Tri-Trophic Stack
Plants, grazers, and predators sharing one landscape. Change how hard the predators hunt, and watch what happens three levels down.
Layer 1 of 2 · the accepted science
The classical picture
Ecologists call a producer, the herbivore that eats it, and the predator that eats the herbivore a three-level food chain. The standard mathematical treatment starts from the Lotka-Volterra predator-prey equations (Lotka, 1925; Volterra, 1926) and stacks one more level on top. With P for plants, H for herbivores and Z for predators, each measured as a density:
Why these terms
- Encounters are products, a P H and b H Z. This is the law of mass action borrowed from chemistry: if animals move around at random, the rate at which grazers meet plants is proportional to how many of each there are. Double either and meals double.
- Plants grow logistically, r P(1 − P/K). Left alone, plants grow at rate r until space or nutrients cap them at a carrying capacity K. Without that cap the bottom of the chain grows without limit, which no real landscape allows.
- Eating is converted at an efficiency, e and c. Only a fraction of what is eaten becomes new consumer. This is where the textbook's loss of energy at each step up enters the equations.
- Consumers die at a constant rate, m and d. Without food a consumer population simply decays.
What the equations predict
Setting all three rates to zero, with every level present, gives the balance point where the three levels coexist. Each line comes from one equation: the predator equation fixes H*, the plant equation then fixes P*, and the herbivore equation then fixes Z*:
The balance point is a real one only if all three values come out positive; if P* or Z* would be negative, the three levels cannot coexist at those rates.
Two textbook results follow directly. First, the herbivore level H* is set entirely by the predator's parameters: its hunting rate b, efficiency c and death rate d. Raise predation and H* falls, and because P* rises as H* falls, the plants go up. That is the trophic cascade written as algebra, the idea behind Hairston, Smith and Slobodkin's 1960 argument that the world is green because predators hold herbivores down. Second, a predator can only establish itself if c b H > d at the herbivore level it finds: when predation is too weak, the top level cannot pay for its own death rate and disappears.
Where the classical equations stop
The equations above assume a well-mixed world in which every animal can meet every other, and they treat populations as smooth quantities that can shrink toward zero forever without reaching it. Real landscapes break both assumptions. Encounters are local, populations are whole numbers, and a population that reaches zero stays there. Ecologists model that second kind of world with stochastic lattice models (in mathematics, interacting particle systems): space is a grid, each site holds one state, and each site changes with a probability that depends on its neighbours. In those models extinction is an absorbing state, a condition the system can enter but never leave, and chance alone can drive a small population into it.
The panel below is a model of that second kind. It has no differential equations inside it, only local probabilistic rules, and the section How this differs from the model in your textbook spells out what that changes.
What you are looking at
A landscape seen from above, divided into patches. Each square is one patch of ground, not one organism. At any moment a patch is in one of four states:
- Green, plants. The producers. They spread into bare ground.
- Gold, grazers. Herbivores. They eat plants.
- Red, predators. They eat grazers.
- Black, bare ground. Nothing living there yet.
The rules
Every patch looks at its neighbours and updates. That is the whole model:
- Bare ground can be colonised by plants from a neighbouring patch.
- A plant next to a grazer can be eaten. The grazer takes over that patch.
- A grazer next to a predator can be eaten. The predator takes over that patch.
- Anything with no food beside it starves, and the patch goes back to bare ground. Grazers need a plant nearby, predators need a grazer nearby.
- Plants also die on their own now and then, which is what stops them from covering the whole landscape.
There are no equations underneath this. Everything you are about to see, including the oscillations, comes from those five rules applied to every patch at once.
Tri-Trophic Stack
Duncan(√5), whose anisotropy 𝒜 is shown above (admissible ≤ 0.02).
Read the counts as cells, not organisms. This is an occupancy model: a predator that eats a herbivore becomes a predator in that cell, so the upper layers accumulate area rather than paying a biomass cost. The readout is not a pyramid of numbers and should not be read as one.
What to notice first
The landscape does not mix. It separates into large single-colour regions: plant country on one side, predator country on the other, bare ground between them.
The grazers are almost never a population. They survive as a thin band along the boundary between the plants and the predators, and that band moves. Watch the gold for a while and you are watching a front, not a population.
That is the biggest difference between what you are seeing here and the standard predator-prey model, and it is worth sitting with before you touch a slider.
Three things to try
1. Push the predators up and watch the plants
Raise predation from 0.20 to 0.55. Leave everything else alone.
What happens: the grazers are cut back hard, and the plants roughly double. Predators up, grazers down, plants up.
A trophic cascade is an effect that travels down through a food web: a change at the top changes the level below it, which changes the level below that.
The predators never touch a plant. They control how much plant there is by controlling the thing that eats it.
2. Take the predators away
Now drag predation down instead: 0.15, then 0.12, then 0.10.
What happens: the grazers irrupt and become the largest group on the landscape, and the plants fall to less than half of what they were. Below about 0.09 the predators cannot hold on and die out. The rest of the landscape does not follow them: the grazers take over, the plants are held down to a small share of the ground, and the two remaining levels carry on without a predator.
