The room it had
Drain a pond slowly and the fish population shrinks to fit it. Drain the same pond to the same level, faster, and everything in it dies.
Same destination. Different outcome. Nothing was crossed in between. There is no level at which an alarm could have gone off, because there is nothing there to trip one: the pond that killed its fish and the pond that did not are, at the end, the same pond.
This is called rate-induced tipping, and in October 2026 Hennessy, Popović and Sattar published the smallest system they could find that does it. Prey, a predator that eats only that prey, and a habitat whose capacity to support them declines on a ramp you control. They worked out the speed limit. Below it the populations follow the shrinking habitat down and settle; above it they fall off the moving state they were tracking and both species go to zero.
The speed limit is a formula, and it has a shape, and the shape says something the formula was not written to say.
Start with the one quantity the model cannot do without. Prey below a certain density stop being able to replace themselves, too few of them to find each other, so the population falls even if nothing further goes wrong. Above that density they recover. Call it a.
Now notice where the prey actually sits. The predator grows when prey are plentiful and declines when they are scarce, so there is one prey density at which the predator exactly breaks even, and the pair settle there and stay. The prey is pinned at that number by the thing eating it. In the model that number is 1, and the floor is a, so the whole margin the system has is the gap between them: 1 minus a. That is the room.
The speed limit comes out as a power series in how far the habitat is pushed, a quantity the paper calls delta. They give the first two terms. I worked out the next four.
Every term past the second one divides by that margin. The third divides by it once, the fourth twice, the fifth three times, the sixth four. The room is in the denominator, and it goes deeper every time.
A series whose terms each divide by the same thing one more time is not really a series in delta at all. It is a series in delta divided by the margin. Nobody wrote it that way. It came out that way, four terms running, and the first two terms do not show it because they have no margin in them to show. So the quantity the mathematics is built from is not how far the habitat moved. It is how far the habitat moved against how much room the animals had.
So there is no such thing as a safe rate of change, full stop. Put two species in two identical ponds and dry both at the same speed; the one with more room between the density it is held at and the density it cannot come back from gets through, and the other does not, and the speed each can survive goes up roughly in proportion to that room. The water is not the variable. The slack is.
And the slack is the number you do not have. Anyone can measure how fast a habitat is degrading; there are satellites for it. The density below which a particular species stops being able to find itself a mate is not in any satellite. It is specific to the animal, hard to establish without watching a population go through it, and ethically unpleasant to determine on purpose.
Which leaves the thing in a shape I keep turning over. The speed is easy and nearly useless on its own. The denominator is what decides the answer, and the denominator is the part nobody can see.
