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The Mathematical Analysis of Biological Aggregation and Dispersal: Progress, Problems and Perspectives

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Dispersal, Individual Movement and Spatial Ecology

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

Abstract

Motile organisms alter their movement in response to signals in their environment for a variety of reasons, such as to find food or mates or to escape danger. In populations of individuals this often this leads to large-scale pattern formation in the form of coherent movement or localized aggregates of individuals, and an important question is how the individual-level decisions are translated into population-level behavior. Mathematical models are frequently developed for a population-level description, and while these are often phenomenological, it is important to understand how individual-level properties can be correctly embedded in the population-level models. We discuss several classes of models that are used to describe individual movement and indicate how they can be translated into population-level models.

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Notes

  1. 1.

    In this section the weight w may be equivalent to the signal S used earlier, or some function of it.

  2. 2.

    These equations are the restriction of (54) to one-space dimension only when the speeds s  ±  are constant, and in that case the moment equations close at the second level for constant λ [73]. We consider the more general case for illustrative purposes.

  3. 3.

    In [41] this estimate appears with the L 2-norm squared, but it is clear from the proof that there should be no square.

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Authors and Affiliations

  1. School of Mathematics and Digital Technology Center, University of Minnesota, Minneapolis, MN, 55455, USA

    Hans G. Othmer

  2. Mathematical Biosciences Institute, Ohio State University, Columbus, OH, 43210, USA

    Chuan Xue

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  1. Hans G. Othmer
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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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Othmer, H.G., Xue, C. (2013). The Mathematical Analysis of Biological Aggregation and Dispersal: Progress, Problems and Perspectives. 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_4

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