Pattern formation triggered by rare events: lessons from the spread of rabies

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Abstract
Understanding of large–scale spatial pattern formation is a key to successful management in ecology and epidemiology. Neighbourhood interactions between local units are known to contribute to large–scale patterns, but how much do they contribute and what is the role of regional interactions caused by long–distance processes? How much long–distance dispersal do we need to explain the patterns that we observe in nature? There seems to be no way to answer these questions empirically. Therefore, we present a modelling approach that is a combination of a grid–based model describing local interactions and an individual–based model describing dispersal. Applying our approach to the spread of rabies, we show that in addition to local rabies dynamics, one long–distance infection per 14000km2 per year is sufficient to reproduce the wave–like spread of this disease. We conclude that even rare ecological events that couple local dynamics on a regional scale may have profound impacts on large–scale patterns and, in turn, dynamics. Furthermore, the following results emerge: (i) Both neighbourhood infection and long–distance infection are needed to generate the wave–like dispersal pattern of rabies; (ii) randomly walking rabid foxes are not sufficient to generate the wave pattern; and (iii) on a scale of less than 100 km times 100 km, temporal oscillations emerge that are independent from long–distance dispersal.