Biology Reference
In-Depth Information
The stability of point A is related to the matrix:
0
1
1
12
1
2
@
A
:
J
¼
d
d
Now,
5
12
d
1
12
d:
det
ð
J
Þ¼
and
trace
ð
J
Þ¼
>
d
Because det( J)
0 for all values of
, we obtain that (1,1) is an unstable
1
12
and stable if
1
12
. The unstable
equilibrium point if 0
< d <
d >
point is also repelling.
Next, we show that the square 0
<
V
<
4, 0
<
O
<
4 is a basin of
attraction.
dV
dt
Note that at V
¼
4
;
0, so a trajectory cannot cross that boundary
dV
dt
¼
going to the right. Similarly, at V
0, so no line can cross that
boundary going to the left. In the same way,
dO
¼
0
;
dt
0ifO
¼
4 and
and
dO
0
V
4
;
dt
¼
0ifO
¼
0, so no solution can cross out of the
1
12
, we see that (1,1) is
square along those boundaries. Thus, if 0
< d <
a repelling point in the basin of attraction 0
4, and,
since there are no other equilibrium points in this basin, according to the
Poincar
´
-Bendixson theorem, we have a periodic solution.
<
V
<
4, 0
<
O
<
VII. A MODEL OF COMPETITIVE INTERACTION
We present a final model describing two species competing for the same
crucial resource. It seems obvious that if one species has a decided
advantage, the other species will become extinct. However, in some
situations, both species can coexist. Figure 2-23 gives the results of two
types of yeast, Sacharromyces cerevisiae and Saccharomyces kefir, competing
for a food supply. The experiment, conducted by G. F. Gause in 1932,
shows mutual long-term coexistence with diminished saturation levels
for both populations.
A well-designed model should be capable of capturing some of these
possible long-term behaviors. As always, we begin with a few general
assumptions:
1. The environment provides a limited food resource.
2. There are only two species competing for this resource.
