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Transport and Anisotropic Diffusion Models for Movement in Oriented Habitats

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Dispersal, Individual Movement and Spatial Ecology

Part of the book series: Lecture Notes in Mathematics ((LNMBIOS,volume 2071))

Abstract

A common feature of many living organisms is the ability to move and navigate in heterogeneous environments. While models for spatial spread of populations are often based on the diffusion equation, here we aim to advertise the use of transport models; in particular in cases where data from individual tracking are available. Rather than developing a full general theory of transport models, we focus on the specific case of animal movement in oriented habitats. The orientations can be given by magnetic cues, elevation profiles, food sources, or disturbances such as seismic lines or roads. In this case we are able to present and contrast the three most common scaling limits, (i) the parabolic scaling, (ii) the hyperbolic scaling, and (iii) the moment closure method. We clearly state the underlying assumptions and guide the reader to an understanding of which scaling method is used in what kind of situations. One interesting result is that the macroscopic drift velocity is given by the mean direction of the underlying linear features, and the diffusion is given by the variance-covariance matrix of the underlying oriented habitat. We illustrate our findings with specific applications to wolf movement in habitats with seismic lines.

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Notes

  1. 1.

    This section is an adaptation of Sect. 4.1.3 from [17]. It was inspired by Dolak and Schmeiser [11] who apply this scaling to chemotactic movement and, while their results do not directly apply here, the methods are the same.

  2. 2.

    This section is an adaptation from [10].

  3. 3.

    A general formula for directional moments, such as \(\int \gamma {\gamma }^{T}d\gamma = \vert {\mathbb{S}}^{n-1}\vert /n\;\mathbb{I}_{n}\) can be found in [16].

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Acknowledgements

Work of Thomas Hillen was supported by NSERC and work of Kevin J. Painter was supported by BBSRC.

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Authors and Affiliations

  1. Centre for Mathematical Biology, University of Alberta, Edmonton, AB, Canada, T6G2G1

    Thomas Hillen

  2. Heriot-Watt University, Edinburgh, EH14 4AS, UK

    Kevin J. Painter

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  1. Thomas Hillen
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Correspondence to Thomas Hillen .

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Editors and Affiliations

  1. Mathematical & Statistical Sciences, University of Alberta, Central Academic Building 632, Edmonton, T6G 2G1, Alberta, Canada

    Mark A. Lewis

  2. Mathematical Institute, Centre for Mathematical Biology, University of Oxford, St. Giles Street 24-29, Oxford, OX1 3LB, United Kingdom

    Philip K. Maini

  3. Department of Mathematics, University of Leicester, University Road, Leicester, LE1 7RH, United Kingdom

    Sergei V. Petrovskii

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Hillen, T., Painter, K.J. (2013). Transport and Anisotropic Diffusion Models for Movement in Oriented Habitats. In: Lewis, M., Maini, P., Petrovskii, S. (eds) Dispersal, Individual Movement and Spatial Ecology. Lecture Notes in Mathematics(), vol 2071. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35497-7_7

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