The strategy for handling all five Great Lakes is to formulate one equation for each of the lakes. Each equation will be similar to the equation developed above. We need to quantify the amount of pollution entering each lake per time period and the amount leaving each lake in each time period. These expressions will be somewhat more complicated than in the single lake example because incoming pollution can come from two sources, polluting industries and other lakes. Regardless of this complication, the basic idea is the same as for the single lake process.
Before we can begin, it is important to develop some systematic notation. First, let us begin by defining symbols for the parameters in the problem. These parameters will be the flow into and between lakes as well as the concentration of pollution in the water flowing into each lake.
We will use subscripts to simplify our notation. The subscripts, usually i but sometimes j, will take on values from 1 to 5. The correspondence between subscript value and lake are given in the following table, along with the volume and inflow for each lake:
The parameters for the five lakes problem are given below. The notation is very similar to the notation used in the single lake example.
The unknowns in this problem are the quantities , the amount of pollution in lake i at time t. There are five lakes, and we develop one difference equation for each, giving us a system of five difference equations.
Each equation can be thought of as stating that the difference in the amount of pollution per year is the difference between the annual pollution inflow into the lake less the annual pollution outflow. The five resulting difference equations are:
where represents the annual flow of water outbound from Lake Ontario.
What remains is to give the values of the parameters in this problem.
The volumes of each lake and the annual inflows have been given above.
Below we present the values of the flows between lakes.
We have purposefully not specified the values of the , the concentration of pollutants in each lake's inflow. These concentrations will vary with each pollutant and scenario studied. You may also want to relax the assumption that the 's are constants, and instead replace them with functions of time to allow you to model gradual increases or decreases in polluting activities.
We have also failed to specify the initial values . These, too, will vary depending on the pollutant being studied and the scenario or time frame being considered.
Before a Stella model can be run the values of the various 's and 's must be specified. Determining these values is a good library research exercise for students (and teachers).