Round and Round, and What Holds It There
Last timeDrawing the States Instead of the Clock
The valley neither settles nor runs away. It circles, because a particular combination of the two counts never changes, which confines the state to a closed curve it cannot leave.
Two lessons ago the valley was nudged off its resting state and the displacement
began to rotate. The slope test could not say where it ends up, and the three
possibilities are genuinely different: it spirals in and settles, it spirals out and
one population dies, or it closes into a loop and repeats forever.
This lesson settles it, and the method is the one at the end of the last lesson.
The quantity the rule may not change
Divide the fox rate by the rabbit rate. The step of time cancels, leaving a relation
between the two counts with no clock in it at all, and that relation can be
accumulated using the previous course. What comes out is a combination of the two
counts whose value never moves.
- the rabbit count, weighted by how efficiently a caught rabbit becomes a new fox
- the logarithm of the rabbit count, weighted by the fox death rate. The logarithm is what the accumulation of a rate proportional to the amount produces, which is why it appears here
- the fox count, weighted by how fast foxes catch rabbits
- the logarithm of the fox count, weighted by the rabbit breeding rate. Each population is weighted by the other's parameters throughout, which is the same crossover seen in the resting state
The path, drawn
Here is the valley in the state picture, with the preserved quantity used as the
check. If the arithmetic had drifted off the curve this figure would not exist.
The lesson stops here
3 more paragraphs to go
You have read the opening. The rest of the argument, the problems that check whether it landed, and the lines worth keeping at the end all come with a plan.
The first lesson of every course in the library reads the whole way through, free, so you can see exactly what the rest of them are.
See the planThe contentsThis is the reading half
Starting the course gives you your own copy of it. Every idea on every page has problems standing under it, marked with a reason rather than a tick, and any sentence you do not believe can be opened and argued with. None of that can happen on a page nobody owns.
The contents