Graze
A meadow of grass and the animals that eat it, solved live as the equations ecology has used for the pair since 1963. The ground is patchy: where it is poor the meadow is calm, and where it is rich the grazers and the grass chase each other in waves. Hold the pointer on the ground to feed it. Feed a calm patch enough and it breaks into rings; let go and it heals in half a minute. The meadow 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 fields are written to this browser so a later visit continues the same computation, step for step. A generation is the time unit of the equations, the grass's own growth time; five of them pass each watched second. The threshold is the carrying capacity above which the coexistence of grass and grazers stops being a steady state and becomes a cycle. The ground's capacity is a fixed patchy field plus whatever the hand has added and time has not yet taken back.
Two 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, because an animal can only eat so fast. Grazers grow by what they eat and die at a fixed rate. Both spread by diffusion, at the same rate, across a meadow with a fence around it. That is the Rosenzweig-MacArthur system of 1963, written here in the dimensionless form the plankton modellers use, with the half-saturation at 0.4, the conversion at 2, the mortality at 0.6, and stepped by Heun's method at a tenth of a generation. Nothing is drawn; the picture is the grass field read as light, with the herd as what eats the light.
The system has one theorem everyone learns, and it is the gesture of this page. Where grass and grazers coexist, the grazers hold the grass at a fixed level, u* = 0.1714 here, whatever the ground can carry. Make the ground richer and the grass does not rise; the grazers do, and past a threshold the pair stops settling and starts to cycle: a bloom of grass, a bloom of grazers, a crash of grass, a crash of grazers, and again. Rosenzweig named it the paradox of enrichment in 1971. The threshold has a closed form, twice the held level plus the half-saturation, 0.7429, and the ground on this page straddles it: roughly two fifths of it is poorer than that and calm, the rest is richer and cycling. Holding the pointer feeds the ground beneath it, raising its capacity by up to 0.9 with a three-generation rise and a hundred-and-twenty-generation decay. Feed a calm patch and it brightens, because the grass climbs toward the new capacity, and then it breaks into rings, because the grazers follow and overshoot. You cannot feed a meadow into peace.
In space the cycles travel. Sherratt, Lewis and Fowler showed in 1995 that a grazer invasion into cycling ground leaves periodic waves in its wake, and Medvinsky and his colleagues showed in 2002 that the waves roll into spirals and the spirals break up far from their cores. The first thing a new visitor sees is that invasion, from one released disc of grazers into a whole meadow of standing grass. The patchiness of the ground is what seeds the spirals; on even ground the wake would be a set of clean rings for a long time.
Seven gate families were written before the code and run against this exact module. Five held on the blind run: the coexistence point is a fixed point to the last bit and an attractor below threshold to a part in a million billion; the oscillation onset in a sweep of capacities brackets the closed-form threshold between 0.740 and 0.745; diffusion on the fenced grid conserves mass to a part in a hundred million million with no overshoot; the invasion front of grass alone runs at 2.057 where the grid's own dispersion relation predicts 2.073 and the continuum predicts 2.000, so this grid runs its fronts three percent fast and says so; and the saved meadow continues bit for bit. Two did not hold as written. The fertilizer missed its closed form by a part in ten million because its field had been stored in single precision to spare the browser's storage; the field is now double precision and the law is met to a part in ten trillion. And the wake gate looked for oscillation at a point four hundred lengths from the release between generations 450 and 650, and found the meadow sitting on its unstable equilibrium to a millionth. It was not wrong to exist; it was early. The invasion leaves a band of calm behind it whose rear moves slower than the front, 1.52 against 1.87 here, and the oscillation reaches that point only when the rear does, at generation 967, after 657 generations of a stillness that the algebra says cannot last and that in the moving frame lasts as long as you like. The rig was rewritten with that prediction on record and held. Both amendments are in the note with their reasons.
what it cannot keep
The meadow cannot be made calm by giving it more. Every gift raises the crest of the next wave, and the only ground on this page that holds still is the ground that was never rich enough to be interesting. What the hand adds, the ground gives back over half a minute, and the rings it started keep travelling long after the reason for them has gone. The grazers will not let the grass stand higher than the level they hold it at. More food does not mean more grass. It means more of them, and then fewer, and then more.