A keystone predator is one whose presence holds a whole community together, so that removing it changes far more than just its own numbers.
Removing the predator does not empty the landscape. It changes what kind of landscape it is: plant country gives way to a world of grazers with the plants kept low, and nothing on the board brings the predator back.
3. Sit on the edge and run it again
Set predation to 0.085 and press Reset basin several times.
What happens: sometimes it survives and sometimes it collapses. Same settings, different outcome. The longer you leave a run going, the more often it ends in collapse, because a population this small only has to get unlucky once.
Near the threshold the predator population gets very small, and a very small population can be wiped out by bad luck alone, even when the average conditions would let it persist. That is one of the main reasons conservation biologists care about population size and not only about habitat quality.
It is also a lesson about doing science. At this setting a single run tells you nothing. You have to run it several times before you know anything at all.
How this differs from the model in your textbook
Lotka-Volterra assumes everything is mixed together. One number for predators, one for prey, and everyone can meet everyone. Here nothing is mixed, and what matters is who is next to whom. The oscillations in the strip are fronts sweeping across the landscape, not a well-mixed cycle.
There is no energy budget. In a real food web roughly ninety percent of the energy is lost at each step up, which is why there is far less predator than plant. This model has none of that: holding a patch costs nothing. That is why the three numbers come out closer together than a real pyramid of biomass would, and why the predator count can look surprisingly high.
And the counts are patches, not animals. A predator that eats a grazer becomes a predator in that patch, so the upper levels accumulate area rather than paying for it.
Layer 2 of 2 · the framework's reading
The wrapped reading
Everything below is written for readers working through Principia Attractum. If you came for the ecology, you have already had it.
This panel is a candidate attractor: a three-layer stack meant to sustain itself through its own internal recursion, not through order handed in from outside. The settled three-level state, with its steady mix of plants, grazers and predators, is the candidate attractor: what the system returns to when nudged. Its basin is the set of starting states that lead there. The Perturb button and the sliders are bounded perturbations: they push the run to a different state and test whether that state is still inside the basin. What you read off is the run's own answer: did the run recover, or did a layer rupture and not come back?
What the runs actually showed
Measured against this exact rule set, the panel's behaviour is consistent with attractor behaviour and shows three distinct phases. None of this is a sovereignty result; it is what the run observables report.
Ignition. A run does not begin at its attractor, it arrives there. From a seeded start the layers overshoot hard and ring: herbivores peak near 2.9× their eventual level within the first ten steps, producers near 1.8× by step 25, and the transient is about twice the amplitude of anything that happens afterwards. It is a single event per reset, not something that recurs.
Bootstrapping interval. The ringing decays over roughly the next sixty to a hundred steps, the stretch between heading for the attractor and settling on it. Readings taken inside that window are not readings of the settled state, and the scrolling plot's vertical scale is still settling with it.
Return to the attractor. Once settled, the structure holds its position against being pushed. Culling 60 to 80% of the predators, twenty-four times across three starting worlds, the run returned to the same state every time, within about 1%. That return, not mere survival, is the part that earns the word attractor.
Rupture. The same structure can fail, and the panel says so when it does. A layer that stays at zero long enough is reported as ruptured rather than recovering, and it does happen: with predation lowered to 0.09, eleven worlds in twelve lost their predators within 2,000 steps. (At the default rates, no settled run in the measurements below ruptured unless every predator was removed.) Recovery and rupture are the same observable read in two directions.
What this is not: it is not an attractlet, which by definition produces none of its own order and does not come back when perturbed. It is also not a sandbox sovereign, which is reserved for a structure clearing all four conditions against a declared economy. This panel exercises two of the four; conditions 3 and 4 are out of scope and supplied respectively, as the card below records. An attractor, and not a sovereign one.
The Shape of the Basin
At the default settings (plant growth 0.06, grazing 0.16, predation 0.20, predator death 0.35) every run that gets through ignition settles into one observed population regime: about 1,920 plant cells, 1,225 grazer cells and 1,245 predator cells out of 8,100, with repeated runs landing within about one percent of each other. That is a statement about the counts. Many different arrangements of cells give the same counts, so it does not show that the cell-by-cell dynamics have only one attractor.
This basin is a matter of odds, not a clean line. The rules are probabilistic, so the same starting state can fall home in one run and fall out in the next. The counts below ("11 of 12") are return rates measured over twelve runs, not a list of which states are inside the basin and which are outside. Where a count is 12 of 12 the state returned in every run tried; where it is lower, the state sits in a band where returning and not returning are both possible.
It is wide in starting states. Worlds seeded with predators anywhere from 0.3% to 40% of the board, grazers from 1% to 60%, or plants from 0.5% to 50% all fell home, 12 runs out of 12 at each setting (11 of 12 for grazers at 1%). How a run begins changes its ignition, and made no difference we could measure to where it ends up.
Its walls sit at the extremes, and they are crossed early. Starts fell out only at the edges of that range: predators seeded at 0.1% of the board, grazers at 0.3% or less, or plants at 80%, each losing three runs out of twelve (and grazers at 1%, one run out of twelve). Every one of those failures happened inside the first sixty steps, during the overshoot of ignition. In the 1,000 steps measured, no run that survived ignition fell out later.
