Predator-prey and the Hopf bifurcation

What you are seeing: the Rosenzweig-MacArthur predator-prey model with Holling Type II response: $\dot{x} = r x (1 - x / K) - a x y / (b + x)$, $\dot{y} = e a x y / (b + x) - d y$. Prey ($x$) grow logistically and are consumed by predators; predators ($y$) grow by eating prey and die at rate $d$. Increasing the prey carrying capacity $K$ destabilizes the coexistence equilibrium through a supercritical Hopf bifurcation at $K_H \approx 0.7$. Below $K_H$ the populations damp to a fixed point; above it they oscillate on a stable limit cycle.

Watch the phase portrait $(x, y)$ on the left and the time series on the right. Sliding $K$ from 0.4 to 2.0 sweeps through the Hopf: small $K$ gives damped spirals; large $K$ gives clean limit cycles whose amplitude grows as $\sqrt{K - K_H}$.