Environmental Engineering Reference
In-Depth Information
N2
3
N1, run 1
N2, run 1
2
3
1.0
2
1.0
0
0
-1.0
-1.0
-2
-2
-3
-3
0
2.0
4.0
6.0
8.0
10.0
12
14
16
-3
-2
-1.0
0
N1
1.0
2
Time
Fig. 6.2 Simulation of (6.8).
Left
:
N
2 over
N
1,
Right
:
N
1 and
N
2 over time. Initial conditions:
N
10
¼
1.0,
N
20
¼
2.0 Parameter:
C
1
¼
3.0,
C
2
¼
4.0
C
1 and
C
2 are positive constants. The variable
N
1 increases exponentially in
proportion to the variable
N
2. The change of
N
2 depends negatively on the size of
N
1. The resulting dynamics is an oscillation, depending on the initial conditions and
the parameter size (Fig.
6.2
). For
C
1
1.0, a sine curve will result when
plotted over time - or a circle when one variable is plotted over the other.
Formal structures like this are used to capture some of the dynamic phenomena
of ecological processes.
¼
C
2
¼
6.3 Differential Equations as a Modelling Approach
for Dynamical Systems
When experimenting with differential equations, it is a frequent experience that
explosions (rapid exponential growth) or a collapse (approximation of zero) occur
unintentionally, if the model was not carefully designed. The art of modelling is to
select quantitative relations in a meaningful way so that they capture relevant and
dominating elements of observable (physical, biological) phenomena. Simplifica-
tions are inevitable, which always brings the possibility of interesting discoveries as
well as irrelevant or trivial results. In cases where only one equation is used,
relevant information about the resulting dynamics can be obtained from a graphical
representation that plots the rate of the change as a function of the size of the
variable. This representation is called
rate level graph
. For one-dimensional sys-
tems, which are described by only one differential equation, the general type of
dynamics can be directly deduced. All intersections of the plot of the rate of change
with the zero line are equilibria. This is because, at that particular variable value,
the changes are zero. If the plot of the rate of change of a variable
N
first decreases
with increasing
N
, then becomes negative for larger
N
, the intersection with the
zero-line is a
stable equilibrium
. This means, if a system in an equilibrium state is
