Start with the following difference equation:
X t þ 1
ð
Þ¼R Ã X t
ð Þ t ¼ 1, 2, 3, 4 . . .
ð2:5Þ
The analytic solution to this equation is:
X t
ð Þ ¼ Xo à R
>
t t ¼ 1, 2, 3, 4 . . .
ð2:6Þ
In STELLA, this difference equation is
ΔX ¼ X t þ 1
ð
ÞÀX t
ð Þ ¼ R À 1
ð
ÞÃX t
ð Þ
ð2:7Þ
which yields what we term Y Numeric in the model of Fig. 2.9.
The continuous form version of this phenomenon is:
ΔY ¼ R Ã Y,
ð2:8Þ
yielding Y Numeric.
The analytic solution to this continuous time equation is:
Y t
ð Þ ¼ Yo à EXP R à t
ð
Þ¼Y Analytic:
ð2:9Þ
X Numeric and X Analytic are the same if DT ¼ 1. As DT approaches 0, these
equations drift apart. But the analytic solution to the difference equation is good
only for DT ¼ 1 (Fig. 2.10). Conversely, when DT ¼ 1, the Y Numeric and Y
Analytic are far apart. Run the model with a DT ¼ 1/1024, and you will find that
numeric and analytical equations converge, as of course they should.
Figure 2.11 shows how much the two numeric solutions agree at DT ¼ 1. Thus
there is a substantial difference between the difference and differential (discrete
vs. continuous) equations when they result in exponential solutions.
Fig. 2.9
38
2 Exploring Dynamic Biological Systems