1550078481-Ordinary_Differential_Equations__Roberts_

Applications of Linear Systems with Constant Coefficients 403

l. The fl.ow rates are constant. (Of course, these rates are not co nstant

but variable, since they are affected seasonally by rainfall, snowfall , and

evaporation.)

2. The volume of each lake is constant. (The volumes of the lakes are
   variable due to the variation in inflow and outflow rates and seasonal
   changes.)


3. P erfect mixing occurs in each lake so that the pollutants are uniformly
   distributed throughout the lake. (Pollutants in t he lakes are not uni-
   formly mixed. Incoming water tends to move from its source to the
   outlet of the lake in a pipeline fashion without mixing. So the clean
   up time for the main part of a lake will be less than predicted while
   the clean up time for isolated, slow moving portions of t he lake will be
   longer than predicted.)


4. Pollutants are dissolved in water and enter and leave the lake by in-
   flow and outflow. (DDT, for example, is ingested by higher predators
   and retained in t heir body fat. These animals are large and not apt to
   leave the lake with the outflow unless they choose to do so. When the
   predator dies most of its body fat is consumed by other organisms. So
   most DDT remains in the biosphere for an extended period of time. As
   a result DDT will remain in a lake in higher quantities than predicted
   by t he model. Phosphorus, on the other hand , causes "algae bloom"- a
   sudden population explosion of algae. Later, the algae dies and settles
   to the bottom of the lake removing some phosphorus in t he process.
   However, this removal is only temporary, since t he decaying process
   returns the phosphorus to t he lake water.)
   We will use t he following notation and information in deriving the system
   of differential equations for the amount of pollution in the Great Lakes.

NON-LAKE

OUTFLOW INFLOW

.LAKE SUBSCRIPT VOLUME RATE RATE

(i) (Vi) mi^3 (ri) mi^3 /yr (Ri) mi^3 /yr

Erie e 116 85 17

Huron h 850 68 15

Michigan m 1180 38 38

Ontario 0 393 99 14

Superior s 2900 15 15

Let
qi(t) be the amount of pollutant (DDT, phosphorus, or mercury) in lake
i at time t;
ci(t) be the concentration of pollutant in lake i at time t; and
Ci be the concentration of pollutant in the inflow to lake i.