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Bull Math Biol. 2016 Oct;78(10):2034-2056. Epub 2016 Sep 27.

Homogenization of a Directed Dispersal Model for Animal Movement in a Heterogeneous Environment.

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Department of Mathematics, Hope College, Holland, MI, 49422-9000, USA.


The dispersal patterns of animals moving through heterogeneous environments have important ecological and epidemiological consequences. In this work, we apply the method of homogenization to analyze an advection-diffusion (AD) model of directed movement in a one-dimensional environment in which the scale of the heterogeneity is small relative to the spatial scale of interest. We show that the large (slow) scale behavior is described by a constant-coefficient diffusion equation under certain assumptions about the fast-scale advection velocity, and we determine a formula for the slow-scale diffusion coefficient in terms of the fast-scale parameters. We extend the homogenization result to predict invasion speeds for an advection-diffusion-reaction (ADR) model with directed dispersal. For periodic environments, the homogenization approximation of the solution of the AD model compares favorably with numerical simulations. Invasion speed approximations for the ADR model also compare favorably with numerical simulations when the spatial period is sufficiently small.


Advection–diffusion–reaction; Diffusion approximation; Directed movement; Homogenization; Invasion speed; Spatial heterogeneity

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