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

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 14 000 km(2) 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 x 100 km, temporal oscillations emerge that are independent from long-distance dispersal.