Biology Reference
In-Depth Information
Definition. An equilibrium point (x
0
, y
0
) of the system of
Eq. (2-8) is called asymptotically stable when all trajectories that start in
some region that contains the point (x
0
, y
0
) converge
to the point (x
0
, y
0
)ast becomes large. A stable equilibrium point (x
0
, y
0
)
that is not asymptotically stable is called neutrally stable.
Asymptotically stable equilibrium points are of particular importance, as
they are most likely to be seen in natural systems. For this type of
equilibria, the long-term behavior of the solution trajectories is
insensitive to small changes in the initial values. Such equilibrium points
are useful for representing the dynamics of many systems in biology,
ecology, or medicine by allowing for ''normal'' variability of the initial
conditions without affecting the long-term evolution of the system. The
following theorem presents a criterion that allows us to determine
whether a given equilibrium point is asymptotically stable.
Theorem. With the notation above, the equilibrium point (x
0
, y
0
) is
asymptotically stable if det( J)
>
0, and trace( J)
<
0.
Example 2-4
.......................
Consider the spread of an epidemic represented by the diagram in
Figure 2-12. This model differs from that defined by Eq. (2-5) in that the
incoming flow to the group of susceptibles is constant, and is not
dependent upon the size of the population.
We may think of parameter
as the rate of immigration into the
group of susceptibles (e.g., periodically a plane full of immigrants
free of the disease arrives at a village where the infectious disease is
spreading).
b
The differential equations describing this model are:
dS
dt
¼a
IS
þ b
(2-9)
dI
dt
¼ a
IS
g
I
:
b
a
SI
g
I
S
I
Immigration
Deaths
FIGURE 2-12.
Block-diagram representing a model for spread of an infectious disease in a system that allows for
immigration at a constant rate b. At any time t, deaths occur at a fixed per capita rate g and leave the
system.