The outer edge is written into the rules. No rule creates a plant, grazer or predator from nothing, so a layer that reaches zero never returns. That edge was authored, not discovered by the run, and no setting moves it. A cull of 100% of the predators crosses it by definition, and did so twelve times out of twelve.
The finding is how close to that edge a run can be pushed and still come back. The Perturb button culls about 80% of the predators. Culls of 80%, 90%, 95%, 98% and 99% were each applied to twelve settled worlds, and all sixty returned to the same state. At these settings a remnant of one predator in a hundred was enough to rebuild the layer every time it was tried. The panel's at risk warning, which fires when the smallest layer drops under 2% of the board, marks the approach to that floor (measured over 104 runs, recorded in the source).
The Morph of the Basin
Predation is the dial that moves the attractor and reshapes its basin. Twelve worlds were run for 2,000 steps at each setting:
| predation | held all three layers | settled levels (plant / grazer / predator) |
|---|---|---|
| 0.55 | 12 of 12 | 3,005 / 740 / 1,535 |
| 0.30 | 12 of 12 | 2,424 / 965 / 1,389 |
| 0.20 | 12 of 12 | 1,914 / 1,226 / 1,244 |
| 0.12 | 12 of 12 | 1,186 / 1,707 / 856 |
| 0.10 | 12 of 12 | 868 / 1,936 / 521 |
| 0.09 | 1 of 12 | predators lost between step 268 and 1,691 |
| 0.085 | 0 of 12 | predators lost between step 71 and 530 |
| 0.05 | 0 of 12 | predators lost by step 54 |
The home drifts first. Lowering predation slides the settled point toward grazers and away from plants, the direction the classical equilibrium H* = d/(c b) predicts, while the predator layer thins.
Then the basin shrinks. The margin that let a run come back from 99% removal narrows. A 99% cull that no settled world failed to recover from at predation 0.20 ruptured one world in twelve at 0.12 and six in twelve at 0.10, where a 98% cull also ruptured four in twelve. The worlds that did return at 0.10 came back less precisely, with layer averages settling up to a third away from where they sat before the push rather than a few percent.
Then it ruptures, and a different attractor takes over. Between predation 0.10 and 0.09 the three-layer state stops holding: twelve worlds in twelve held at 0.10, one in twelve at 0.09. That is a transition region, not a measured critical point. It was found with twelve runs per setting on a 90 by 90 board updated in lockstep, over a 2,000-step window, and at 0.09 predators were still being lost as late as step 1,691, so a longer window could move it. The predators go first in every case, 59 collapses out of 59 across the settings from 0.05 to 0.09. This is qualitatively like the classical condition c b H > d failing, where the top layer can no longer pay for its own death rate, though the lattice rule is not those equations and does not compute that quantity. What remains is not an empty board. In fourteen worlds at predation 0.07 and 0.085, followed to step 3,000 after losing their predators, grazers settled as the largest group (about 1,750 to 2,300 cells) with plants held low (about 370 to 750 cells), and none of the fourteen lost a second layer. The system left the basin of the three-level attractor and settled on a two-level one.
Predator death narrows it from the other side. From 0.20 to 0.50 all twelve worlds held at every setting. Below that the predators are strong enough to overshoot during ignition: at 0.10 three worlds in twelve lost their grazers, and at 0.02 four in twelve, all within the first 81 steps. Worlds that got past ignition held.
What the morph can and cannot say here. Every shape described above is held up by supports supplied from outside: plants regrow into bare ground at an authored rate with no inflow, and the clock updates every cell in lockstep. No maintenance cost is represented anywhere in the model. Moving a slider rewrites the rules, and the runs settle on whatever attractor the new rules have. That is the reading the disclosure card below records as condition 4 supplied. Whether a stack can hold its shape while paying a represented cost for it is the question this panel cannot reach and the sibling panel The Modeled Economy is built to ask.
STACK §5.2 · BPP §10.8.1). An authored design claim, not a measurement.
STACK §5.2). A vertical layering, not a metaphor: the predator layer's persistence rides on the herbivore layer, and that on the producer layer.
SOV §7.2), what this
panel supplies, not how it scored- exercised 1 · Recursion lock. The three-layer loop either closes (producers recolonize, herbivores graze, predators crop herbivores) or it does not, and you can watch it fail to close.
- exercised 2 · Internal recurcline persistence. The Perturb control culls about 80% of predators and the run reports recovered or ruptured, which is a basin-recovery capacity in the AC₄/AC₅ sense. Caveat: The recovery runs on temporal order supplied by the global clock at u=1, so it is not shown to be self-produced.
- out of scope 3 · Boundary retention. No boundary is modeled. Cell states are trophic roles, not a membrane, and the lattice wraps.
- supplied 4 · Maintenance-bearing continuation. Nothing pays to persist. Producers colonize empty cells at an authored growth rate with no modeled inflow, so the bottom of the stack is replenished for free.
No sovereignty verdict is claimed here or anywhere in this gallery. The Sovereign column is empty, and that emptiness is the honest reading: no panel here has been shown to produce its own order.