Environmental Engineering Reference
In-Depth Information
dPrey
dt
¼
Prey
K
ð
Prey
Þ
C
1
C
2
Prey
Pred
(6.15)
K
dPred
dt
¼
C
2
b
Prey
Pred
C
3
Pred
The form used in the prey equation is equivalent to the logistic curve, as given in
(
6.9
), which can be verified by multiplying and replacing
C
1/
K
by
C
2. Since the
behaviour is more easily understood if the size of the environmental capacity can be
directly specified as a constant, the form in (
6.15
) is frequently used in models.
Figure
6.7
presents a simulation result of (
6.15
).
Oscillations with Increasing Amplitude
The system described in (6.16) exhibits oscillations with increasing amplitude,
successively moving away from an unstable stationary state. Since the amplitude
grows exponentially, the effect can be seen best by starting the system close to
the equilibrium (Fig.
6.8
). In this case, the Lotka-Volterra equations are extended
by a saturation term. This term describes increasing predator growth rate with
increasing prey population, up to a saturation level. The same form is frequently
used to describe enzyme kinetics (Michaelis-Menten equation). One constant
specifies the maximum achievable rate (here:
C
11) and the other one the half
saturation concentration (here: population size at which half of the maximum rate
occurs,
CS
).
a
b
400
300
Prey
Pred
300
200
200
100
100
0
0
0
100
200
300
400
500
600
700
800
0
100
200
N1
300
400
Time
Fig. 6.7 Damped oscillations obtained from a simulation of (6.15). (a) Display of Pred and Prey
over time. (b) Display of Pred over Prey. Initial conditions:
Prey
¼
400,
Pred
¼
100, Parameter:
C
1
¼
0.1,
C
2
¼
0.001,
C
3
¼
0.1,
b
¼
0.7,
K
¼
2,000
