A generalized three-level Lotka-Volterra food chain — grass G, rabbits R, foxes F — integrated with RK4 at Δt=0.01: G˙=rGG(1−G/K)−aGRGR,
R˙=eGRaGRGR−aRFRF−dRR,
F˙=eRFaRFRF−dFF,
with rG=1.0, K=1.0, aGR=1.2, eGR=0.7, aRF=1.0, eRF=0.5, dR=0.4 and dF scrubbed live via mouse-Y in [0.18,0.55]. The left panel scrolls the three populations through time; the right panel renders the trajectory in (G,R,F) space via an oblique 2-D projection (x,y)=(G+0.45F,R+0.30F), where you can see the system spiral toward a stable focus, a limit cycle, or — if dF is pushed high enough that foxes can't survive — collapse onto the two-species (G,R) sub-attractor. The headline is the trophic cascade: click anywhere to instantly halve the fox population and watch the wave propagate downward. Released from predation, rabbits boom; the rabbit boom flattens grass; only once foxes have rebuilt on the rabbit surplus does the chain settle back to its attractor — a textbook example of top-down control reaching all the way to the producers.