Journal of Mathematical Biology

, Volume 28, Issue 5, pp 529–565

# The velocity of spatial population expansion

• F. van den Bosch
• J. A. J. Metz
• O. Diekmann
Article

## Abstract

We consider the velocity with which an invading population spreads over space. For a general linear model, originally due to Diekmann and Thieme, it is shown that the asymptotic velocity of population expansion can be calculated if information is available on: (i) the net-reproduction, Ro; i.e. the expected number of offspring produced by one individual throughout its life, and (ii) the (normalized) reproduction-and-dispersal kernel, β(a, χ − ξ); i.e. the density of newborns produced per unit of time at position χ by an individual of age a born at ξ By means of numerical examples we study the effect of the net-reproduction and the shape of the reproduction-and-dispersal kernel on the velocity of population expansion. The reproduction-and-dispersal kernel is difficult to measure in full. This leads us to derive approximation formulas in terms of easily measurable parameters. The relation between the velocity of population expansion calculated from the general model and that from the Fisher/Skellam diffusion model is discussed. As a final step we use the model to analyse some real-life examples, thus showing how it can be put to work.

## Key words

Space time Integral equation Dispersal Asymptotic velocity of propagation Approximation formulae Cumulant generating function Fisher/Skellam diffusion model

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## Authors and Affiliations

• F. van den Bosch
• 1
• J. A. J. Metz
• 1
• O. Diekmann
• 1
• 2
1. 1.Institute of Theoretical BiologyState University of LeidenLeidenThe Netherlands
2. 2.Centre for Mathematics and Computer ScienceAmsterdamThe Netherlands