Biology Reference
In-Depth Information
O
The null clines for O are:
¼
d
a
g
O
¼
0
;
and V
e
:
The null clines are shown in Figure 2-19.
They intersect inside the region
f
ð
V
;
O
Þ :
V
>
0
;
O
>
0
g
only when
A
d
e
<
a
b
:
We prove below that when this condition is satisfied, and thus the
null clines intersect, the intersection point A is a stable equilibrium for
model (2-12).
0
V
e
a
b
Before we do this, we want to discuss the biological meaning of the
condition
d
FIGURE 2-19.
Null clines for the model described by Eq. (2-12).
e
<
a
because our mathematical arguments suggest that when
b
it is not satisfied, the model may exhibit radically different long-term
behavior.
b ¼
K
;
and so
a
First, recall that
K—the carrying capacity for the vole
population in isolation (i.e., in the absence of owls). The condition
d
b
¼
e
<
a
b
now takes the form
d
e
<
K
;
or
d < e
K, equivalently. Recall that
d
is the
owls' per capita death rate when no voles are present and that
V(t) is
the owls' birth rate due to the presence of voles. Because V(t) can never
exceed its carrying capacity K (assuming no immigration), we have V(t)
<
e
K represents the owls' maximal per capita growth
rate controlled by food resources.
K. Thus, the term
e
d < e
The condition
K then requires that the owls' per capita death rate
caused by a lack of food be lower than the owls' maximal per capita
birth rate caused by the presence of food. We shall see below that under
this condition, the vole and owl populations both stabilize around
nonzero equilibrium values. When this condition is not satisfied and
d > e
V(t), i.e.,
the owls' per capita death rate exceeds their per capita birth rate. The
owl population will then die out. We shall see below that this is exactly
what the model predicts.
K, this means [because V(t)
<
K] that at any moment t,
d > e
Proceeding with the mathematical analyses of the equilibrium states, our
previous notation yields:
dV
dt
¼
V
2
f
ð
V
;
O
Þ¼a
V
b
g
OV
(2-13)
dO
dt
¼
g
ð
V
;
O
Þ¼d
O
þ e
OV
:
We calculate the partial derivatives for the functions f (V,O) and g(V,O)
to get:
