This note demonstrates the GROWFN applet shown at the right. This is a small module that I wrote originally to test and demonstrate some Java programming techniques. It also shows some features of the modified Schumacher function, and general properties of yield, MAI and CAI curves.
There are three functions drawn on the graph, in blue , red, and green. There are also two active areas of the graph which can be controlled with a mouse. The dotted blue line can be moved up or down by positioning the mouse over it and dragging with the left button. This controls the asymptote (maximum value) of the blue curve. The red spot can similarly be dragged with the mouse in any direction, and will change the shape of all the curves.
The graph represents stem volume growth over time, with volume in m3/ha on the left axis. This is depicted by the blue line, which plots the modified Schumacher function (Lundqvist, 1957):
V = A.exp(-m.t-k)
{eq.1}
In this equation, V is volume, t is time, and A, m and k are coefficients. A is the asymptote, or limiting volume as t→∞. The m and k parameters can be estimated from equations {3}and {4} below. Moving the dotted blue line directly adjusts the asymptote on the blue line.
The red line shows current annual increment (CAI), which is the derivative of equation {1}, and is given by the formula:
dV/dt = k.m.A.exp(-m.t-k).t-(k+1)
{eq.2}
The green line shows mean annual increment (MAI), defined simply as V/t. Both the CAI and MAI curves are read off the right-hand axis, in m3/ha/yr. The red spot falls where the CAI and MAI lines cross, which always happens to occur at the maximum value of MAI (why?). If the coordinates of the maximum MAI point are called (t*, V*), then it can be shown algebraically that the coefficients m and k in equation {1} are given by:
k = -1/ln(V*/A)
{eq.3}
m = t*k/k
{eq.4}
Note that as the red spot is moved up and to the right, the value of m will increase until the function becomes uncomputable. The calculated A, m and k values should appear on the status bar of the browser.
If you would like to use this applet for teaching on demonstration purposes, it can be downloaded here. It can be inserted into your web page with the following HTML code:
<applet code="growfn.class" width=400 height=400 align="right">
GROWFN applet copyright (c) Denis Alder, 2002
Source: http://www.denisalder.com/D02.htm
</applet>
The width, height and align properties can be varied as needed. The applet may not be used for any commercial purpose or on any private, secure or subscription website other than that of a school or university without the express permission and agreement of the author. The following link must be inserted onto the web page where the applet is displayed, and must be visible at the same time as the applet:
<a href="http://www.denisalder.com/D02.htm" target="_blank">www.denisalder.com</a>
If you are using any of the above formulae in an academic paper or student assignment, the following citation should be given:
Alder, D (2002)Technical notes on the modified Schumacher function. http://www.denisalder.com/D02.htm.
If you have any other queries about this applet or any of the material presented above, please email me.