Stalk
A predator let into the meadow of Graze. Grass, the animals that eat it, and the animals that eat those, solved live as the three-level food chain Hastings and Powell wrote down in 1991, on Graze's patchy ground raised by 0.3, for a reason given below. Without predators all of this ground cycles. Hold the pointer to release predators beneath it: they hold the grazers down and the grass stands tall behind them, and where the ground is rich enough the meadow never repeats itself. It grows only while it is watched, and it is kept between visits.
what the meadow can say about itself
Watched time is the only time this meadow has: the integrator advances while the tab is visible and stops when it is not, and the three fields are written to this browser's database so a later visit continues the same computation, step for step. A generation is the grass's own growth time; five of them pass each watched second. Grass stands where it is more than three times the level the grazers alone would hold it at. The Lyapunov exponent is not measured live; it is the number the gate board measured for these parameters at unit capacity, printed so the word chaos has a figure beside it. The chaos band, 0.95 to 1.15, is where a single cell of the chain came out chaotic in the same sweep; it is a measurement on this integrator, not a theorem.
Three fields on one ground. Grass grows toward the capacity of the ground under it and is eaten by grazers at a rate that saturates when grass is plentiful, the Holling type II response. Grazers grow by what they eat, die at a fixed rate, and are eaten in turn by predators at a rate that saturates the same way. Predators grow by what they eat and die slowly, forty times more slowly than the grazers, which is the slow clock of the whole piece. That is the Hastings-Powell chain of 1991 in the authors' own dimensionless form, with their parameters (a1 = 5, b1 = 3, a2 = 0.1, b2 = 2, d1 = 0.4, d2 = 0.01), the grass's capacity replaced by Graze's patchy field raised by 0.3, and all three spreading by diffusion across a fenced meadow, the predators twice as fast as the rest, which is a choice and not the paper's, since the paper has no space. Heun's method at a tenth of a growth time. Nothing is drawn; the picture is the grass read as light, the herd as what eats the light, and the predators as the few dark marks whose count follows their density.
The first theorem is the one the gesture performs. Where all three coexist, the predators fix the grazers at a level set by the predators' own bookkeeping, 0.125 here, whatever the ground can carry, exactly as the grazers in Graze fixed the grass. Freed of the grazers, the grass stands: at unit capacity the two-level chain holds the grass at 0.105 and the three-level chain at 0.819, and measured over a hundred thousand growth times of the moving system the means are 0.204 without predators and 0.745 with. Hairston, Smith and Slobodkin argued in 1960 that this is why the world is green; Paine called such a predator a keystone in 1966; the picture everyone carries is the wolves returned to Yellowstone in 1995 and the willows coming back along the rivers, which is used here as a picture, since the field evidence for that particular cascade is still argued over. Release predators into the waves and a still, bright ground spreads behind them at the speed the equations set, a fifth of a cell per growth time on calm ground.
The second theorem is what happens inside the bright ground. Hastings and Powell found that at these parameters the chain has a chaotic attractor, the teacup: fast grass-grazer oscillations riding a slow predator tide, never repeating. On this page that is not a mood but a number. The gate board computed the largest Lyapunov exponent on the module's own integrator by the Benettin method, fifty thousand growth times after a five-thousand transient: 0.0069 per growth time at b1 = 3.0, positive in both halves of the run, and −0.0167 at the control b1 = 2.0, where the chain settles to a point. Swept from 1.8 to 3.2, the exponent crosses zero between 2.2 and 2.4 and dips back to zero near 2.6, one of the periodic windows in the paper's own bifurcation diagram. The successive maxima of the predator density, 2,117 of them in a hundred thousand growth times, spread over 8.03 to 10.46 with a coefficient of variation of 0.10; at the control there are none. A trajectory that never repeats is drawn here as ground that never settles: the tide of predators rises and falls over a minute or two of watching, the waves of grass and grazers under it over seconds.
The ground is not Graze's exactly, and the reason is the piece's own finding. The first run on Graze's ground, which carries between 0.55 and 1.05, watched the predators cover the meadow and then watched the meadow stop: by generation sixteen hundred the grass sat at 0.62 across the field, varying from place to place by a tenth and from moment to moment not at all. A single cell of the chain, swept against the capacity of its ground, explains it: at 0.7 the chain settles to a point, at 0.8 and 0.9 it cycles small, at 1.0 and 1.1 it is chaotic, and from 1.2 upward it cycles large. Most of Graze's ground lies where the three-level chain settles, and diffusion pulls the rest along. So the ground here runs from 0.85 to 1.35: the poorest patches cycle small, thirty-eight percent lies in the chaotic band, the richest cycle large, and the mature meadow is bright ground crossed by dark travelling loops rather than a still plain. It is Rosenzweig's paradox at the third level. The ground that fed the grazers into waves is, once the predators arrive, ground on which everything comes to rest; only richer ground keeps the chain moving.
Seven gate families were written before the code and run against this exact module. All seven held on the blind run: the interior equilibrium is a fixed point of the discrete map to the last bit and repels, with eigenvalues −0.611 and 0.039 ± 0.075i; the chaos gate and the maxima gate above; the cascade; diffusion on the fenced grid conserves each of the three fields to a part in a hundred million million with no overshoot, at the largest predator diffusivity the explicit step keeps positive; the release matches its closed form to the last bit, and the predator front runs at 0.1892 cells per growth time where the Fisher formula gives 0.1934 and the Bramson correction for a front this young gives 0.1865; and the saved meadow continues bit for bit through the browser's store. The store is new. Graze kept its fields as text in a drawer that holds about five megabytes, and a third field would not have fit, so both pieces now keep their fields as numbers in the browser's database, and Graze moved first.
what it cannot keep
The predators cannot be taken back. What the hand lets out lives, hunts and spreads at the pace the equations give it, and the ground it makes still is not calm; it is the slow face of a motion that never repeats. The grazers will not let the grass stand, the predators will not let the grazers keep it down, and none of the three can hold a level for long. More predators do not mean more peace. They mean a longer wave.