# Logistic map

### 31 May 2012

I want to describe a little bit of biology-inspired mathematics that will demonstrate how a simple system can lead to complicated results. The following approach is applicable to many different animals, but I will choose to talk about the miner bee.

Miner bees are solitary, and so don’t live in colonies. Their generations are non-overlapping: female miner bees lay their eggs in burrows before dying; the next year the new bees emerge from underground, fly around, mate, laying their own eggs, and die. And so the cycle repeats.

Now for some maths. Say we count the number of miner bees in Britain in 2012 but that this is costly, and we can’t afford to do it again in 2013. How do we estimate how many bees there will be? Well we can assume that the number of bees in 2013 will have some relation to the number of bees in 2012. For example, we can estimate the number of bees in 2013 as the 2012 count multiplied by some number (a linear map):

We don’t have to stop there. We can use the same relation to calculate the 2014 estimate given the 2013 estimate, the 2015 estimate given the 2014 estimate, and so on. Now the value of $r$ is critical. If $r$ is too small ($% $) then the number of miner bees will always decrease, and eventually we will get an extinction event. Conversely, if $r$ is too large ($r>1$) then the number of bees will explode (we’ll be swarmed with bees).

An alternative assumption about how the miner bee population varies year-by-year could be to assume the logistic map:

Here we have the previous linear map multiplied by another term ($1 - X_t$) which models the concept of carrying capacity, the idea that only so many miner bees can happily live with the food and space available.

What happens this time as we look into the future? Again this depends on the value of $r$. If $r$ is too small we get another extinction event. For larger $r$ there is now an optimum bee population size that will persist for all time, and no matter the starting number of bees, we will always tend towards this optimum population size.

As we increase $r$ further we see a curious phenomenon—a bifurcation—when a small quantitative change makes the system display a large qualitative change. In this case, the optimum population size is replaced by a optimum population size that oscillates every 2 years! So the even years will have one optimum, and the odd years another. By increasing $r$ further, we get additional bifurcations, to cause oscillations to happen every 4 years, then every 8 years, and so on. In fact, we can find values of $r$ that will give oscillations of any number of years: e.g. 20, 59, 1254.

Then there are also values of $r$ that give chaotic results. Here I mean chaos in a very specific sense: the number of bees predicted for each year will look random, but it’s not because we know the rule that generates these predictions! This also means that any small error in reported in the bee census in 2012 will magnify up to large differences in subsequent years.

Chaos is something I find really interesting. There are suggestions that our brains are chaotic, and seizures happen when they lose that chaos. It’s also where the term ‘the butterfly effect’ originated, but it’s another sort of insect and so I’ll stop here.