Biology Reference
In-Depth Information
@
f
f
dO
¼g
@
V
¼ a
2
b
V
g
O
V
@
@
g
@
g
V
¼ e
O
¼d þ e
:
O
V
@
@
E
XERCISE
2-12
(a) Show that the equilibrium states for the model defined by
0
.
a
b
;
d
e
;
a
g
bd
Eq. (2-13) are (0,0),
;
and
ge
d
e
;
a
g
bd
(b) Show that for the equilibrium point A
¼
(x
0
,y
0
)
¼
;
ge
we have:
@
f
y
0
Þ¼
bd
e
f
dO
ð
@
y
0
Þ¼
gd
e
V
ð
x
0
;
x
0
;
@
@
g
Þ¼
ae bd
g
@
g
V
ð
x
0
;
y
0
O
ð
x
0
;
y
0
Þ¼
0
:
@
@
Using the results obtained in Exercise 2-12, we form the Jacobian J and
d
e
;
a
g
bd
use it to analyze the stability of the equilibrium point A =
ge
(see Figure 2-19). Calculating the determinant and the trace of this
and trace
2
Þ¼da
bd
a
b
d
Þ¼
bd
e
matrix, we obtain: det
ð
J
< db
ð
J
:
e
e
0if
a
b
>
d
Now we see that trace( J)
<
0 and det( J)
>
e
;
which is
necessary for the null clines in question to intersect. This shows that
for the model defined by Eq. (2-12) is
d
e
;
a
g
bd
the equilibrium point
ge
asymptotically stable. The point of using arbitrary constants was to show
that as long as
d
e
<
a
b
;
this equilibrium point in the modified Lotka-
Volterra model is always asymptotically stable, regardless of the specific
parameter values.
3. More on Classifying the Equilibrium States
We now revisit the classification of equilibrium states for the model
defined by Eq. (2-8).
Recall that a point (x
0
, y
0
) is an equilibrium point if and only if
f (x
0
, y
0
)
¼
¼
0. An asymptotically stable equilibrium point
is one that attracts solutions any time a trajectory passes sufficiently
close. Unstable equilibrium points may attract some trajectories, but not
all. Among unstable equilibrium points is a special class called repellers.
Any trajectory coming close to a repeller is forced away from it.
Figure 2-20(B) shows a repeller (for simplicity, the trajectories are shown
as straight lines).
0 and g(x
0
, y
0
)
