Lake Superior's position at the upstream end of the Great Lakes (a position shared with Lake Michigan) makes it one of the easier cases to consider. Let us say we are interested in how quickly Lake Superior will clean itself given an existing pollutant is controlled to the point where very little of it is being introduced into the lake. In particular, suppose we are interested in how long we must wait until the amount of pollution is 5% of its level when pollution controls went into affect.
Suppose that 1000 tons of the pollutant are in the lake when controls go into effect. If we call this point in time t=0, then this initial condition can be stated as
The relevant flow parameters for Lake Superior are . The volume of Lake Superior is . From this we see the initial concentration of pollution in the lake is 0.345 tons of pollution per cubic mile. For this level of contamination to have been achieved, the concentration of pollutant in the inflow must have been considerably higher than this during the years of pollution. The recent controls have reduced the concentration of pollution in the inflow to a level of 0.01 tons of pollution per cubic mile of incoming water.
Taking these quantities and placing them into the difference equation we see:
A good way to model this with the Stella software is to use converters for all the problem parameters rather than to combine all these values into numerical constants as we did in the last version of the difference equation. Keeping the various parameters separate in converters allows us to quick change these values to determine their effects on the overall system.
By using Stella to plot for many values of t we can determine when the pollution in the lake drops to 5% of its original value, namely 50 tons.