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

Part of the Lecture Notes in Mathematics book series (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.

Keywords

  • Jump Process
  • Wait Time Distribution
  • Smoluchowski Equation
  • Signal Gradient
  • Complementary Cumulative Distribution Function

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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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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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