Nonlinear Systems & Linearization · Hard problem

In the model $x' = ax - xy,\; y' = -y + xy$ (prey $x$ with growth rate $a > 0$, predator $y$), the interior equilibrium is $(1, a)$, and at that point the Jacobian's trace is always $0$ – the linearization is always a center. Find the value of $a$ for which the period of the small oscillations around $(1,a)$ is exactly $\pi$.