Wednesday, December 2, 2009

a curious population model

Dave Richeson of Division by Zero has recently posted a very nice GeoGebra applet for the interesting discrete logistic equation.

\[p(n + 1) = Mp(n)(1 - p(n))\]
The discrete logistic equation is a simple model of population growth within a closed environment. p(n) is a decimal between 0 and 1 representing the fraction of the maximum population that has been reached (initial seed value of p(0) = 0.001), and M is the "Malthus factor" a multiplier that represents the fertility of the population (value between 1 and 4). Different values of M determine whether the population dies out, achieves a stable level, or fluctuates.

This simple discrete dynamical system is also easy to implement and explore in Fathom. You only need a slider (for the M value) and a couple attributes (you may want to add others as you explore).

1. In a new Fathom document, create a slider M
2. Add a new Collection, and create an attribute generation, and an attribute population.
3. Provide this formula for generation: caseIndex - 1
4. Provide this formula for population: if (generation=1){0.001, M*prev(population)*(1-prev(population))
5. Add cases in to represent the generations of the population (say 20 or so)
6. Create a graph with population as the y attribute and generations as the x attribute
7. Explore how the graph changes as M varies between 0 and 4.

An example Fathom file is here.

When the population is stable, you have a nice curve like one below.

When the Malthus factor is higher, the population becomes less stable - overcrowding and then dying back.

This population model is mentioned in Mark Haddon's, The Curious Incident of the Dog in the Night-time, whose protagonist is a young mathematician with some behavioural difficulties. See the MAA online review of the book here, which includes pointers to other places to learn about the logistic equation.