Environmental Engineering Reference
In-Depth Information
C 2* N 1* N 2 is modified ( 6.17 ) and added to the predator as well as the prey
equation
C 2
N 1
N 2
(6.17)
A þ N 1
N 1
A the predation efficiency is reduced to 50%. This rate decreases for
smaller N 1. Further modifications are required: (a) the introduction of a saturation
function into the growth term. An appropriate selection of constants can ensure that
the oscillations close to the equilibrium point are unstable (this can lead to an
increasing amplitude around the equilibrium, as was seen in ( 6.16 ) and Fig. 6.8 );
(b) the introduction of a logistic term (capacity limitation), preventing an infinite
increase of the oscillations, as it was seen in ( 6.15 ) and Fig. 6.7 . Then, the following
differential equation system results ( 6.18 ), where increasing and decreasing oscilla-
tions can be observed (Fig. 6.9 ).
At N 1
¼
a
b
700
6000
Prey
Pred
600
5000
500
4000
400
3000
300
2000
200
1000
100
0
0
0
100
200
300
400
500
600
700
800
0
1000
2000
3000
4000
5000
6000
c
d
Time
N1
600
5000
Prey
Pred
500
4000
400
3000
300
2000
200
1000
100
0
0
0
100
200
300
400
500
600
700
800
0
1000
2000
3000
4000
5000
Time
N1
Fig. 6.9 Simulation results for (6.18). Left :(a) and (c) show the display of predator ( N 2) and prey
( N 1) over time. Right :(b) and (d) show the trajectories in the phase space, Predator ( N 2) over prey
( N 1). If we use the initial condition N 1
¼
6,000, N 2
¼
160, the amplitude decreases towards a limit
cycle ( top : a and b). If we use the initial condition N 1
¼
1,000, N 2
¼
160, the amplitude increases
towards a limit cycle ( bottom : c and d)
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