Homogenization analysis of invasion dynamics in heterogeneous landscapes with differential bias and motility



Animal movement behaviors vary spatially in response to environmental heterogeneity. An important problem in spatial ecology is to determine how large-scale population growth and dispersal patterns emerge within highly variable landscapes. We apply the method of homogenization to study the large-scale behavior of a reaction–diffusion–advection model of population growth and dispersal. Our model includes small-scale variation in the directed and random components of movement and growth rates, as well as large-scale drift. Using the homogenized model we derive simple approximate formulas for persistence conditions and asymptotic invasion speeds, which are interpreted in terms of residence index. The homogenization results show good agreement with numerical solutions for environments with a high degree of fragmentation, both with and without periodicity at the fast scale. The simplicity of the formulas, and their connection to residence index make them appealing for studying the large-scale effects of a variety of small-scale movement behaviors.


Reaction–diffusion–advection Homogenization Spatial heterogeneity Directed movement Invasion speed Residence index 

Mathematics Subject Classification




I would like to thank Darin R. Stephenson and Charles A. Cusack for useful discussions relating to this work. This work also benefited from discussions during a meeting of the SQuaRE, “Homogenization techniques in ecological and epidemiological models,” supported by the American Institute of Mathematics SQuaREs program. Finally, I would like to thank two anonymous reviewers for their useful feedback on this manuscript.

Compliance with ethical standards

Conflict of interest

The author declares that he has no conflict of interest.


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© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Department of MathematicsHope CollegeHollandUSA

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