Environmental Engineering Reference
In-Depth Information
equilibrium point where no changes in population size occur. The coordinates of the
equilibrium point are defined by ( 6.12 ) and ( 6.13 ).
At all other points of the phase space both variables of the system change over
time. The isoclines separate the phase space into four sectors: One where both
populations grow, one where both decline, and two other sectors where either prey
or predator increase while the other decreases (Fig. 6.5 ).
The analytical result obtained in Fig. 6.5a can be refined, if the values are calculated
grid-wise and the resulting direction of prey and predator vector is drawn. This yields
the example of a direction field in Fig. 6.5b . It can be seen, that the system oscillates.
The oscillation occurs for all initial combinations of predator and prey population
sizes; however, the amplitude depends on the starting value (Fig. 6.5 , Fig. 6.6 ). Only
for a starting point precisely at the equilibrium, there would be no subsequent change
in the values of the variables.
a
b
Pred
C 1 / C 2
Prey
C 3 /b * C 2
Fig. 6.5 (a) Lotka-Volterra isoclines. The vectors indicate in which direction prey and predator
populations develop in the phase space. (b) Example of a direction field for the Lotka-Volterra
model (6.11). The following values were used: C 1
¼
0.1, C 2
¼
0.001, C 3
¼
0.1, b
¼
0.1. The
model equilibrium occurs at Pred
¼
100 and Prey
¼
1,000, x -axis: 0.0
2,000, y -axis: 0.0
200
...
...
a
b
N2
400
300
300
200
200
100
100
0
0
0
100
200
N1
300
400
0
100
200
Time
300
400
Fig. 6.6 Simulation the Lotka-Volterra system (6.11). (a) Display of Pred and Prey over time,
(b) Display of Pred over Prey (trajectory drawn in the phase space). 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
 
Search WWH ::




Custom Search