Volume 3, Number 4, Fall 1995
THE EFFECTS OF DISPERSAL
ALONG ENVIRONMENTAL GRADIENTS ON
THE DYNAMICS OF POPULATIONS
IN HETEROGENEOUS ENVIRONMENTS
FETHI BELGACEM AND CHRIS COSNER
Abstract. In this paper we study the effects of adding a
term describing drift or advection along environmental gradients
to reactiondiffusion models for population dynamics with
dispersal. The basic models are linear or logistic equations
with diffusion and with a spatially varying linear zero order
term describing the local population growth rate. The drift
terms are constructed from the gradient of the local growth
rate and thus describe directed movement of the population
up or down the gradient of the growth rate. The effects of
drift depend critically on boundary conditions. If the boundary
of the region inhabited by the population acts as a barrier,
then sufficiently rapid movement in the direction of the gradient
of the growth rate is always beneficial. If the boundary
is lethal to the population, then movement up the gradient of
the growth rate may be either beneficial or harmful depending
on the specific situation.
The analysis is performed by observing the effects of drift
on the principal eigenvalues of certain elliptic operators. The
eigenvalues determine whether a given model predicts persistence
or extinction for the population it describes. The
eigenvalues are estimated via a change of variables which permits
the use of variational methods even though the original
problems are not selfadjoint.
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