From: len@math.nwu.edu (Len Evens)
Newsgroups: sci.environment
Subject: Definition of growth rate
Date: 27 Sep 92 15:08:54 GMT
Article-I.D.: news.1992Sep27.150854.28456
Organization: Dept of Math, Northwestern Univ


Since there has been some confusion which seems to have resulted,
at least indirectly, from something I posted, let me try to clarify
a bit of elementary differential equations.   This is relevant in
using a continuous model (which is of course an oversimplification)
in studying population.

Let  x=x(t)  stand for the population size
as a function of time (which is to be determined).  In common usage,
`growth rate' is defined to be the rate of change divided by the
population size, i.e.,  (dx/dt)/x.  (It is usually expressed as a
percent per year in human population studies.)   For simple exponential
growth of the form  x = Ce^{rt}  with  r  constant, this turns out to
be  r.   For linear growth  x = Ct + C_0  it would be   C/(Ct + C_0).
(However,  as one poster pointed out, the concept may not be too
useful in case of linear growth.)

To determine the effect of different assumptions about growth rate
one considers the differential equation
dx/dt = r x
where in principle  r  could be a function both of  x   and  t.
The simplest models usually assume it is just a function of x,
e.g.,  r = a = const   or    r =  a - bx.    The former assumption is
called the Malthusian model and results in exponential growth (if a > 0).
The latter assumption
is called the logistic law and results in  x  appearing initially
to grow exponentially and then later approaching  x = a/b
asymptotically.  (This is under the assumption that  x < a/b  to
start.)    One could of course consider other models with other
plausible assumptions about the form of  r  as a function of x
and t.

It might also be possible to define growth rate in the following way.
Assume  x = f(t) = e^{r(t) t}, i.e., take  r(t) = (ln f(t))/t.   This
gives a different answer from  (dx/dt)/x  in general.   I don't remember
seeing that defintion anywhere except in a posting in this news group.
I don't believe it is common, but perhaps I am wrong.

I would hope the above comments are non-controversial and could be
accepted by informed people of all ideological bents.  About the
only conclusion about public policy one can draw from the mathematics
is that population will grow faster than some exponential curve 
(so eventually will exceed
some reality imposed limit) unless the growth RATE approaches zero
(or become negative).   One can also do a more refined analysis of
the form  if  r(t) >  r_0 > 0  for some specified time period, then
at the end of that period, the population will be greater than some
calculated amount.   One can then argue about whther that population
is too large.   The current dispute, at least among those who understand
the `arithmetic', is whether to promote active measures to encourage
birth control worldwide or to let things take their course in the
expectation that growth rates will decline because of other factors.

Finally, I should note that the above discussion refers only to population
models since in such models the rate of increase  dx/dt is affected by
what happens to individuals and so, at least as a first approximation,
should be proportional to x.   It also ignores the issue of
immigration and emigration, which I think is appropirate for
world population.    There are other problems in which such
assumptions would not make sense.   For example,
according to the IPCC report on climate change, observational data
is consistent with the theory that CO_2 concentration has been growing
exponentially since the beginning of the industrial revolution.
However, in this case, the model has to be quite a bit more complicated,
since dx/dt wouldn't depend in a simple way (to a first approximation)
on x.

Leonard Evens      len@math.nwu.edu      708-491-5537
Dept. of Mathematics, Northwestern Univ., Evanston, IL 60208
Newsgroups: sci.environment,talk.environment,sci.bio,or.general,or.misc,sci.engr.chem