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Spatial Models

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Data-driven Modelling of Structured Populations

Abstract

Before IPMs were used for size-structured populations, they had a long history as models for spatially structured populations and for spatial spread of infectious diseases and genes. We review some of those models and two recent applications to questions of population persistence. The merger of demographic and spatial structure within one model came much later. The new modeling issue in spatial models is the dispersal kernel that describes individual changes in location. We explain and demonstrate some of the approaches used to model dispersal and estimate the parameters of movement kernels. A major theme in the theory is the difference between bounded and unbounded spatial domains. Models with a bounded spatial domain are “normal” IPMs, and spatial location is just one more trait that differs among individuals. Models with unbounded spatial domain are different. Their long-term behavior is characterized by traveling wave solutions instead of convergence to a stable population structure. We explain how the long-term rate of population spread and its sensitivities can be calculated, and used to determine the factors governing the spread rate of an invasive species.

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Notes

  1. 1.

    Calling k d the dispersal kernel is established in the literature so we use that term even though “kernel” usually refers to P, F, G, or K in this book.

  2. 2.

    In case you’ve forgotten: if \((x_{1},x_{2},\ldots,x_{n})\) is a set of independent observations, and the likelihood of a single observation x is L(x), then the likelihood of the set of observations is \(L(x_{1})L(x_{2})\cdots L(x_{n})\) and the log likelihood is \(\log L(x_{1}) +\log L(x_{2}) + \cdots +\log L(x_{n})\).

  3. 3.

    We thank André Grüning, University of Surrey, for suggesting this example and showing us a proof of the result. It follows from the Limit Set Trichotomy (see Hirsch and Smith 2005) for maps which are monotone, strongly sublinear, and order-compact. If the r th iterate of the dispersal kernel is positive, then the r th iterate of the map (8.1.3) is monotone and strictly sublinear because f has those properties, and order compactness follows from the Arzela-Ascoli Theorem when the dispersal kernel is smooth.

  4. 4.

    The use of c for population spread rate is so consistent in the literature that we use it in this chapter, even though it conflicts with our use of c(z′) for offspring size distribution in other chapters.

  5. 5.

    Alas, this is one more different meaning for the letter s. Because it is so widely used for wave shape we do the same – but it is only in this chapter that s means wave shape.

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Authors and Affiliations

  1. Ecology and Evolutionary Biology, Cornell University, Corson Hall, Ithaca, NY, USA

    Stephen P. Ellner

  2. Animal and Plant Sciences, University of Sheffield, Yorkshire, UK

    Dylan Z. Childs & Mark Rees

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  1. Stephen P. Ellner
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  2. Dylan Z. Childs
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  3. Mark Rees
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Ellner, S.P., Childs, D.Z., Rees, M. (2016). Spatial Models. In: Data-driven Modelling of Structured Populations. Lecture Notes on Mathematical Modelling in the Life Sciences. Springer, Cham. https://doi.org/10.1007/978-3-319-28893-2_8

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