Volume 10, Number 4, Winter 2002
SPATIAL DYNAMICS OF THE DIFFUSIVE LOGISTIC EQUATION WITH A SEDENTARY COMPARTMENT
K. P. HADELER AND M. A. LEWIS
Abstract. We study an extension of the diffusive logistic
equation or Fisher's equation for a situation where one part
of the population is sedentary and reproducing, and the other
part migrating and subject to mortality. We show that this
system is essentially equivalent to a semilinear wave equation
with viscous damping. With respect to persistence in bounded
domains with absorbing boundary conditions and with respect
to the rate of spread of a locally introduced population, there
are two distinct scenarios, depending on the choice of parameters.
In the first scenario the population can survive in sufficiently
large domains and the linearization at the leading edge
of the front yields a unique candidate for the spread rate. In the
second scenario the population can survive in arbitrarily small
domains and there are two possible candidates for the spread
rate. Analysis shows it is the larger candidate which gives the
correct spread rate. The phenomenon of spread is also investigated
using travelling wave theory. Here the minimal speed of
possible travelling front solutions equals the previously calculated
spread rate. The results are explained in biological terms.
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