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


Dr Tom Sherratt, Department of Biological Sciences, University of Durham, Durham DH1 3LE, UK (fax 0191 3742417; e-mail


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.


Understanding the processes that influence the spatiotemporal dynamics of natural populations is an important goal of ecology, and a field of research with enormous practical application (Caldow & Racey 2000). It is now well known that dispersal among local populations will, under some conditions, enhance the persistence of the ensemble population (Hassell, Comins & May 1991; Sherratt & Jepson 1993; Baillie et al. 2000), and that variations in local density can in theory arise even in a physically homogeneous habitat (Bascompte & Solé 1995). Much recent interest on spatiotemporal dynamics has centred on elucidating how and why local populations separated by different distances fluctuate in synchrony (Ranta et al. 1995; Ranta, Kaitala & Lundberg 1997; Grenfell et al. 1998). Other studies have focused on determining the conditions under which travelling waves, structures that propagate over space without change in shape, are generated and whether there is any evidence of such phenomena in the natural world (Dunbar 1983; Murray, Stanley & Brown 1986; Ranta & Kaitala 1997; Sherratt, Eagan & Lewis 1997).

In this paper we briefly review current data on the spatiotemporal dynamics of field voles Microtus agrestis L. in a large coniferous forest (Kielder Forest, Northumberland, UK). We then quantitatively evaluate the validity of several possible explanations for the observed spatiotemporal phenomena, in particular the observed synchrony domain. Synchrony has a major influence on the likelihood of metapopulation extinction (Sherratt & Jepson 1993), and therefore the development of a deeper general understanding of the causes and consequences of synchrony may have implications for applied disciplines such as conservation and epidemiology. Our primary aim has been to gain a better understanding of factors that influence the spatiotemporal dynamics of field voles and to draw general insights from these relationships. However, as field voles are abundant in the transitory islands of grassland that develop after clear-cutting of forest stands, we hope that this research will contribute to a better understanding of the interrelationship between forest management practice and microtine dynamics.

The paper has two main sections. First, we briefly describe our study area and present a summary of our current data on the spatiotemporal dynamics of field voles in this region. Secondly, we present a series of partially parameterized models that seek to test the possibility that the observed patterns can be explained as a simple consequence of linking a series of otherwise independently oscillating populations.

In the context of this special issue, this work augments the approaches to population modelling taken by other contributors (e.g. Cowley et al. 2000; Pettifor et al. 2000; Wadsworth et al. 2000).

Kielder Forest

A detailed description of the study area is provided in our companion paper (Petty et al. 2000). Kielder Forest is a large (620-km2) commercial plantation in northern England (55°13′ N, 2°33′W), dominated by Sitka spruce Picea sitchensis Bong. and Norway spruce Picea abies L. The forest is subdivided for management into large blocks, defined by valley systems: Kielder, Kershope, Redesdale and Wark blocks. Harvesting of timber (approximately 1000 ha year–1) generates clear-fell patches of 5–100 ha. They are subsequently planted (on average 1 year after felling) and the majority of these patches progress through grassland (dominated by Deschampsia caespitosa, Holcus spp., Agrostis spp.) to thicket stage after 12–15 years. Field voles are found in these ephemeral habitats, but are absent from dense spruce plantations that lack grass cover (Thomson 1996).

Spatiotemporal dynamics of field voles in Kielder Forest

Vole densities have been estimated using a simple vole sign index (VSI) at a minimum of 13 grassland sites distributed over Kielder Block in March, May/June and September for the past 14 years (Petty 1992; Lambin et al. 1998). This index, which involves an assessment of vole signs in 25 quadrats, has been shown to be a reliable measure of field vole abundance, and has been calibrated against field vole densities calculated from mark–recapture techniques (Lambin, Petty & MacKinnon 2000). In the spring and autumn of 1996 and 1997 and the spring of 1998, 147 grassland sites, covering not just Kielder Block but the adjacent Kershope, Redesdale and Wark Blocks, were similarly assessed using the VSI.

