Environmental Engineering Reference
In-Depth Information
Fig. 6.1 The pool metaphor - how differential equation models describe quantitative relations.
Any change of the pool size (the size of the variable) implies a flow. How much of a flow per unit
of time occurs, is considered to be regulated (by valves). This regulation is described as a function,
which can depend on the size of the variable itself, on other variables or flows, or on external
conditions
of the valves that regulate the amount of inflow and outflow per unit of time.
As functions, inflow and outflow can depend on specific constants, including the
pool size itself (or other pools), and other (externally determined) functions. For
ecological applications, variables typically represent populations in relatively
homogeneous and constant environments.
With some elementary examples we can see how the metaphor and the formula-
tion of equations relate. First, we look at the simplest equations, combine and
expand them to model successively more complex dynamic phenomena. A constant
inflow only (i.e. with no outflow) would lead to a constant increase of the pool size
per unit of time. Constant inflow and outflow would lead to a net change rate -
either positive or negative, depending on the relative flows (or no change if the
inflow equals the outflow). If the state variable size feeds back to influence inflow or
forward to influence outflow in exact proportion to its current pool size, exponential
growth or decline will be modelled. In (6.1)-(6.3),
N
denotes the pool size state,
C
is
a constant
0, and
t
is the time.
Equation (
6.1
): Constant rate of increase of a pool (variable) per unit of time
>
dN
dt
¼
C
:
(6.1)
Equation (
6.2
): Constant decrease of a pool (variable) per unit of time and
dN
dt
¼
C
:
(6.2)
