Use of coupled oscillator models to understand synchrony and travelling waves in populations of the field vole Microtus agrestis in northern England

1. Earlier studies have reported that field vole Microtus agrestis populations in Kielder Forest, UK, exhibit typical 3–4-year cyclical dynamics, and that the observed spatiotemporal patterns are consistent with a travelling wave in vole abundance moving along an axis south-west–north-east at approximately 19 km year–1. One property of this wave is that nearby populations fluctuate more synchronously than distant ones, with correlations falling lower than the average for the sampling area beyond approximately 13 km. 2. In this paper we present a series of models that investigate the possibility that both the observed degree of synchrony and the travelling wave can be explained as a simple consequence of linking a series of otherwise independently oscillating populations. Our ‘coupled oscillator’ models consider a series of populations, distributed either in a linear array or in a two-dimensional regular matrix. Local population fluctuations, each with a 3–4-year period, were generated using either a Ricker equation or a set of discrete-time Lotka–Volterra equations. Movement among populations was simulated either by a fixed proportion of each population moving locally to their nearest neighbour populations, or the same proportion being distributed via a continuous geometric function (more distant populations receiving less). 3. For a variety of different ways of generating cycles and a number of different movement rules, local exchange between oscillating populations tended to generate synchrony domains that extended over a large number of populations. When the rates of exchange between local populations were relatively low, then permanent travelling waves emerged, especially after an initial invasion phase. There was a non-linear relationship between the amount of dispersal and the domain of synchrony that this movement generated. Furthermore, the observed spatiotemporal patterns that emerged following an initial invasion phase were found to be highly dependent on the extreme distances reached by rare dispersers. 4. As populations of voles are predominantly distributed in grassland patches created by clear-cutting of forest stands, we estimated the mean patch diameter and mean interpatch distance using a geographical information system (GIS) of the forest. Our simplified models suggest that if as much as 5–10% of each vole population dispersed a mean of 178 m between clear-cuts per generation, then this would generate a synchrony domain and speed of wave in the region of 6–24 km (per year), which is reasonably consistent with the observed synchrony domain and speed. Much less dispersal would be capable of generating this scale of domain if some individuals occasionally moved beyond the nearest-neighbour patch. 5. While we still do not know what causes the local oscillations, our models question the need to invoke additional factors to explain large-scale synchrony and travelling waves beyond small-scale dispersal and local density-dependent feedback. Our work also suggests that the higher degrees of synchrony observed in Fennoscandian habitats compared with Kielder may be due in part to the relative ease of movement of voles in these former habitats. As our work confirms that the rates of exchange among local populations will have a strong influence on synchrony, then we anticipate that the spatiotemporal distribution of clear-cuts will also have an important influence on the dynamics of predators of voles.