The Math of Lake Pollution
Martha Torstenson, Math Teaching Fellow

In math class, we explored the way that pollution might flow through a chain of lakes using mathematical modeling. The central goal of this lesson was to understand a way in which math can be used to help understand conservation issues.

Lake pollution is a topic that is close to home here in the Northwoods of Wisconsin, near the shores of Lake Superior. The Great Lakes can be thought of as one slow-moving river, where the waters of Lake Superior, and any pollutants they carry, flow through Lakes Michigan and Huron, to Lake Erie, to Lake Ontario and then out to the Atlantic Ocean.

We began our class by discussing the types of pollution present in the Great Lakes. We looked at maps from of the concentrations of phosphorus, nitrogen, copper, mercury, PCBs, and sediment throughout the Great Lakes. We then turned our attention towards one of the most important steps of the mathematical modeling process, making assumptions.

Some important assumptions that we made included assuming that the volume of the lakes remains constant over time and that the pollutant remains evenly mixed in the water and does not degrade, disappear, or settle out of the water column. These assumptions fit better for some pollutants than for others. Looking at the concentration maps, we noticed that these assumptions might work fairly well to describe the movement of phosphorus, nitrogen, PCBs, and mercury, which seem to be fairly evenly mixed and to be flowing downstream, with higher concentrations in Lake Ontario than in Lake Superior.

We then moved into using difference equations to model a simple 3 lake system in which a lake clean up project stopped new pollution from entering the system. Difference equations use an initial condition and a recursive rule describing the difference between last year and this year to determine the amount of pollution each year. 

After creating a model, and investigating how it played out over a 5 year period, we discussed an interesting result. In the first year, the pollution in the lake furthest downstream increased, before starting to decrease in the second year. We discussed how this phenomenon could be explained by our model, and how this might be communicated to concerned stakeholders. I appreciated the chance to remind our students that the math they are working hard to learn every day can be utilized to understand the world around them and influence effective conservation action.