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Structured Populations

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Integrodifference Equations in Spatial Ecology

Part of the book series: Interdisciplinary Applied Mathematics ((IAM,volume 49))

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Abstract

So far, we have treated populations as homogeneous: all individuals were assumed to be identical with respect to reproduction and dispersal. We only modeled the dynamics of a single density function. In reality, most populations are heterogeneous in many ways. Individuals differ with respect to age, size, gender, and other attributes, and their reproductive and dispersal behavior may depend on these attributes. The nonspatial dynamics of populations with complex life cycles have been successfully described by matrix models. In this chapter, we introduce and study spatially explicit matrix models to generalize the simple IDE to stage-structured populations. We present an in-depth analysis of the critical patch-size problem and the spreading speed for these equations, including several proofs that we omitted in the scalar case in earlier chapters. Throughout the chapter, we use a simple two-stage model for juveniles and adults to illustrate the theory. We close with an overview of the rich literature of applications of structured IDEs to real-world systems, in particular to species invasions.

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Notes

  1. 1.

    The Perron–Frobenius theorem holds under the more general condition that B is irreducible. A matrix is irreducible if it is not conjugate to a block upper-triangular matrix (Caswell 2001). A typical example for a reducible matrix arises in populations with post-reproductive stages. Since these stages do not contribute to reproduction, their contribution to the pre-reproductive stages is zero. If we order the variables in the equations such that the post-reproductive stages are first, the resulting matrix will be block upper-triangular. In terms of the life-cycle graph, a matrix is reducible if there is a proper subset of vertices from which there are no edges (transitions) to nodes outside that subset.

  2. 2.

    A function on some interval is piecewise continuous if the interval can be written as a finite number of subintervals such that the function is continuous on each open subinterval and has a finite limit at each endpoint of each subinterval.

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  1. Mathematics and Statistics, University of Ottawa, Ottawa, ON, Canada

    Frithjof Lutscher

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Lutscher, F. (2019). Structured Populations. In: Integrodifference Equations in Spatial Ecology. Interdisciplinary Applied Mathematics, vol 49. Springer, Cham. https://doi.org/10.1007/978-3-030-29294-2_13

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