Nonlinear Systems & Linearization · Challenge problem

The Sel'kov model of glycolysis with $b = \tfrac12$ is $x' = -x + ay + x^2y,\; y' = \tfrac12 - ay - x^2y$, with parameter $a > 0$. It has one equilibrium, whose Jacobian always has positive determinant. Find the value of $a$ at which the trace of the Jacobian is zero (the Hopf bifurcation), in simplest radical form.