Lambin et al. (1998) examined the data from VSI assessments within Kielder Block at a total of 21 sites (13 full time series from 1984 and more recently initiated assessments). Overall, densities showed cyclic dynamics, with peaks occurring every 3 or 4 years. However, oscillations in different locations were not entirely synchronous (positively cross-correlated). By fitting an appropriate generalized additive model, Lambin et al. (1998) concluded that the observed spatiotemporal dynamics were consistent with a linear periodic travelling wave in abundance. This wave was estimated to move at 19 km year–1, along a south-west to north-east axis (78° from north). The same statistical model has recently been fitted to the shorter-term (2·5 years) but more widespread survey of field vole density. These data similarly support the contention of a travelling wave, with a comparable speed (14 km year–1) and direction (66° from North). In both studies, the overall cross-correlation between time series decreased with increasing separation between sites, although the presence of a travelling wave-like pattern meant that the strength of this relationship varied with the direction of projection. The more recent survey covered a sufficient area to allow the authors to assess the distance beyond which cross-correlations were lower than the average for the sampling area. This was estimated as 15 km using an isotropic Mantel correlogram. The estimated distance in the direction of the wave was slightly less, at approximately 13 km.

Explanations for synchrony and travelling waves

While occasional field voles are capable of long-distance movement, the majority disperse less than 100 m (Sandell et al. 1990). It is therefore extremely unlikely that the observed synchrony domain of several kilometres can be directly explained by long-distance movement of voles alone. Perhaps regular small-scale local movement can, over time, gradually even out differences among populations, thereby generating larger-scale synchrony? Although this possibility has recently been commented upon (May 1998), the phenomenon has received relatively little attention (but see for example Ruxton 1996; Kaitala & Ranta 1998; Bjørnstad, Stenseth & Saitoh 1999).

To quantify the relationship between the degree of local movement and the extent of synchrony, we considered what happens when otherwise independently oscillating populations become partially coupled by a small degree of local movement. To begin with, we considered the consequences of linking a series of populations, each of which exhibited dynamics according to the time-delayed discrete-time Ricker equation (Kaitala & Ranta 1998):

NT + 1 = NT exp(r – a1NT –  a2 NT1)(eqn 1)

where NT is the population size at time T. When the reproductive rate coefficient r = 2·2 and the density-dependent terms a1 = 0·05 and a2 = 0·05, then it is easy to show that, once transients die out, an isolated population will fluctuate with regular cycles that are repeated every 4 years, analogous to microtine dynamics (Kaitala & Ranta 1998). The precise point in the cycle that is reached at any time T depends on the starting population size.

Ricker dynamics, linear array, local dispersal

Our first model considered 1000 distinct populations, distributed in a linear array of discrete cells. Each of these populations was assumed to fluctuate in size according to Ricker dynamics with 4-year cycles (r = 2·2; a1 = a2 = 0·05). At each generation before reproduction, a proportion of m individuals in each cell moved to populations in their neighbouring cells (m/2 to each). In this instance, the array was considered dissipating, in that m/2 of individuals in the terminal cells were lost from the system at each iteration. Even in the spatial model, local populations still went through regular 4-point oscillations under the conditions we investigated, with similar attractors. Starting the simulation with random population sizes in each cell (0–100), we determined the median number of adjacent cells whose populations were in the same point in the cycle after 1000 generations. Figure 1 shows that when m = 0, then the median size of a cluster of adjacent cells with local populations that were simultaneously at the high point in the cycle was 1. However, as m increased then an increasing number of populations in adjacent cells tended to oscillate in phase, clearly demonstrating that local movement can generate very much larger-scale synchrony.

Figure 1.

