Agriculture Reference
In-Depth Information
Introduction
is
X
(
t
), and the initial number (
t
= 0) is
x
0
.
In a deterministic view:
dX t
dt
Mathematical models are increasingly
used in biology (Yeargers
et al
., 1996) and
agriculture (France and Thornley, 1984) to
make quantitative inferences and predic-
tions about complex systems under vari-
ous discrete or continuous inputs. Most
biological models have a deterministic
structure. That is, they make definite pre-
dictions for quantities without any associ-
ated probability distribution. Successive
runs of the model yield identical out-
comes if initial states, parameter values
and inputs are the same. Although such
models are useful for understanding the
complex interactions that occur in biol-
ogy, they lack a fundamental characteris-
tic of biological systems: variation. Re-
gardless of the precision of measurements,
replications of an experiment even when
done on the same subjects do not yield the
same identical outcomes. In contrast sto-
chastic models contain random elements
and probability distributions within the
models themselves. Thus, stochastic models
help us understand not only the mean
outcome of a system, but also the reasons
as to why such outcomes vary under ap-
parently identical situations. In this chap-
ter, we first present a very simple bio-
logical process and model it using both a
deterministic and a stochastic approach.
This serves to illustrate the enhanced
value of stochastic models. This is fol-
lowed by a section identifying the sources
of stochasticity in models. Methods used
to identify important elements are then
presented followed by methods of valid-
ation of stochastic models.
()
=lm
(
) ()
Xt
(5.1)
Integrating Eqn 5.1, we get:
X
(
t
) =
x
0
e
(
l
−
μ
)
t
(5.2)
In Eqn 5.2, the slope of the process through
time is entirely determined by λ - μ. If λ - μ
> 0 (i.e. λ > μ), this indicates exponential
growth. If λ - μ < 0 (i.e. λ < μ), this indicates
exponential decay. In fact, the solution to
Eqn 5.2 depends exclusively on (λ - μ). For
example, the solution if λ =
20
and μ =
19
is
identical to the solution if λ =
2
and μ =
1.
Thus Eqn 5.2 is fundamentally overparame-
terized: experimental data can only provide
information about (λ - μ), and Eqn 5.2 must
be restated as:
X
(
t
) =
x
0
e
rt
(5.3)
Where ρ = (λ - μ). Experimental data can
only provide information on the difference
between birth and death rates.
A stochastic view of the linear birth-
death process takes a very different approach.
First, the number of bacteria varies discretely.
That is, the number of bacteria can only take
integer values at all times. Second, stochas-
ticity is introduced to the system because
each bacterium has its own probability of
giving birth and of dying over each unit
of time. Simulated results for a decaying
process where
x
0
=
50,
λ =
3
and μ =
4
are
presented in
Fig. 5.1
. At
t
=
2,
the deter-
ministic model yields
X
= 6.77, whereas the
stochastic model yields predictions ranging
from
3
to
20.
In an actual experiment, a bac-
terial count of 6.77 could never be observed;
bacterial counts can only take an integer
value. The prediction of 6.77 represents the
mean, that is, the expected value over a very
large number of replicates. The stochastic
approach actually models the expected vari-
ation among replicates, something that is
completely absent in the deterministic model.
The time to extinction is infinity (∞) for
the deterministic model, but takes values
ranging from 2.2 to undetermined (i.e. not
extinct by
t
= 5) for the stochastic model.
If the experiment were actually performed,
Linear Birth-Death Process
Suppose that we are interested in modelling
the number of bacteria in a bacterial colony
(Wilkinson, 2006). We define the birth rate,
λ, as the average number of offspring pro-
duced by each bacterium in a unit of time.
Likewise, the death rate, μ, is defined as the
average proportion of bacteria that die in a
unit time. The number of bacteria at time
t
