<abstract xmlns="http://www.w3.org/1999/xhtml">

In this article, we study the integrability and the non-existence of periodic orbits for the planar Kolmogorov differential systems of the form
<disp-formula>
<alternatives>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_tmmp-2022-0011_eq_001.png"></graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mover accent="true"><mi>x</mi><mo>˙</mo></mover><mo>=</mo><mi>x</mi><mrow><mo>(</mo><mrow><msub><mrow><mi>R</mi></mrow><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msub><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow><mo>)</mo></mrow><mo>+</mo><msub><mrow><mi>P</mi></mrow><mi>n</mi></msub><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow><mo>)</mo></mrow><mo>+</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mo>,</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mover accent="true"><mi>y</mi><mo>˙</mo></mover><mo>=</mo><mi>y</mi><mrow><mo>(</mo><mrow><msub><mrow><mi>R</mi></mrow><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msub><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow><mo>)</mo></mrow><mo>+</mo><msub><mrow><mi>Q</mi></mrow><mi>n</mi></msub><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow><mo>)</mo></mrow><mo>+</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mo>,</mo></mrow></mtd></mtr></mtable></mrow></math>
<tex-math>\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 }</tex-math>
</alternatives>
</disp-formula>
where <italic>n</italic> is a positive integer, <italic>Rn−1</italic>, <italic>Pn</italic>, <italic>Qn</italic> and <italic>Sn</italic>+1 are homogeneous polynomials of degree <italic>n</italic> − 1, <italic>n</italic>, <italic>n</italic> and <italic>n</italic> + 1, respectively. Applications of Kolmogorov systems can be found particularly in modeling population dynamics in biology and ecology.
</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(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, 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.

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

Department of Mathematics, Faculty of Science

Tatra Mountains Mathematical Publications

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

In this article, we study the integrability and the non-existence of periodic orbits for the planar Kolmogorov differential systems of the...

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

Published Online: Nov 29, 2022

Page range: 145 - 154

Received: Jul 18, 2022

DOI: https://doi.org/10.2478/tmmp-2022-0011

Keywords
Kolmogorov differential systems, first integral, invariant algebraic curve, periodic orbit

© 2022 Sarbast Hussein et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

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

Published Online: Nov 29, 2022

Page range: 145 - 154

Received: Jul 18, 2022

DOI: https://doi.org/10.2478/tmmp-2022-0011

KeywordsKolmogorov differential systems, first integral, invariant algebraic curve, periodic orbit

© 2022 Sarbast Hussein et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Keywords
Kolmogorov differential systems, first integral, invariant algebraic curve, periodic orbit