Agriculture Reference
In-Depth Information
variance, or regression coefficients. These types of models are obviously related, but
there are also real differences between them.
Mathematical Models
When knowledge of a process is sufficiently concise that it can be represented with
mathematical relationships, the relationship can be referred to as a mathematical
model of the plant, crop, or process of concern. A mathematical model is defined by
a series of equations, input factors, parameters, and variables aimed at characteriz-
ing the process being investigated. Statistically derived mathematical relationships
are normally used to represent most processes. However, they do not contain infor-
mation about mechanisms that cause events to happen; but only represent what
happened.
Explicitly speaking, this type of model grow out of equations that determine how
a system changes from one state to the next (differential equations) and/or how one
variable depends on the value or state of other variables (state equations). These can
also be divided into
-
numerical models and
-
analytical models.
Numerical Models
Numerical models are mathematical models that use some sort of numerical time-
stepping procedure to obtain the models behavior over time. The mathematical
solution is represented by a generated table and/or graph.
Analytical Models
Analytical models are mathematical models that have a closed form solution, i.e.,
the solution to the equations used to describe changes in a system can be expressed
as a mathematical analytic function. For example, a model of personal savings that
assumes a fixed yearly growth rate (
r
) in savings (
S
) implies that time rate of change
in saving d(
S
)/d
t
is given by,
d(
S
)/d
t
=
r
(
S
)
This example is also used to describe numerical models so that numerical and
analytical models can be compared and contrasted more easily.
The analytical solution to this differential equation is
S
=
S
0
×
EXP(
r
·
t
)
where
S
0
is the initial savings,
t
is the time. This equation is the analytical model of
personal savings with fixed growth rate.
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