Last modified: 2014-03-28

#### Abstract

Landscape fragmentation has huge ecological and economic implications and affects the spatial dynamics of many plant species. Determining the speed of population spread in fragmented landscapes is therefore of utmost importance to ecologists.

An effective way of determining spread rates is by constructing and analysing a mathematical model. We use spatially explicit, discrete time, stage-structured integrodifference equations (IDEs) to model spreading populations. IDEs are deterministic models which accurately reflect many species' life cycles and dispersal patterns and can incorporate spatial heterogeneity and demographic structure.

Analytical expressions for spread-rates in IDEs are useful as they allow us to understand the qualitative behaviour of the model and dependencies on particular parameters. They can be derived exactly for spatially homogeneous landscapes and have been approximated for a particular type of dispersal kernel, and landscapes in which the scale of variation is much smaller than the dispersal scale.

We propose an analytical approximation to the wave-speeds of IDE solutions with periodic landscapes of alternating good and bad patches, where the dispersal scale is greater than the extent of each good patch and where the ratio of the demographic rates in the good and bad patches is small.

The approximation is derived by linearising the IDE model and considering periodic travelling wave (PTW) solutions with different decay-rates. For each decay-rate, the speed of the travelling wave is the principal eigenvalue of an integral operator. We find an asymptotic approximation to this wave-speed through regular perturbation theory and by approximating dispersal from the good patches as dispersal from a point source. We find that for biological populations governed by the non-linear IDE, the spreading speed is given by the minimum wave-speed of the PTWs.

We formulate this approximation for the Gaussian and Laplace dispersal kernels and for stage structured and non-stage structured populations, and compare the results against numerical simulations. We find that the relative error of the approximation (compared to numerical results) is small for the range of parameters considered, and therefore that the approximation is highly accurate.

For both dispersal kernels, habitat loss is found to reduce invasion speed, however, the effect of landscape connectivity, as classified by the landscape period, on invasion speed differs between the dispersal kernels. For the thinner tailed Gaussian kernel, increased landscape period corresponds to slower invasions, whereas for the thicker tailed Laplace kernel, increased landscape period results in faster invasions.

Therefore, to accurately predict the speed of an invasion, it is of utmost importance to accurately determine the dispersal kernel, especially when the landscape is fragmented.

M. A. Gilbert, S. M. White, J. M. Bullock, E. A. Gaffney, Spreading Speeds for Stage Structured Plant Populations in Fragmented Landscapes (2014) http://dx.doi.org/10.1016/j.jtbi.2014.01.024