From: Dave Rusin [rusin@math.niu.edu]
Date: Wednesday, March 13, 1996 11:37 AM
To: rmcgehee@TECLink.Net
Subject: Re: reexpression
Someone, surely, will give you the lecture that it's inappropriate simply
to ask for any function which has a similar graph; what you ought to
do is make a model of the situation which explains why, for example,
the number of deaths is (like the number of births) proportional to a
product N*(N_0 - N) (N=number of individuals). I hope you will consider
such an approach -- if you have a sense of what drives your function in
the first place, it's easier to describe appropriate functions to fit
to your data.
Having said this, let me offer a suggestion. You know the function
2/(1+e^-t) increases from values nearly zero (for t < < 0 ) to the
value of 1 ( for t > > 0 ). Of course you can add a time delay and
a time scaling factor and get the same general description; that is, you
can use any function of the form
2/(1 + exp(-a(t-b)) ).
Here we have a > 0 to get a curve whose graph has the increasing shape
described above, but of course we could look at functions with a < 0
and get the mirror image of those curves: curves which decrease from
1 to 0 as t ranges over all real values. This suggests a way to
get a curve with your requirements: simply multiply two such functions
together -- one with a > 0 and the other (corresponding to a larger
value of b) with a < 0. For example, the function
4/ [ (1+exp(-(t+10))) * (1+exp(+(t-10))) ]
is nearly zero if t is large in magnitude; it rises near t=-10 to
values just less than 1; stabilizes; drops near t=+10 back to nearly zero.
This is just the kind of curve you would get if, during the extinction
phase, your population is dying off at a rate proportional to the
product N*(N_0 - N) as I suggested before.
dave