Environmental Engineering Reference
In-Depth Information
Equation (
6.3
): Several constant factors can be aggregated
dN
dt
¼
ð
C
1
þ
C
2
C
3
Þ ¼
C
:
(6.3)
Equation (
6.4
): Characteristic for exponential growth is the strict linear propor-
tionality of the rate of increase with the size of the pool. When N is plotted over
time, we see that the increase accelerates with time.
dN
dt
¼
C
N
:
(6.4)
Equation (
6.5
): For the exponential decrease we have a negative slope of the rate
and a decelerating decrease when pool size is plotted over time.
dN
dt
¼
C
N
:
(6.5)
Equations (
6.4
) and (
6.5
) describe flows which depend only on the pool size and
a constant. As a result, exponential growth or exponential decline, respectively, will
occur.
Now, we extend the functions used in the equations to successively greater
complexity and discuss the dynamic results. In (6.6) constant increase and expo-
nential increase are combined. In principle, we can use any function which allows
us to determine the rate of change - with one or more variables interacting, for
example exponential growth and negative quadratic decrease (
6.7
).
Equation (
6.6
): Exponential growth together with constant increase
dN
dt
¼
C
N
þ
C
:
(6.6)
Equation (
6.7
): Exponential growth with negative quadratic decrease. This is the
so-called logistic growth function (see below)
dN
dt
¼
N
2
C
1
N
C
2
:
(6.7)
We can simultaneously model several variables that can be either independent of
each other, or coupled; that is, mutually influence each other. One of the simplest
pairs of coupled equations, that is, having two interacting variables and interesting
dynamics, is
dN
1
dt
¼
C
1
N
2
(6.8)
dN
2
dt
¼
C
2
N
1
:
