Biology Reference
In-Depth Information
According to the Lotka-Volterra model, both predator and prey
population levels oscillate with time, with predators lagging behind
prey. A typical time-trajectory plot and the corresponding phase plot are
presented in Figure 2-18. The closed phase trajectories indicate strictly
periodic cycles that repeat indefinitely. The trajectories do not converge
to an equilibrium, and no trajectory drifts off to infinity. Instead, any
change in the initial conditions gives rise to a new closed phase
trajectory [see Figure 2-18(C) and (D)].
300
250
200
150
100
50
0
0
200
400
600
800
1000
1200
A
Time
V:1
300
250
E
XERCISE
2-11
200
150
(a) Determine the null clines for V and O in the Lotka-Volterra model
from Eq. (2-11).
100
50
0
B
0
20
40
60
80
100
120
140
O
(b) How many equilibrium states does the Lotka-Volterra model
have? List them.
300
250
200
(c) Draw a V versus O phase diagram indicating the directional fields
in each of the regions between the null clines (see, for example,
Figure 2-10).
150
100
50
(d) Based on your answers to (a)-(c) above and Figure 2-18, would
you characterize the nontrivial equilibrium state of the Lotka-
Volterra model as stable, unstable, or neutrally stable?
0
C
0
20
40
60
80
100
120
140
O
300
250
200
150
2. A Predator-Prey Model with Limited Growth
One weakness of the Lotka-Volterra model is that, in the absence of
predators, it allows voles to multiply exponentially. A modified version
of the model (2-11) that hypothesizes logistic growth for the voles with
carrying capacity K would be:
100
50
0
D
0
20
40
60
80
100
120
140
O
FIGURE 2-18.
Typical time plots and phase plots for the Lotka-
Volterra model. Parameter values: a ¼ 0.06,
d ¼ 0.02, g ¼ 0.001, e ¼ 0.0002. Panel A: Time
trajectories for the vole (solid line) and owl
(dashed line) populations with initial population
sizes: O
0
¼ 40, V
0
¼ 250; panel B: Phase
trajectory with initial population sizes O
0
¼ 40,
V
0
¼ 250; panel C: Phase trajectories for initial
population sizes V
0
¼ 250 and, from outside in,
O
0
¼ 40, O
0
¼ 60, and O
0
¼ 80; panel D: Phase
trajectories for initial population sizes O
0
¼ 40
and, from outside in, V
0
¼ 250, V
0
¼ 200, and
V
0
¼ 150.
0
@
1
A
V
dV
dt
¼ a
V
K
V
2
1
g
OV
¼ a
V
b
g
OV
(2-12)
dO
dt
¼d
O
þ e
OV
;
b ¼ a
a
where
now represents the inherent per capita net growth
rate for the voles in the absence of owls. We again assume owls eat only
voles, and, if there were no voles, owls would die at a constant per capita
rate.
/K and
We describe the analysis of the phase plane, leaving the details as an
exercise. The null clines for V, where
dV
dt
¼
0
;
are:
¼
a
g
b
V
¼
0 and O
V
:
g