Results of simulations in which the dynamics of otherwise independently oscillating populations become coupled by a degree of local movement. A linear array of 1000 populations was assumed, with each population exhibiting 4-year cycles according to the Ricker equation. The initial population sizes in all simulations were set using a random real between 0 and 100. Graphs show the overall median and interquartile range (n = 10 in all cases) for the median size of a cluster of adjacent populations that are all at peak phase in their cycle at t = 1000 generations. When the proportion of the population (m) moving to neighbouring populations was increased, so did this simple measure of the extent of local synchrony.

Clearly, the initial population of voles immigrating into a recently established clear-cut patch will not be an entirely random value, but in part will be determined by the density of voles in neighbouring patches. To account for this, we considered an abstract representation in which cells were initially colonized by a process of invasion, and investigated the degree of synchrony that was established once all cells were occupied and a steady state had been achieved. The model considered 500 cells, each initially containing no individuals except the terminal cell (cell 1), which was seeded with a random number (0–100) of individuals. Default values were as before. Repeated simulations, each for 1000 generations, consistently showed that the steady-state dynamics (here assumed to occur after at least 800 generations of transients) of close populations were more likely to be synchronous than distant populations. Furthermore, a periodic travelling wave, characterized by a cross-correlation signature in which synchrony first decreased with increasing distance, then increased, was found to propagate along this network of cells (Fig. 2a–c). This was not a transient phenomenon, as analysis of the final 200 generations in 10 000 iteration simulations revealed identical signatures. Furthermore, the travelling waves and synchrony domains (taken here as the distance when cross-correlations were first zero) were largely unchanged when the treatment of terminal populations was modified (to taurus or absorbing with only m/2 dispersing from end cells). The wavelengths of the waves generated were found to depend in a complex way on the proportion of individuals moving to neighbouring cells at each iteration (m). When movement rates were relatively low (5% moving), then the closest distance to populations in cells that were oscillating with high degrees of synchrony (measured peak to peak) was relatively high (approximately 164 cells; Fig. 2a). As the rate of movement increased, then the distance between peaks decreased and so did the synchrony domain (Fig. 2b). However, when local movement was very high (20% moving) then oscillations at all locations tended to be positively correlated (hence no travelling wave emerged), and the sensitivity to initial conditions increased (Fig. 2c). When the rate of movement was set below 5%, then unusual cross-correlation signatures emerged in this model, rapidly fluctuating from mean negative to mean positive correlations over very short distances.

Figure 2.

Synchrony and travelling waves following an initial period of invasion. A linear array of 500 populations was assumed, with each population oscillating with 4-year cycles according to the Ricker equation. Asymmetrical starting conditions were generated through setting all population sizes as 0, but seeding the terminal cell with a random (0–100) number of individuals. Each graph shows the mean cross-correlations for different distance classes, which were calculated by estimating the average product–moment correlation in the final 200 generations of a 1000-generation simulation, calculated over all possible pairs of populations within a central range. Five separate simulations are shown for each set of conditions. (a) 5% movement to nearest neighbour (cells 130–370 used for analysis); (b) 10% movement (cells 220–280 analysed); (c) 20% movement (cells 190–310 analysed).

Ricker dynamics, linear array, geometric dispersal

To evaluate whether our model was sensitive to the precise movement rules that were adopted, we investigated the resultant spatiotemporal dynamics when migrants could disperse to other cells, beyond their closest neighbour. Specifically, we followed Kaitala & Ranta (1998) by assuming that the proportion of individuals moving from cell i with NT individuals to cell j could be represented by an exponential function of the distance between these cells (dij) over all cells s, namely:

