Chaos in Logistic Models Transition to chaos in a family of logistic growth sequences. Presented in Elementary Math Models class, April 21, 2015

Fish Pop Model Let an = population in 100,000’s Diff eqn an+1 = m(1 – an) an Look at various combinations of m and a0 Note that L = 1 so mL = m

m = 3.2 a0 = .7 and .4

m = 3.4 a0 = .7 and .4

m = 3.5 a0 = .7 and .4

m = 3.8295 a0 = .9000 and .9001

New Kind of Graph Transition to Chaos Compare long term behavior of models differing only in the value of m Horizontal axis shows values of m Compress sequence graph to a single vertical line, located above the value of m Discard first 100 terms, plot next 500, so that only long term behavior is shown First example: m = 3.2

Bifurcation Diagram Detailed survey of results for 0 ≤ m ≤ 4 Shows long term behavior for ALL logistic growth models in one graph Shows transition to chaos Shows surprising visual patterns in the chaotic behaviors A new kind of order – not in the evolution of any one model, but in relations among many different versions of the model

Closeup on Chaos Expanded view for for 3.5 ≤ m ≤ 4

High Resolution Image Expanded view for for 2.8 ≤ m ≤ 4 by Robert Devaney, Boston University