Volume 11, Number 2, Summer 2003
WHERE TO PUT DELAYS IN POPULATION
MODELS, IN PARTICULAR IN THE NEUTRAL
CASE
K. P. HADELER AND G. BOCHAROV
Abstract. Hutchinson's model or the delayed logistic
equation explains oscillations in populations by delayed response to exhaustion of nutrients. The exploration of periodic solutions
and global behavior of nonlinear delay equations has started from this model. Another delay model for oscillations in populations is the blowfly equation which has been thought
to be better suited to explain experiments. Also delay differential
equations of neutral type have been proposed as population
models. The GurtinMacCamy system of a partial differential
equation with boundary condition models populations structured
by age with birth and death rates depending on age and
on total population size. If written as a renewal equation then
piecewise constant coefficients, even with delta peaks, are admissible.
It is shown that these "realistic caricatures" of age
structure models are equivalent to delay equations, in general of
neutral type. In the standard retarded case (as opposed to the
neutral case) the resulting systems have the form of the blowfly
equation. Hence the latter seems indeed better justified than
the delayed logistic equation. In the case of neutral equations
the age structure approach presented here is a rigorous derivation
of such population models. Also in the retarded case the
view on oscillations is drastically changed since Hutchinson's
model and the blowfly model have distinct stability properties.
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