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 $\sqrt{K - K_H}$ .

Figure 1. Rosenzweig-MacArthur model with Hopf bifurcation. Method: RK4 with analytic equilibrium markers.

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WHAT TO TRY

Raise the prey carrying capacity K: the stable coexistence point loses stability through a Hopf bifurcation, and the populations break into a growing limit cycle. Enriching the environment destabilizes it, the paradox of enrichment.
Watch the phase portrait and the time series together: below the bifurcation the spiral winds into the fixed point, above it the spiral winds out onto a closed predator-prey cycle.
Lower K back down: the limit cycle shrinks and collapses back onto the stable fixed point, the populations settling instead of oscillating.