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[HM] history of logistic eqn variations
Posted:
Dec 14, 2005 11:58 PM
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Dear History of Mathematics List Members,
For several years I have been an admiring reader of this forum and
have used the discussions in a history of mathematics course that I have
recently started teaching. Thank you to the organizer and partipants.
I and my two colleagues Anne Noonburg and Ben Pollina are preparing
a pedagogically oriented mathematics paper that uses an
example of the logistic equation with periodic harvesting
dy/dt = ry(1 -y/K) - Asin(2*Pi*t) to present techniques for studying more
general differential equations of the form dy/dt = f(t, y) where f is
periodic in t.
(Such questions are connected with the still unsolved 16th Hilbert problem.)
This led us to the history of the logistic equation. Sharon
Kingsland's fine book Modeling Nature clarifies the history of dy/dt =
ry(1 - y/K) with its discussion of Verhulst, Pearl, and Lotka.
We are wondering though about the later history as it concerns our
problem. We came up with the following three questions.
a. How did the study begin and develop of the logistic equation with
constant harvesting: dy/dt = ry(1 - y/K) - H?
b. When did scientists notice the bifurcation value H* = K/4, such that
for H < H* the population survives but for H > H* the population becomes
extinct no matter the initial population? It is our impression that this
somewhat counterintuitive result has been used in presentations to the
general public. E.g. The presence of a stable population of fish on the
Grand Banks does not mean that you can increase the fishing rate by a little
and still have a stable population.
c. When did scientists start to look at time dependent or periodically time
dependent parameters in the logistic equation; r = r(t), K = K(t), H = H(t)?
This question may seem more appropriate for a list devoted
to the history of (population) biology, but perhaps this erudite readership
could be of help.
Thank you very much.
Sincerely yours,
Diego Benardete
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