Biology Reference
In-Depth Information
2. Show that the system for the equilibrium state is
a
x
¼
F
ð
y
Þ
b
y
¼
G
ð
x
Þ
3. Solve the above system for x and show that the resulting equation:
1
a
G
ð
x
Þ
x
¼
F
b
has only one positive solution. To do this, prove first that the
function in the right-hand side of the above equation is monoto-
nously decreasing. Do the same for y.
4. Investigate the stability by calculating the determinant and trace of
the matrix
F
0
ð
a
y
Þ
:
G
0
ð
x
Þb
Finalize your reasoning by demonstrating that in the equilibrium
state the trace - (
F
0
ð
G
0
ð
a þ b
) < 0 and the determinant
ab
y
Þ
x
Þ >
0.
The picture changes considerably in the presence of delays, because even
a single nonzero delay (as in Eq. (10-14)) might change the properties of
the steady state,
2
that may, for a certain range of delay values, become a
repellor. In the latter case, there will exist a unique asymptotically stable
periodic solution (which encircles the fixed point in the phase space)
acting as a global limit cycle by attracting all trajectories, except the one
originating from the fixed point (see the theorem of Poincar
´
-Bendixson
in Chapter 2).
Although Poincar
´
-Bendixson's theorem gives a sufficient condition for
the existence of a limit cycle, the verification of these conditions is often
nontrivial, and we shall not focus on this question here. Instead, we
examine the periodic solutions of two specific realizations of the
networks shown in Figure 10-14. Each of these examples has a unique
periodic solution and a unique repelling fixed point (Figure 10-15, right
panel). We note that oscillations may be quite sensitive to changes in the
model parameters and examine the system's response to external
influences, such as changes to sensitivity, antibody infusion, and
exogenous hormone infusion, expressed as appropriate modifications to
the mathematical models.
2. The particular sensitivity analysis is nontrivial and is beyond the scope of
this textbook. It consists of investigating the real part of eigenvalues, which are
roots of equation containing a transcendental term, involving the delay. For more
details, see Farhy and Veldhuis (2004).
