One-parameter family of cubic Kolmogorov systems with an isochronous center (Q1924834)

We show that the one real parameter family of the cubic Kolmogorov system \[ \begin{aligned} \dot x &= x (1- a- 2x (1-a)- a(x^2- y^2)),\\ \dot y &= y(- (1- a)+ 2y (1- a)- a(x^2- y^2))\end{aligned} \] has, for \(a> 1\), an isochronous center of period \({{2\pi} \over {\sqrt {a-1}}}\) at the point \((1/2, 1/2)\).