Chaos on Purpose

Occupy Math enjoys the show The West Wing. One of the best episodes is 20 Hours in America. Near the end of the episode, the deputy chief of staff is talking about chaos theory and says:

But it has to do with there being order, and even — great beauty in what looks like total chaos.

This week’s Occupy Math looks at a particular type of chaos, logistic chaos.

Chaos is often confused with randomness. Chaos is often not random at all.

We will work up to chaos a bit gradually, through models. The first mathematical model that students encounter in school is exponential growth – the behavior you get when the rate at which a population is growing is proportional to the current size of the population. The problem with exponential growth is that it’s only correct very temporarily. An elegant mathematical proof of this fact is that the entire mass of the universe has not been converted to bacteria yet. Exponential growth always hits limits. The alcohol in beer is, from the point of view of the yeast that made it, a toxic by-product that constitutes a serious limit on their exponential growth.

It is easy to make a better model. Set the rate of growth to be proportional to the current population times the difference between the maximum capacity of the environment and the current population. That way, as you approach the maximum, the growth rate drops smoothly to zero. The new model includes a term for carrying capacity – the maximum population the environment can support. This new model is called logistic growth. Let’s graph exponential and logistic growth curves and compare:

The exponential model shoots off toward infinity – the logistic model grows to meet the carrying capacity.

But, I hear you cry, where is the chaos in all this? The logistic growth model in the graph above has a hidden assumption: that the new population members show up a few at a time. For people or bacteria this isn’t a bad assumption. Salmon and mayflies, however, have a very specific narrow breeding season – which lets them shoot past the carrying capacity. The original logistic model uses continuous mathematics while species with a breeding season use discrete mathematics. The discrete logistic model is still that next year’s population is the current population times the growth rate times the fraction of the carrying capacity currently occupied. The problem is what happens for large growth constants. Occupy Math has tabulated the population over time for several different growth constants. The actual population size is printed along the bottom of each diagram. They all start at 200 out of a maximum possible of 1000.


With the growth constant at 2.9, the population settles to a single value.

When we increase the growth rate to 3.1, the population settles to jumping between two values. The same thing happens for a growth rate of 3.3, except the jumps are bigger.

A growth rate of 3.61 causes a good deal of hopping around. Remember – no random numbers are involved, these are crisp, deterministic computations.

The next step is to diagram how the jumping around happens. The diagram below shows the population levels that are possible for each growth rate from 2.5 to 4.0. Very much to the surprise of the people that discovered it, this is a really complicated diagram. Let’s take a look.

For growth rates below 3 the dynamics settle down to one population level. From 3 to a little below 3.5 the populations jump between two values – we saw this in the plots for 3,1 and 3.3 above. After the area with two population levels there is a double split to four levels and then – very shortly – another split to 8. The distance between splits is decaying – in fact decaying so fast that an infinite number of splits happen up to the vertical white band (well, mostly white) where the model starts jumping between six values. Occupy Math calls this the chaos wall. There is even more weird – it is hard to see without a building-sized diagram, but there are an infinite number of these chaos walls before we get to a growth rate of four. The chaos of the logistic model appears for the values of the growth rate where the hopping around of the population gets out of control (becomes unpredictable in practical terms).

Every single value of the growth rate gives a population that settles into a predictable, repeating pattern – but those patterns can cycle through thousands or millions of population level before they repeat. Not randomness, but huge chaos. Contrast this with the simple continuous logistic model up near the top of this post. Having a breeding season can make population dynamics incredibly more complex. The great statistician and mathematician George E. P. Box observed that:

All mathematical models are wrong. Some are useful.

This means that all models simplify things, but some oversimplify. Where does this land in the real world? Countries regulate their fish catch, and fish often have breeding seasons. There are a number of cases where a country set the catch for a species based on the continuous logistic model when the discrete one would have been appropriate – and managed to set the catch for next year to a number larger than the actual population. The fish involved were almost wiped out. This suggests that the continuous logistic model was not useful for this particular type of fishing policy.

In an earlier post, Things of Prime Importance, Occupy Math showed how cicadas make themselves bad prey animals by having a life cycle with a long, prime-number length. The irregularity of the population cycles that arise from the logistic model are another way to be bad prey – an undependable food source. Biology does not give a species enough control to pick a growth rate right on top of a chaos wall – but it can pick a growth rate in a region with a lot of jumping around.

When Occupy Math was thinking of doing a post on chaos the natural target is the butterfly effect, the most famous example of chaos, but logistic chaos is easier to explain. Chaos shows up in other places. The beating of a healthy heart is chaotic. When your heartbeat becomes regular you are about to have a heart attack. Logistic chaos shows up all over the place as well. The time between drips of a dripping faucet, for example, obeys the rules of logistic chaos. Do you know of another place where chaos appears in the real world? Comment or tweet and let Occupy Math know about it.

I hope to see you here again,
Daniel Ashlock,
University of Guelph,
Department of Mathematics and Statistics

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