , Beihang University , Beijing , China
Abstract
Abstract
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.