Phase portraits of a family of Kolmogorov systems depending on six parameters
Consider a general $3$-dimensional Lotka-Volterra system with a rationalfirst integral of degree two of the form $H=x^i y^j z^k$. The restriction ofthis Lotka-Volterra system to each surface $H(x,y,z)=h$ varying $h\in\mathbb{R}$ provide Kolmogorov systems. With the additional assumption thatthey have a Darboux invariant of the form $x^\ell y^m e^{st}$ they reduce tothe Kolmogorov systems \begin{equation*} \begin{split} \dot{x}&=x \left( a_0-\mu (c_1 x + c_2 z^2 + c_3 z)\right),\\ \dot{z}&=z\left( c_0+ c_1 x + c_2 z^2 +c_3 z\right). \end{split} \end{equation*} In this paper we classify the phaseportraits in the Poincar\'e disc of all these Kolmogorov systems which dependon six parameters.
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