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The denominator is effectively a normalization constant that allows the total proportion of individuals dispersing from any cell to be set at m. The exponential coefficient c is a dispersal parameter, with small c corresponding to long dispersal distances. Numerically, if c = 0·1 then approximately 60% of dispersers would move to the populations 1–10 cells away, while approximately 20% of dispersers would move to the populations 11–20 cells away. Similarly, if c = 1 then the vast majority of dispersers would move to the populations only 1–2 cells away. One difficulty with the above routine as a representation of dispersal is that under a pure exponential distribution, a very small number of individuals would move across the entire matrix in one iteration, however large the system. Colonization events at these extremes are perhaps best modelled as a stochastic process, but an alternative approach is simply to round down extremely small proportions of the long-distance dispersers beyond a fixed critical threshold, to zero in the calculation. This routine is necessarily arbitrary, but it has the advantage of being based on a simple deterministic relationship (hence the effects of dispersal are more easily seen), and it is certainly more realistic than allowing for spread over infinite distances. In the majority of our simulations our cut-off was set arbitrarily at c dij = 30 (numerator and denominator). Thus for example, when c = 1 then dispersal was limited to a maximum distance of 30 cells, but when c = 0·1 then dispersal was limited to 300 cells. Crucially, in all of our simulations this criterion allowed the vast majority (≫ 99·9%) of all dispersal events, but prevented individuals moving all over the system in a single time step.

As before, a linear model of 500 cells that were each initially colonized by a process of invasion was assumed, the only difference being that we invoked this exponential movement algorithm. Repeated simulations over the range 0·0001 ≤ m ≤ 0·1 and 0·1 ≤ c ≤ 1 indicated that the correlation–distance relationship was highly sensitive to c (Fig. 3), but insensitive to m. In essence, extending the tail of the dispersal distribution significantly extended the domain of synchrony. Therefore, in these invasion scenarios at least, the resultant spatiotemporal dynamics was more dependent on the rate of colonization of empty cells rather than the precise numbers that got there.

Figure 3.

Effect of the dispersal algorithm. These simulations involved identical conditions (500 cells, 1000 generations, invasion dynamics) as Fig. 2 except that dispersers from each population were distributed according to a negative exponential function to all other populations, rather than being distributed locally. The total proportion of individuals moving from each population per generation (m) was set at 0·01, but the nature of the distribution was altered by changing the size of the coefficient c, with high c corresponding to long dispersal distances (flatter distribution).

Ricker dynamics, two-dimensional array, local dispersal

In reality, organisms are rarely, if ever, limited to dispersing in a single spatial dimension. To address this issue, we investigated the synchrony that developed when populations were distributed in a regular matrix of cells rather than in an array. Once again, oscillating populations were linked through limited dispersal. As our primary aim was to establish the synchronizing effects of very localized dispersal, we began by reverting to our earlier assumption that movement was limited to nearest neighbours, in this instance m/8 to each of the eight neighbouring populations. Under a variety of start conditions (random and invasive start), travelling waves were evident, propagated in a variety of directions (Fig. 4). Interpreting the synchrony domain statistics and establishing the speed of waves that are propagated is a challenge because of the diversity of shapes and the complex patterns of interference between waves. This was particularly the case when the start conditions in each cell were set at random rather than being generated by an invasion. Here we simply note that as m was increased then the degree of clustering of local populations in identical phases also tended to increase. Repeated simulations for the above default conditions with m = 0·05 and m = 0·1, indicated that the approximate wavelength peak–peak along the axis of propagation of the most obvious travelling waves was approximately in the order of 12–16 cells.

Figure 4.

Effects of coupling oscillators in two dimensions. In this simulation populations were distributed in the cells of a 41 × 41 matrix and all the cells but one were empty at t = 0. An invasion was initiated by seeding the central upper edge cell with a random number of individuals. A combined proportion of m = 0·05 of each local population moved to their eight nearest neighbours per generation and edge cells were assumed to be dissipating. The local dynamics in each cell was generated using a Ricker function with 4-year cycles. The map shows the densities in each cell of the matrix at t = 1000 generations, with darker cells indicating higher densities.

