Re: [Phys-L] Rodent Population Model Behavior.

[Paul Nord <paul.nord@valpo.edu>]

Predator-prey models aren't the only ones that produce these Feigenbaum
diagrams.  (Bringing this discussion back to physics...)

Dripping faucets do something similar as a function of flow rate:
http://fy.chalmers.se/~f7xiz/TIF081C/drippingfaucet.pdf

We had some students set this up and get frustrated when they adjusted
things and reported, "It drips at a constant rate and then, sometimes, if
we bump the table like this [bump] it starts dripping at this other rate."
The bifurcation in the Feigenbaum diagram is the first indication of the
onset of chaos.  I think that their apparatus didn't allow for much higher
flow rates and they never got into the really chaotic regions.

Paul

On Wed, Apr 18, 2018 at 10:28 PM, brian whatcott <betwys1@sbcglobal.net>
wrote:

> I looked over a used book sale at the local library last week.  I chose
> just one book: "A Mathematician Reads The Newspaper" (BasicBooks 1995)
>
> My interest was captured by a little piece on a population dynamics
> model explored by May & Feigenbaum. The iterative model they used was
> indeed simple - a logistic curve generator: X' = R * X ( 1 - X )
>
> X' next year's normalized population.
>
> X this year's normalized population [0..1]
>
> R is a parameter [0..4]
>
> Examples: for X = 0.1 and R = 1.5 The population stabilizes at 0.333
> after some years. This steady state population is invariant to the
> starting X value.
>
> When R  = 3.2 the population alternates between two values:
>
>   when R = 3.5 the population steps between four values;
>
> Slightly larger R values continue to double the number of population
> states until R   = 3.57 when the population size becomes chaotic.
>
> https://imgur.com/a/63jox
>
>
> Enjoy!
>
> Brian W
>
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