3. Pollution in lakes is a major ecological problem for today's society. Using a simple model, we analyze the increase or decrease of pollution in some of the Great Lakes based on data from Rainey [1]. A basic mathematical model for the concentration of a pollutant, c(t), in a well-mixed lake is given by the differential equation:
where k is the concentration of pollutant entering the lake at a rate r (which is also the rate water leaves) and V is the volume of the lake.
a. Assume that a new pesticide is used and is found entering the Great Lake ecosystem via the rivers at a concentration of 5 ppb. (Thus, assume that k = 5.) Assume that the lakes are initially clear of the pollutant, so c(0) = 0. The volume of lakes Superior and Michigan are 12,200 and 4,900 km 3, respectively. The flow rates r are 65.2 and 158 km 3/yr for the respective lakes. Solve the differential equations for each of the lakes. Determine how long it takes for the pollutant to reach 2 ppb in each lake.
b. Now consider what happens if the lake is already polluted with this pesticide. Assume that each of the Great Lakes has 4 ppb of the pollutant, so c(0) = 4. If some law is enacted that removes all of the pesticide from the river sources, then k becomes zero in the differential equation. Solve the differential equations for the Great Lakes you were given in Part a. with these changes, then find how long it takes for the pesticide level to fall to 10% of its original amount (4 ppb). Give the concentrations at times t = 5, 10, and 20 years.
c. More realistically, it takes time for the pesticide to decay from the fields where it was used. Assume that the concentration of the pesticide entering via rivers is a decaying function of time with k(t) = 4e-0.15t, so the differential equation becomes
Starting with c(0) = 4 and h = 0.5 (yr), use the Improved Euler's method to simulate this differential equation for 20 years for each of the Great Lakes listed above and graph the solutions. Give the concentrations at times t = 5, 10, and 20 years. Compare these answers to the ones in Part b.
d. Briefly discuss what you observe about build up and loss of pollutants in the Great Lakes that you studied. Give at least one strength and one weakness of the mathematical model used in these problems.