Introduction
Classic ecological theory assumes that population dynamics result from interacting organisms in time but in a non-spatial context (e.g., Lotka-Volterra model). However, these predictions are modified when accounting for restricted species movement by including space and dispersal (Levin 1974). When interactions between pairs of species, broadly fitting the definition of activator-inhibitor (such as predator-prey, parasite-host, etc.), result in local cycles, incorporating space and accounting for restricted dispersal can give rise to spatio-temporal patterns (de Roos et al . 1991; Bjørnstadet al . 2002; Sherratt 2001; Johnson et al . 2006). These dynamic spatial patterns can take various forms, ranging from chaos (Liet al . 2005) to perfect synchrony (Blasius et al . 1999) and much in between.
Causes of synchrony have been attributed to climate conditions (the Moran effect, e.g., Bogdziewicz et al . 2021), dispersal of individuals, and trophic interactions. While the Moran effect is often suggested as the cause of synchrony (e.g., Fay et al . 2020), microcosm experiments have strongly implicated an interaction between dispersal of organisms and their trophic interactions through the differential depletion of denser than average prey/host populations as a potent cause of synchrony (Vasseur & Fox, 2009; Fox et al . 2013).
Synchrony itself exists on a spectrum. Of note are periodic travelling waves (termed partial synchrony in Fig. 1B), whereby the oscillations of population cycles seemingly travel across space over time, either in a single constant direction (i.e., anisotropic, henceforth termed planar wave , e.g., Lambin et al . 1998; Berthier et al . 2014; Bjørnstad et al . 2002) or in all directions (i.e., isotropic; henceforth termed radial wave , e.g., Johnson et al . 2004), at a given speed. For population cycles linked via a travelling wave, all populations experience the same cycle, but do so at potentially different times. For such populations, with increasing distance between them, the cycle will become increasingly asynchronous until eventually returning to the same cycle phase. Conversely, a perfectly synchronised cycle (termed true synchronyin Fig. 1C) is merely one where the wave speed is practically infinite (Jepsen et al . 2016; Sherratt 2001). In a cycle with true synchrony, all populations in a landscape exhibit the same phase of the cycle simultaneously with no spatio-temporal lag. The opposing end of the synchrony spectrum would be populations that are disconnected and cycle completely independently of each other (termed true asynchrony in Fig. 1A).
While travelling waves appear to be routinely detected when datasets are sufficient, there remains much uncertainty. Namely, what form a travelling wave will take when spreading across a natural landscape, what features determine the source location(s) of the wave(s), and whether activator-inhibitor dynamics play a role in the underlying mechanisms generating travelling waves?
Theoretical simulations of travelling waves unfolding in homogenous landscapes suggest the spread should be radial. However, real world landscapes include habitat heterogeneity (but see Johnson et al . 2006). Intriguingly, spatial inhomogeneity can lead to the formation of both radial and planar waves via either variation in productivity, connectivity or interactions with dispersal. However, theoretical work explicitly investigating the role that heterogenous landscapes have on travelling waves, via inclusion of physical features (e.g. lakes), suggest that waves may originate from these structures with an imparted directionality (Sherratt et al . 2002; Sherratt et al . 2003). If the feature preventing isotropic dispersal is itself linear, then the resulting form of the wave would be expected to be planar. Because heterogeneities are ubiquitous in real landscapes and affect both dispersal and productivity, theory offers no prediction on what pattern should unfold in any real-world landscapes and arguments on any match between empirical pattern and theory have been post-hoc.
Empirical research projects, which by their nature occur in heterogenous environments, have often used planar wave parameterisations to describe the observed travelling waves in cyclic populations (Lambin et al . 1998; Berthier et al . 2014). Such a mismatch between generally predicted (i.e., radial) and observed (i.e., planar) patterns may have two interpretations. The first may be that the apparent planar waves are simply a feature of observing a radial wave at too small a spatial scale (feasible given the substantial data requirements [Koenig 1999]). Alternatively, observed planar waves may reflect real world conditions which some simulations fail to account for (e.g. heterogenous landscapes with regards to the distribution of both habitats and organisms). Thus, true planar waves may arise due to approximately linear physical features in the landscape. Building on Sherratt and Smith’s (2008) theoretical work, which suggested physical features may generate travelling waves, Berthier et al . (2014) invoked quasi-linear physical features in their landscape as being potentially responsible for planar waves in cyclic montane water vole populations. However, because of necessary theoretical assumptions for how physical features interact with organisms (resulting in boundary conditions which are hard to quantify empirically) Berthier et al . (2014) could not ascertain which of two plausible features were responsible. This reflects the challenge of translating theoretical assumptions into real-world characteristics and vice-versa.
An alternative to physical features generating travelling waves is the suggestion that they are generated in foci with particular features. Such features include: areas with high densities (Bulgrim et al . 1996); areas where predators were introduced (Sherratt et al . 1997; Gurney et al . 1998; Sherratt et al . 2000; Sherratt 2001; Sherratt 2016) and areas of high population connectivity or habitat quality (Johnson et al . 2006). The epicentre hypothesis posits that travelling waves recurrently form via epicentres. These epicentres reflect regions in space with defined characteristics (e.g., highly connected populations in high quality habitats) that give rise to waves. Johnson et al . (2006) invoked the epicentre hypothesis to explain travelling waves in cyclic larch bud moths (Johnson et al . 2004), whereby they proposed that waves emanate from regions with high-quality, well-connected populations which then spread to more distant populations, resulting in partially synchronous cycles.
