In the Great Lakes, water flows from Lake Superior and Lake Michigan, through Lake Huron and Lake Erie into Lake Ontario. Water from Lake Ontario flows into the St. Lawrence Seaway and into the ocean. Therefore, pollutants which flow into any of the lakes get mixed with the water and continue their flow along with the water through the system. As a native of Rochester, NY, which is on Lake Ontario, I am naturally concerned with reduction of the level of pollutants in the system.

To simplify our study, we will limit or concern to the last 3 lakes, Huron, Erie and Ontario. Each year, 11% of the water from Lake Huron flows into Lake Erie, 36% of the water from Lake Erie flows into Lake Ontario, and 12% of the water from Lake Ontario flows into the St. Lawrence Seaway. It has been demonstrated that action to improve the water quality in the lakes can be effective. At one time, Lake Erie was declared a dead lake due to the level of pollutants. But strong action reduced that level and allowed the lake to revive.

To assess what might happen if strong action were taken to reduce the level of pollutants in the system, assume that suddenly no new pollutants were entered into the system. This is a possible stage in a multi-stage modeling process. Assume that presently there are 3500 tons of pollutants in Lake Huron, 1800 tons of pollutants in Lake Erie and 2400 tons of pollutants in Lake Ontario. Assume further that the pollutants are uniformly dissolved in the water, so that the flow rate of pollutants from one body of water to another is the same as the flow rate of water as a whole.

1. To model this situation, write a system of differential equations modeling the above situation. let H(t) denote the amount of pollutants in Lake Huron, E(t) the amount of pollutants in Lake Erie, and O(t) the amount of pollutants in Lake Ontario after t years.

2. Find the matrix of the system, all eigenvalues and corresponding eigenvectors.

3. . Describe what would happen to the pollution levels in 5, 10, and 20 years and over the long run.