406
Chapter 9
The shape of the oscillation depends on the size of the perturbation and on the parameter values in Eqs. (1) -
(3). Figure 9.15A shows very regular oscillations for a small initial disturbance
&X,
= 0.025 of the steady
state
X!Q
= 0.125,
X20
= 0.250,
S0 =
0.8125.
T
is 8.97 (dimensionless), which is close to the limiting
value 2tt /
ij-l - 2k-Jl =
8.886. For
&Xt
= 0.425 (Fig. 9.15B), the oscillations are severely distorted
with sharp peaks of
X,
and
X2.
The cycle time has increased to 13.87.
Figure 9.16 is similar to the phase plot Figure 9.11, but instead of the
saddle point
in Figure 9.11 we now
have
limit cycles.
Figure 9.16 collects the essential information concerning the undamped oscillations that
result after perturbing the above steady state with different AAV For the present, simple kinetics the shape of
the phase-plane plots can be calculated analytically. The substitution
yj
ln(X;), and
y2
= InfX?) is
introduced in the cell balances:
^0
- fi\
1
Pixwen
(
8
)
dy2
dO
= P2Y„Sfey'
-1
(9)
The two equations are differentiated once more with respect to
0
and subtracted. After substitution of
en
and
eyi
from Eq. (8) to Eq. (9) in the resulting equation, one obtains
Figure 9.16. Coexistence of prey and predator. Phase plane diagram of
Xt
and
X2
for different values of
the initial disturbance AAi. The extrema for the limit cycle marked A are shown in the table. Limit cycle
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