Nonlinear Systems & Linearization · Superchallenge problem

The Rosenzweig–MacArthur predator–prey model is $x' = x\left(1 - \dfrac{x}{K}\right) - \dfrac{xy}{1 + x},\; y' = y\left(\dfrac{x}{1 + x} - \dfrac13\right)$ with carrying capacity $K > \tfrac12$. Its coexistence equilibrium is stable for small $K$ and loses stability through a Hopf bifurcation as $K$ grows (the "paradox of enrichment"). Find the critical value of $K$.