Wednesday, 6 October 2021

Bifurcation on Bright Beck

 Bright Beck runs down into Stickle Tarn from behind Pavey Arc. Its ravine looks like an eroded fault or dyke. Near the top, the ravine splits as pictured by Wainwright in High Raise 4 - a bifurcation.

I haven't looked at bifurcation theory for years but in honour of the Nobel Prize for complexity, here goes. This theory is based on the work of Verhuist, a Belgian mathematician who in the 1840s used negative feedback to stabilise the explosive population growth of Malthus's theory. The equation he came up with is now called the logistic equation. It links the population in one generation with the population in the generation before. If x is the population size, xnext = rx(1-x). In the 1970s, Robert May found that if r=2.6, he got a stable population but above that population oscillation happens, initially between two different population sizes that swap from generation to generation predictably. This change in behaviour from one population size to two is the bifurcation.