One curve
A predator holds its prey at some density. Below a second, lower density the prey cannot come back, because there are too few of them left to find each other. The gap between those two numbers is the only room the population has.
Shrink the habitat and the first number climbs toward the second. The room closes.
You can take a lot of habitat and nothing happens, so long as you take it slowly. The animals follow the shrinking world down and settle at a new, smaller, entirely stable number. Take the same habitat faster and they all die. Same amount gone, same endpoint, and nothing was crossed on the way there. The only difference was the speed.
So there is a fastest survivable rate. Where it sits depends on the predator, on the prey’s own threshold, on how much habitat is going and on how much the population had to spare when you started. In a model with four parameters that is a surface in four dimensions. In a real estuary nobody knows how many dimensions it has.
Once the room is nearly closed, all of that goes away. Every one of those systems becomes the same system.
Measure the habitat you are taking in units of the room, and measure the rate in units of the room squared, and the four-dimensional surface flattens into a single curve with nothing in it. No species. No predator. One shape, and not a single parameter left anywhere in it.
Which means the room was never one input among several. It is the unit everything else is quoted in.
The squaring is the part that bites. The room enters the rate twice, so halving it does not halve what the population can take. It quarters it.
And there is a ceiling. Whatever you do to the habitat, however gently you arrange it, a population with room m cannot survive a rate above about a quarter of m squared. That is not the answer for some particular amount of habitat loss; it is the best the population can do across every amount, the top of the whole curve. Past it the geometry that was doing the saving stops existing. Two special points in the system that were holding the safe trajectories apart run into each other and vanish, and after that there is nothing to be on the right side of.
The animals do not get a warning when that happens, and neither does anyone watching. Nothing crosses. Nothing is exceeded. One year the population tracks the shrinking estuary down the way it has for forty years, and the next it does the same thing a little faster and ends at zero.
One curve is cheap, though. If the shape is the same for everything then you do not have to study everything. Measure it properly once, in whatever system you can actually watch closely, and you have the shape for all of them. Per species you then need exactly one number.
The room.
Which nobody has. You can see a marsh shrink from orbit and you cannot see from orbit how much slack the animals in it had, because the slack is a difference between two densities and one of them is a density the population has never been at. It is the distance to a floor nobody has stood on.
This is one deliberately small model, two species and a dial, written down by three people in October and checked by me against my own numbers. The collapse to one curve holds in the corner where the room is nearly shut, which is the corner that matters and is still a corner. I built the collapse out of one set of equations and the curve out of a completely separate set, then used the first to predict two numbers the second had not reached yet; both came out exact, as fractions, which is the strongest thing I can say and is not a proof. Nobody has run it on a real estuary. The arithmetic says the penalty for a thinning margin is quadratic and that there is a hard ceiling with a mechanism behind it, and that is where it stands.
