Agriculture Reference
In-Depth Information
60
50
40
30
20
10
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
t
Fig. 5.1.
Five realizations of a stochastic linear birth-death process together with the continuous determinis-
tic solution (
x
0
=
50,
λ
=
3,
μ =
4;
see text for model description).
there would be replicates where the bacterial
colony becomes extinct before
t
=
5,
and there
would be replicates where there are still a
few individuals left at
t
=
5.
No experiment
could actually wait until
t
=
∞
to actually
verify that all colonies over an infinite num-
ber of replicates become extinct. In essence,
the average colony under the deterministic
view never becomes extinct.
The deterministic solution depends
only on (λ - μ). The same deterministic curve
as seen in
Fig. 5.1
would be obtained if
λ =
30
and μ =
31.
This is not the case for the
stochastic model, whose solution depends
explicitly on both λ and μ: (λ - μ) controls
the essential shape, whereas (λ + μ) controls
the volatility. If five replicates of the experi-
ment were actually performed, the spread of
the various outcomes at a given time would
allow the estimation of λ + μ, whereas the
overall trajectory would allow for the esti-
mation of λ - μ. Estimation techniques re-
quire high-quality, calibrated, high-resolution
time-course measurements of levels of a
reasonably large subset of model variables.
This is difficult to obtain. Methods for such
estimation are complex and currently based
on Bayesian inference coupled with Markov
chain Monte Carlo methods (Gibson and
Renshaw, 1998). Research in this area is
very active and substantial progress should
be made in the near future. The point, how-
ever, is that when using the same experi-
mental data a deterministic model can only
estimate the difference between the birth and
death rates, whereas the stochastic model
yields separate estimates of both λ and μ.
This illustrates the additional knowledge
gained from using a stochastic view of the
system. It also demonstrates that one cannot
fit a deterministic model to available experi-
mental data and then use the inferred rate
constant in a stochastic simulation. As we
shall see, there is more to stochastic model-
ling than just appending a random error to a
deterministic model.
Sources of Stochasticity in Models
A univariate model (one that has only one
dependent variable) has the following gen-
eral structure:
Y
=
f
(
X
,
b
) +
e
(5.4)
