A class of quintic Kolmogorov systems with explicit non-algebraic limit cycle

Various physical, ecological, economic, etc phenomena are governed by planar differential systems. Subsequently, several research studies are interested in the study of limit cycles because of their interest in the understanding of these systems. The aim of this paper is to investigate a class of quintic Kolmogorov systems, namely systems of the form
\begin{equation*} \begin{array}{c} \overset{.}{x}~=x~P_{4}\left( x,y\right),\\ \overset{.}{y}~=y~Q_{4}\left( x,y\right), \end{array} \end{equation*}
where $P_{4}$ and $Q_{4}$ are quartic polynomials. Within this class, our attention is restricted to study the limit cycle in the realistic quadrant $\left \{ (x,y)\in\mathbb{R}^{2};~x>0,~y>0\right \}$. According to the hypothesises, the existence of algebraic or non-algebraic limit cycle is proved. Furthermore, this limit cycle is explicitly given in polar coordinates. Some examples are presented in order to illustrate the applicability of our result.