Lotka–volterra dynamics, one- and two- dimensional arrays, local dispersal

If cycles were generated through a more specific model, then would a similar set of conclusions be reached? The discrete-time Lotka–Volterra equations can be considered under some conditions to represent either the coupled dynamics of a predator and prey population, or the dynamics of healthy and infected individuals:

NT + 1 = NT + rNT (1  –  (NT/K))  – a NT PT(eqn 3)
PT + 1 = PT + f NT PT – b PT(eqn 4)

where PT = size of predator population (or infected individuals), K = carrying capacity, a = search (or infectivity) coefficient, f = fecundity coefficient (or infectivity if a = f), and b = survivorship coefficient. For parsimony, we let these equations reflect dynamics generated by disease. Under particular conditions (e.g. r = 1·8, a = f = 0·01, b = 1·5, K = 300), the equations generate cyclical dynamics in the number of healthy and surviving infected individuals, with a period of approximately 7–8 generations. Not surprisingly, whether the infected population goes extinct before the cycles are reached depends on the starting conditions. If healthy and infected hosts both reproduce twice a year (certainly a more realistic assumption for voles than the assumption of reproduction once a year), then these cycles are effectively repeated approximately every 3–4 years.

We began by assuming there were 500 cells containing discrete populations, distributed in a linear array as earlier. Again, all cells but the terminal cell (cell 1) had no individuals, while cell 1 was seeded with 70 healthy and 50 infected hosts. Both healthy and infected hosts moved at an identical rate m to their nearest neighbour population. Figure 5a shows a typical cross-correlation signature, providing clear evidence for both a domain of synchrony and a travelling wave. Hence, it is clear that our qualitative conclusions are relatively insensitive to structural changes in the way the oscillations themselves are generated. To evaluate the dynamics generated by the Lotka–Volterra dynamics in two dimensions, we simply altered the specific dynamic that was assumed in our earlier two-dimensional simulations. Once again, travelling waves were apparent, even at extremely low rates of local movement (Fig. 5b), with wavefronts sometimes emerging in spirals.

Figure 5.

(a) Cross-correlation signature in the dynamics of healthy and infected hosts different distances apart for m = 0·1. The conditions were the same as for Fig. 2, but local oscillations were generated by Lotka–Volterra dynamics with 4-year cycles (two generations per year). (b) Coupled map lattice with 33 × 33 populations. The simulation conditions were the same as in Fig. 4, but the map shows the local densities of populations at t = 1000 generations for m = 0·001 when local dynamics were generated by 4-year Lotka–Volterra cycles.

Our models are certainly too simplistic and sensitive to underlying assumptions to furnish exact predictions, but it is instructive to consider the results in the context of the average size of a patch of suitable vole habitat in Kielder Forest (all blocks), and the average distance between these patches (Fig. 6a,b). The wavelengths of travelling waves in the majority of our simulations tended to fall in the region of approximately 30–170 cells (approximately four times the domain of synchrony). With a real mean patch size of 14·3 ha (equivalent to 426·7 m diameter if a circular patch is assumed), and a mean nearest neighbour distance between nearest edges of 177·7 m, a synchrony domain of 10–40 patches corresponds (very) approximately to between 6 km and 24 km. Therefore, assuming a 4-year cycle, 5–10% entirely local movement (or considerably less if some dispersers can travel beyond their nearest neighbour patch) will be capable of generating waves that travel at speeds in the order of 4·4 km year–1 to 25·5 km year–1.

Figure 6.

Frequency distribution of (a) area of clear-cut patches less than 12 years old, and (b) distance between the edges of the nearest neighbour clear-cut patch, as calculated from our GIS of Kielder Forest using Forestry Enterprise records up to 1995.


Our study did not evaluate the properties of every different possible model permutation (one-dimensional/two-dimensional; Ricker/Lotka–Volterra; nearest neighbour dispersal/exponential dispersal; random start/invasive start) but nevertheless clear trends emerged from our preliminary analyses. Several studies have demonstrated that small amounts of movement between cities can synchronize infections from disease (Bolker & Grenfell 1996; Lloyd & May 1996; May 1998). In this study we found that large-scale synchrony can readily emerge from small-scale dispersal alone, so long as differences between populations are cumulatively reduced through a combination of small-scale movement and local density-dependent feedback.

