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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 reaction-diffusion 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 self-adjoint.

 

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