Simulation and analysis of spatial pattern formation at multiple scales in relation to plausible dispersal kernels.

Michael Shaw, University of Reading, UK.

e-mail: m.w.shaw@reading.ac.uk

Dispersal mixes populations and also allows populations to explore new territory and forage for sparsely populated areas. The key feature determining the balance between these is the dispersal kernel. For spores or pollen dispersing through the air a reasonable generic form is

this may also apply to seeds and less obviously to animals. This is a combination of an exponential term, set essentially by wash-out and death, with a well-defined scale set by h, and a power-law term with an exponent b set by the characteristics of wind-flows. A maximum upper scale is set by the circumference of the earth; for most purposes there is also a local scale set by the size of a host plant, below which this form does not apply, but also below which all available dispersal sites will usually be saturated with propagules. For spores and pollen h is of the order of 10-1000 km, depending on the fragility of the propagule; b is typically well below 2. Fortunately most of the interesting generic features don’t depend on how b and l vary with direction.

A consequence of this form is that if h is comparable to the scale of a region (such as a continent) expansion of a new genetic form or a new pest into new territory will take the form of rapid but sporadic expansion across the whole potential range, followed by consolidation into it, rather than a “wave of advance”. The result will be a quasi-fractal pattern of occupied and unoccupied territory, becoming closer to a fractal the smaller b is. Thus for example, repeated discovery of fungicide resistance need not imply recurrent mutation.

A second consequence is that the notion of “isolation distance” makes sense only for specific contamination thresholds. If there is a possibility of a genuine undesirable consequence of, for example, growing a GM crop, there is no specific area outside which vigilance can be relaxed or contamination can be said to be zero. Current isolation distances are very small by comparison with h, and make sense only if b is larger than is likely.

A third consequence is that evaluating disease or pest management strategies, especially those depending on biodiversity, is hard using conventional trials. This is because the actual outcome will often depend on the pattern and density of suitable hosts over the whole relevant scale set by h.