Quantitatively, the predicted domains of synchrony tended to extend to approximately 10–40 cells when 5–10% of all individuals present in each cell dispersed to their neighbouring cells each generation, although the precise values were model dependent. If clear-cuts are on average 178 m apart, then this rate of exchange would translate into a synchrony domain in the region of 6–24 km, which is within the range of that observed. But is our figure of 5–10% dispersal consistent with the observed rates of movement of field voles in Kielder forest? During 4 years of mark–recapture studies of field voles in Kielder, in which approximately 6000 individuals were tagged, J.L. Mackinnon (personal communication) caught three individuals (0·05%) that had dispersed between clear-cuts, in this instance a distance class of approximately 2 km. This observation is likely to underestimate the real proportion of voles that move such distances, as there were several other clear-cuts in the vicinity that were not sampled. Nevertheless, by fitting a simple exponential model we can estimate the scale of dispersal required to generate this pattern. If approximately 5% of all individuals moved a distance class of 600–800 m (as we have tended to assume in our simulations, centre to centre), while 0·05% of individuals moved 2000–2200, then the fitted exponential model would imply that approximately 85% of individuals stay in their natal patch, with the majority of dispersal < 500 m. Given the low numbers that disperse long distances, gaining good data on the extreme movements of small mammals is difficult, but these figures do not seem particularly unrealistic.

Our simulations have indicated that much less dispersal would be capable of generating extensive synchrony domains if some individuals occasionally moved beyond the nearest-neighbour patch. Several authors (Mollison 1991; Gurney et al. 1998) have similarly highlighted the importance of the dispersal kernel for shaping the resultant spatiotemporal dynamics. From manipulating the movement algorithm in our invasion models, it was evident that the resultant spatiotemporal dynamics were more dependent on the rate of colonization of empty cells than the precise numbers that got there: once a cell was colonized, cycles could begin no matter how many initial colonists there were.

Our models have also demonstrated that there is not necessarily a simple positive relation between the degree of local dispersal and the domain of synchrony. This was a rather unexpected result, but it is likely to arise from the complex relationship between movement and the ‘higher order’ phenomena such as travelling waves that this movement helps to generate. Specifically, our model showed that when the degree of local movement was gradually increased from relatively low levels, it first brought the peaks of the travelling wave closer together, decreasing the domain of synchrony. As dispersal was increased to even higher levels, the domain of synchrony then began to increase. Although higher-order spatiotemporal phenomena have not been reported in the Hokkaido data set, it is interesting to note that the dynamics of the Japanese wood mouse Apodemus speciosus had a lower domain of synchrony than the grey-sided vole Clethrionomys rufocanus in this area, despite the fact that it was considerably more mobile (Bjørnstad, Stenseth & Saitoh 1999).

Like Kaitala & Ranta (1998), we found that travelling waves readily emerged when oscillators became coupled. Indeed, travelling waves are a common signature of this form of reaction–diffusion kinetics (Murray 1989; Gurney et al. 1998). Interestingly, the waves appeared only for relatively low rates of movement between local populations: too much movement and the dynamics became largely homogenized. This general observation may help explain apparent differences in microtine spatiotemporal dynamics in different parts of the world. For example, after 12–18 years of age, close-grown spruce forests in Britain have little ground vegetation suitable for voles. Therefore, clear-cuts provide islands of suitable vole habitat within an otherwise inhospitable forest matrix. In contrast, in Fennoscandia, forests are more open due to a high component of Scots pine Pinus sylvestris, wide intertree spacing and a greater proportion of thinned and mature stands. Here, ericaceous and grassy vegetation persists under trees, so providing habitat for voles, which is contiguous with grass-dominated vole habitat on adjacent clear-cuts. Perhaps, therefore, the relative ease of movement of microtines in Fennoscandian habitats compared with British commercial spruce forests such as Kielder, may be a further factor explaining why domains of synchrony appear much greater in these regions (Steen, Ims & Sonerud 1996; Lambin et al. 1998; Bjørnstad, Stenseth & Saitoh 1999).

