Environmental Engineering Reference
In-Depth Information
a
b
N1, run1
N2
N2, run1
700
300
280
600
260
500
240
400
220
300
200
200
180
100
160
0
0
140
0
100
200
300
400
500
600
700
800
100
200
300
400
500
600
700
c
Time
d
N1
N1,run1
N2
N2,run1
9000
400
8000
7000
300
6000
5000
200
4000
3000
100
2000
1000
0
0
0
0
50
100
Time
150
200
1000 2000 3000 4000
N1
5000 6000 7000 8000 9000
Fig. 6.15 Hopf bifurcation: The system described by (6.18) exhibits a limit cycle. If the parameter
C
3 is decreased from
0.1
(original version shown in Fig.
6.11
)to
0.01
, the limit cycle vanishes and
a stable equilibrium emerges (
upper two figures
). The same happens in the equation if the
parameter is increased to
0.4
(
lower two figures
). In between there are transition points where
the phase shift between approximating a stable equilibrium and a limit cycle occur
models require a numerical approach via simulation. When assessing an ecological
context with analytical approaches, the level of natural complexity has to be
reduced for tractability. A negative consequence of this simplification might be
the loss of the dynamical interaction structures. A prominent exception to this
dilemma is represented by the trophic level analysis using steady state models as
explained in Chap. 5.