What is a mathematical model?

06 Feb 2013

Imagine you’re a biologist. Let’s say you specialise in ecology – how populations of animals interact with their environment.

You’ve decided to start a research project investigating a particular endangered species – let’s say a type of vole. This vole has one major predator – lets say a type of weasel.

Because the voles are endangered, it’s very important to have a good estimate of the number that are alive in the wild. But these voles live in dense rainforest and are very secretive. We have a rough idea of the size and location of their territory.

How might we find out how many there are?

We can’t possibly find every vole. We can do a small experiment – cover part of their territory with traps and count how many voles we see. We can then multiply up this number to represent the whole of their territory.

There are a number of issues with this approach. Among them are the assumptions that our knowledge of the area of their territory is correct, and that the voles are spread evenly over this territory.

In trying to work out the total number of voles, we’ve had to make assumptions – or simplifications. But we hope that we have a good enough method of working out something we can’t measure from something we can. This we call a “model”. A model of the real world if you like. And we’re use maths – so it’s a “mathematical model”.

We mean “model” in a similar sense to a model aeroplane. We’ve ignored some of the hard bits – made it smaller – focused on the important features. We can then go that extra step and use our model to find out something new about the real world.

Now skip forward twenty years. You’ve been counting the number of voles – and their predator, the weasels – every year. You think the data is accurate – and you’ve entered it into a spreadsheet.

One very simple mathematical model would be to assume that if the vole population is decreasing, it will continue to decrease at the same rate. That sounds fair. However when you look at the data – you find that you can’t easily predict what will happen next year! The numbers of voles isn’t decreasing each year – some years it even increases dramatically!

This is a problem. What do we do?

We need to take the number of weasels in the year – or perhaps the previous year – into account. And that’s when you need to build a more complicated mathematical model – one which includes all the things you know (and maybe some things you have to find out or even guess!). In this way we can make a model that makes a prediction of a future event.