Fig tree diagram

For systems that can exhibit periodic and chaotic behavior, it is interesting to know how a transition to chaos is heralded (e.g. periodic heartbeat - chaotic ventricular fibrillation). A period doubling (bifurcation) can often be observed before the transition to chaos. This phenomenon is particularly visible in the so-called fig tree diagram.

The development of the grasshopper population can be described approximately using the logistic equation. This assumes that the number of offspring is proportional to the number of grasshoppers present. However, if the food supply is fixed, a proportion of the grasshoppers die before laying their eggs due to a lack of food. The lack of food is in turn proportional to the current population. The resulting equation can be written as x(n+1) = A x(n) * ( 1 - x(n) ) with only one parameter A. The number x(n) represents the relative size of the population in year n. The fig tree diagram describes the development of the population as a function of the parameter A. For small A, an equilibrium population is reached after a few years (one line in the diagram). For larger A, period doubling occurs and the population fluctuates between two values or even 4, 8, 16, etc. different population sizes (branches of the lines). For even larger values, chaotic behavior occurs (right-hand side of the diagram).