It is important to note that all of the models we have presented in this paper are primarily deterministic (few elements of chance). In the real world, populations will experience intrinsic stochasticity due to the unpredictable nature of births and deaths, as well as extrinsic noise from changes in the environment. The population trajectories in both the Ricker and Lotka–Volterra appear relatively robust to minor stochastic shocks. However, it has long been recognized that if all populations show the same density-dependent structure, and if shocks were simultaneously spread over a number of cells (e.g. through weather), then large-scale synchrony can emerge (the ‘Moran effect’; Moran 1953). The question of how correlated these environmental shocks need to be before the entire system becomes synchronous is a subject of increasing interest (for a recent review see Hudson & Cattadori 1999). Here, we have restricted ourselves to considering the potential synchronizing effects of dispersal alone.

Repeated simulations of our models have demonstrated that travelling waves occur for both random and invasive starting conditions: even under random start conditions there will be cases where, by chance, a large population is surrounded by several smaller populations, hence setting up a minor invasion front. Nevertheless, the waves that are generated from random conditions showed complex patterns of interference, while those generated through an initial inhomogeneity, such as an invasion, proved much easier to detect and characterize statistically. Other sources of inhomogeneity in the real world include ‘impermeable’ landscape features (such as the large water body in our study, Kielder reservoir) and we are currently exploring the extent to which obstacles such as this can organize and shape spatiotemporal dynamics. Similarly, clear-cuts will rarely be evenly dispersed in a commercial forest, so coupling between local populations will not be constant. The effects that this will have on the observed spatiotemporal dynamics clearly deserve further investigation.

While this paper has concentrated on elucidating the characteristics of permanent waves set up in the wake of an invasion, the assumptions of random starting conditions and those set by invasion are clearly two extreme ends of a continuum. We are therefore in the process of developing and exploring more realistic models that effectively combine the two sets of assumptions by allowing new and ephemeral habitat patches to be intermittently created (such as might be generated by clear-cutting) and colonized from established occupied ones in the proximity. Now that the underlying properties of the system in its abstract form have been elucidated, it has become clear that more detailed spatiotemporal models could be of considerable value in guiding forest management decisions. For instance, the rate of dispersal of voles between local populations is likely to be strongly influenced by the spatial pattern of clear-cutting. As we have seen above, systems with reduced contagion among local populations are more likely to fluctuate asynchronously. One implication of asynchrony in prey dynamics is that it may allow wide-ranging predator populations to persist when they would not otherwise do so. Therefore, the spatiotemporal distribution of clear-cuts is likely to have important knock-on implications for the density of the predators of small mammals, such as raptors.

In summary, on the basis of these simple parameterized models we tentatively suggest that the synchrony domain observed in Kielder can be explained by a small degree of movement between otherwise independently oscillating local populations, without the need to invoke additional synchronizing factors. Ironically, the qualitative predictions of our models depend crucially on the assumption of relatively limited, rather than high, dispersal. If too much movement is assumed then properties such as the travelling wave disappear and the synchrony domain becomes highly extensive. We are still unclear about the underlying cause(s) of oscillations in the vole populations and for this reason adopted several different ways of generating cycles. However, on the basis of our models and a consideration of the observed synchrony domains, we predict that the scale of the factors generating oscillatory dynamics are more likely to be local (e.g. disease or predators such as weasels Mustela nivalis) than regional (e.g. nomadic vole-eating raptors) or global (e.g. sunspots).


We thank the Natural Environment Research Council and the Scottish Executive Rural Affairs Department for funding under the NERC/SERAD Thematic Programme on Large-Scale Processes in Ecology and Hydrology (reference GST 02/1218), and Ottar Bjørnstad for helpful comments. We would also like to sincerely thank Forest Enterprise for allowing us to work in Kielder, for providing accommodation, and for access to the Forest Inventory data that we have used in our GIS analysis.

Received 23 January 1999; revision received 23 September 1999