INTERACTIVE MODEL · VOLUME 1, ISSUE 1 · POPULATIONS AND CHANCE

The Extinction Gambler

A live model accompanying “A Population That Grows on Average Can Still Die Out”

Set a mean offspring number above 1 and the population grows on average. Below you can watch what that average is worth to a few hundred small populations at once. Every line is one population, every step is one generation, and a line that touches the floor is gone for good.

Model 1: three hundred populations, one setting

Pick the average number of offspring per individual, pick how variable reproduction is, pick how many individuals you start with, then press run. The counter underneath keeps the tally of how many populations have died, with a 95 per cent interval, and compares it with the exact answer from the branching-process theory.

Mean offspring 1.15 Reproduction Starting size 20 Populations run 300 Generations 100

offspring variance: – generation: 0 extinct: 0 measured: – exact: –

Press run. Lines are drawn on a logarithmic scale, so a straight rising line is steady exponential growth.

Two things are worth doing here. Set the mean to 1.15 with geometric reproduction and a starting size of 5, and watch how ordinary the disaster looks: no crash, just a run of unremarkable generations that happen to go the wrong way. Then raise the starting size to 22 and run it again. The mean growth has not changed at all.

Model 2: where the risk drops below five per cent

The second model runs the same experiment across a range of starting sizes and plots the extinction probability against the size. The smooth line under the points is not a fit. It is the exact answer, obtained by iterating the offspring probability generating function, which is what makes this whole subject checkable: the simulation has something true to be compared against.

Mean offspring 1.15 Horizon 100 generations Replicates per point 400

progress: idle threshold N* for 5 per cent: Poisson – geometric – lottery –

The thresholds update the moment you move the mean, because they come from the exact calculation rather than from the simulation.

What this model is not

There is no carrying capacity here, so a population that escapes grows for ever, and we stop drawing it at 4,000 individuals. There is no environmental variation: no good years, no bad ones, no droughts. Individuals never interact, so nothing in this model knows about mates, crowding or food. Removing all of that is deliberate, because it isolates the one effect we wanted to measure, and it is also the reason these thresholds are far below the minimum viable population sizes quoted in conservation biology.

Two details of the arithmetic, for anyone reading the source. Totals for populations of more than 40 individuals are drawn from the exact compound distribution (a Poisson mixed over a gamma for the negative binomial cases, a binomial for the lottery), and Poisson draws with a mean above 40 use a normal approximation, which cannot affect extinction because extinction only ever happens from small numbers. The random number generator is seeded, so the same settings give the same picture until you press new seed.