5.1 An Environmental Pollutant
Consider a region in the natural environment where a water borne pollutant enters and leaves by stream flow, rainfall, and evaporation. A plant species absorbs this pollutant and returns a portion of it to the environment after death. A herbivore species absorbs the pollutant by eating the plants and drinking the water, and it return a portion of the pollutant to the environment in excrement and after death. We are given (by ecologists) the initial concentrations of the pollutant in the ambient environment, the plants, and the animals together with the rates at which the pollutant is transported by the water flow, the plants and the animals. A basic problem is to determine the concentrations of the pollutant in the water, plants, and animals as a function of time and the parameters in the system.
Figure 5.1 depicts the transport pathways for our pollutant, where the state variable x1 denotes the pollutant concentration in the environment, x2 its concentration in the plants, and x3 its concentration in the herbivores. These three state variables all have dimensions of mass per volume (in some consistent units of measurement). The pollutant enters the environment at the rate A (with dimensions of mass per volume per time) and leaves at the rate ex1 (where e has dimensions of inverse time). Likewise the remaining
Figure 5.1: Schematic representation of the transport of an environmental pollutant. rate constants a, b, c, d, and f all have dimensions of inverse time. The scenario just described is typical. We will derive a model from the law of conservation of mass: the rate of change of the amount of a substance (which is not created or destroyed) in some volume is its rate in (through the boundary) minus its rate out; or, to reiterate, time rate of change of amount of substance = rate in − rate out.
This law is easy to apply by taking the units of measurement into account. Often we are given the volume V of a container, compartment, or region measured in some units of volume, the concentration x of some substance in the compartment measured in amount (mass) per volume, and rates k of inflow and ! of outflow for some medium carrying the substance measured in units of volume per time. The rate of change dx/dt in this case, where t denotes time measured in some consistent unit, has units of amount per volume per time. The rate of outflow ! times the concentration x has units of amount per time. Thus, we must multiply the time-derivative by the total volume to achieve consistent units in the differential equation