### ! Growth Models: Polynomial (including Straight Line), Exponential, Gompertz Curve

Ref: DR97900

#### There are several models that portray the growth of organisms and systems.

1.     The simplest is a straight line (S.L.) fitted to historical data. The S.L. is a particular case of a polynomial. Other polynomials can be fitted to historical data. A S.L. assumes a constant quantity increase or decrease per period. This is usually only realistic for a few periods. When this model is used for time value calculations, the value of a series of increasing or decreasing payments can be estimated using the expressions for a gradient series.

2.     An assumption of a constant fractional increase of the current population from period to period gives exponential growth. This can be estimated using the compound interest model for future value. Exponential growth is only realistic as long as there appears to be no limits to growth. Many systems appear to grow in this fashion for the initial periods until some capacity constraint begins to take place.

3.     A model that takes into account capacity constraints is the GOMPERTZ:

Y = kabx

Y <- k x a * b * x (APL notation)

Depending on the values of k, a, b or their logs the curve may take on a variety of shapes.

When log a < 0, and 0 < b <1 the curve has an upper asymtote defined by the upper limit on capacity k or whatever. >

When log a < 0, and b > 1 the curve starts at the upper limit k & decreases with a lower asymptote.

When log a > 0, and 0 < b < 1 the curve give a negative exponential decay from the maximum to a lower limit k.

Log a > 0, and b > 1 produces exponential growth beginning at k.

The x = 0 position of all these curves is about the 1/3 or 2/3 points.