# Chaos theory and population cycles

I have just begun to read about Chaos theory and have come across the statement that "Period three implies chaos."

My question: Does any odd period imply chaos or only 3? If so, how can populations of cicadas that cycle every 17 years exist?

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Can you give a bit more background? What period 3? –  Chris Feb 12 at 18:20
@chris Period 3 means a cycle of 3 years. –  biogirl Feb 12 at 18:23
Ahh, now. Lifecycles based in the number 3 overlap with several others - 6, 9, 12, yearly, and so on. If the lifecycle is only 17 years, then a parasite needs to have exactly the same lifecycle (its a primer number so there are no other fractions) to meet its host. –  Chris Feb 12 at 18:25
@Chris Ummm..I have read that before and I am not trying to ask the reason why 17 year cycle are beneficial, i am asking how can they EXIST in the first place ? –  biogirl Feb 12 at 18:27
@Chris It will become clearer to you what exactly am asking after reading this –  biogirl Feb 12 at 18:28

The chaotic behaviour you are referring to (at least the one described in your link in the comments) is a property of the discrete version of the logistic equation, where you get chaotic dynamics at growth rates above ~3.55 (see the logistic map). The behaviour of this equation has been described in a classic paper by Robert May (1976). As you increase growth rate (a in your link) from one you go from a stable attractor (approached directly or by dampened occilations) to cycling behaviour (between 2 to 4 to 8 states as growth rate increase) to chaotic dynamics, which is shown in this bifurcation diagram.

And to be clear, nothing of this has anything to do with period/cycle length per se (in the sense of species interactions or lagging feedback loops), but is a property of the model and the population growth rate. The statement about period 3 implying chaos probably refers to a brief window around r ~3.83 ($1+\sqrt{8})$ where you get cycling between three values, and at higher values then this you only find chaotic behaviour (which you get at lower values as well though).

Addition: I now realize that the statement "Period three implies chaos" comes from the theoretical paper with the same name (Li & Yorke, 1975). This paper proves that all one-dimensional models (not only the equation above) that has a period 3 cycle will also show chaotic behaviour. This proof is a special case of the Sharkovsky's theorem, which is older but was unknown to Li & Yorke at the time.

There is also a question at MathSE that deals with the same problem, and some of the answers there link to useful resourses:
http://math.stackexchange.com/questions/2901/period-of-3-implies-chaos

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