By Dominique Douglas-Smith, Australian National University

Known characteristics of an environmental system can be converted into equations.  These equations form a mathematical ‘model’ of the system.  This model can be used to predict how the system will behave under different scenarios.

Let’s look at an example from the field of hydrology (water science).  Consider a catchment of water: a lake, river, the ocean.  The simplest catchment is a bucket.  Water enters the bucket through the top.  Let’s suppose there’s a hole in the side, allowing water to flow out, as well.  Hydrologists call this bucket a ‘store’ and in equations it is represented by the letter S.

We can model how water flows out of the bucket by an ordinary differential equation (ODE).  This equation is a relationship between the rate of change of storage over time and the storage itself.  It looks like this:

The left-hand side of the equation is a derivative (hence the term, ‘differential’).  S(t) is the amount of water in the bucket at time ‘t’.

‘a’ is a ‘constant of proportionality.’ It tells us how the store is responding to an input of water. If a is negative, this indicates that water is flowing inwards, rather than out of the hole in the side (this is, of course, physically impossible).  If a is 0, there is no flow (like a dam).  If a is really big (think, infinite), then the bucket would empty as soon as water entered it. So usually, we choose a value for a that is greater than 0, but not too large.

The solution to this equation will be a function that tells us how much water is in the bucket at a particular time.  Think of a function like a rule that tells us how much of one thing (e.g. storage volume) we have, depending on the amount of another thing (e.g. time).

So there we have it! We’ve modelled the flow of water from a bucket.  What’s great about this is the model can be used for a range of purposes.  One area is water quality.  Looking at data, mathematicians have worked out relationships between flow rates and concentrations of chemicals, pathogens, algae, et cetera, in the water.  So by having an accurate model of the flow, we can predict how the water quality will change under different climate conditions.

Dominique Douglas-Smith was one of the recipients of a 2017/18 AMSI Vacation Research Scholarship.

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