Biomedical Engineering Reference
In-Depth Information
Figure 1
. Symmetry breaking in a competition model. Here the parameter 8 measures the
difference between the two populations (with a carrying capacity normalized to one). This
parameter is plotted against the competition parameter, measuring the inhibitory effect of one
population on the other.
Coexistence takes place and is stable provided that C < 1. Otherwise, one of two
exclusion points,
(
N
1
*
,
N
2
*
) = (
K
,0),
[7]
(
N
1
*
,
N
2
*
) = (0,
K
),
[8]
will be reached. Which population wins depends only on the initial conditions:
the first population that increases in size over the other will take over. This is
illustrated in Figure 1, which depicts the difference, 8 = |
N
1
-
N
2
|, between both
populations at equilibrium starting from two different initial conditions in which
each population has a slightly large population than the other.
Using different competition rates C, we observe a sharp transition at C
c
= 1.
For low subcritical competition levels, populations coexist and have the same
size. At high competition rates, exclusion takes place. Once an initial difference
between the populations is created, exclusion takes place. For this symmetric
system, the two possible choices define two attractors of the dynamics. The
situation is qualitatively shown in Figure 2. Here the original state is represented
by the ball in the middle of the two valleys, representing the two possible attrac-
tors. The ball can roll down in two different directions, breaking the original
symmetry and choosing one of two possible outcomes.