Related to uncertainties with what generates a wave, is the ambiguity of theory on resulting direction of travel relative to the source. It has been suggested for the larch budmoth that waves travel outwards from epicentres (Johnson et al . 2006), resulting in expanding radial travelling waves. Conversely, alternative studies have suggested the opposite may occur, whereby waves begin at hostile environment boundaries (i.e. where individuals die if entered) and contract inwards towards a central location (Sherratt 2003; Sherratt & Smith 2008). There has been no empirical research with an analytical approach that explicitly tested for such expanding or contracting waves.
As a wave spreads, via dispersal and trophic interaction (Vasseur & Fox 2009), the cycle spreads across a landscape from one population to the next, each population in turn experiences the same cyclical successions of activation or inhibition of growth rates. Such changes to a population’s growth rate are, in part, dependent on neighbouring populations. For instance, inhibition may represent the spread of agents such as pathogens or predators from one population to the next, resulting in local populations being suppressed as the respective wave passes. Theoretical expectations of travelling waves have been supported by empirical evidence from a variety of fields, all of which can be considered to have such activator-inhibitor relationships; e.g., herbivore-plant, predator-prey, parasite-host, (Lambin et al . 1998; Moss et al . 2000; Johnson et al. 2004; Biermanet al . 2006; Mackinnon et al . 2008; Berthier et al . 2014), susceptible-recovered, (Grenfell et al . 2001; Cummingset al . 2004), death and regeneration (Sprugel 1976), and cellular biochemistry (Müller et al . 1998; Bailles et al . 2019) amongst others. Within such systems, it is the cumulative impact of both activator and inhibitor that gives rise to the overall cyclic pattern.
The conceptualisation of population cycles as activation-inhibition, as well as the wealth of theoretical literature considering the role of such activation and inhibition accompanied by restricted dispersal in spatial patterns (Levin 1974; de Roos et al . 1991; Bjørnstadet al . 2002; Sherratt et al . 2000; Sherratt 2001; Johnsonet al . 2006) implies that statistical representation of empirical data might decompose the overall pattern in growth and retrieve evidence of two contributing travelling waves, promoting and inhibiting growth, respectively, as found in non-ecological travelling waves (Kapustinaet al . 2013; Martinet et al . 2017). Additionally, the interplay between dispersal abilities of activator and inhibitor have been suggested as a component which leads to the formation of radial and planar waves (Johnson et al. 2006).
Building on exceptional data, this study evaluates a suite of hypotheses, which are flexible phenomenological descriptions of travelling waves, representing theoretical or empirical work or their logical extensions. Given the richness of our dataset, we are able to lessen requirements for simplified caricatures and consider more complex forms. Our approach considers an initial demarcation between radial and planar waves, including whether the radial waves contract or expand. These hypotheses are further divided to represent either a single travelling wave or multiple (as simulated in Johnson et al . 2006), each in turn split into whether multiple waves are isolated from each other by physical features or coalesce into a single pattern reflecting activator-inhibitor dynamics. To do so, we used abundance indices of a rodent crop pest from a study site spanning > 35,000 km2 over seven years. We find evidence of a single cumulative spatio-temporal pattern consisting of two expanding radial travelling waves, which we propose may arise due to activator-inhibitor dynamics.
Materials and Methods
Study species
The common vole (Microtus arvalis) is a small rodent inhabiting natural grasslands and agricultural ecosystems in Europe. It is prey for both specialist and generalist predators alike (Mougeot et al . 2019) and is the host of multiple direct and vector transmitted pathogens (Rodríguez-Pastor et al . 2019). Common voles are a frequent farmland pest causing both crop damages and disease spillovers during population outbreaks that occur every 3-4 years (Jacob & Tkadlec 2010; Mougeot et al . 2019; Rodríguez-Pastor et al . 2019). Common voles have been extensively monitored for pest management across our study site (> 35,000 km2) since 2011.
Study site
We (ITACyL) collected data on vole abundances in Castilla-y-León (CyL), NW Spain. CyL is a large (94,226 km2), relatively flat semi-arid agro-steppe plateau encircled by mountains and bisected east to west by the ca. 25-150 m wide Duero River (Fig. 2). As a result of land-use changes (ca. the 1970s), common voles colonised the plateau from the adjacent mountain ranges in the north, east and south (Luque-Larena et al . 2013, Jareño et al . 2015). Within the wider region, common voles are believed to occur at higher densities within the plateau than in the surrounding mountains, likely due to the region’s agricultural practices (Roos et al . 2019). A particular area in the centre of CyL (Tierra de Campos) is known to practitioners as problematic due to early, large or persistent outbreaks.
While not a perfectly homogenous landscape, the plateau likely presents a ”best real-world match” for conditions used in most theoretical research, which do not account for landscape features (but see Sherratt & Smith 2008). However, there are two complicating physical features: the Duero river and surrounding mountain ranges. If physical features are related to the form of a wave, we may expect either planar waves that travel north and south due to the river or a contracting radial wave resulting from the encircling mountains.
Data collection
We made use of a widely employed calibrated abundance index method, based on vole presence, to monitor vole abundance at large spatial scales (Roos et al . 2019; Jareño et al . 2014). Transects, up to 99 m in length (dependent on the field’s length), were carried out in linear stable landscape features (field, track or ditch margins) to estimate vole abundance from winter 2011 until autumn 2017 (\(n\ =\ 42,973\)). Margins are known to be reservoir habitats for voles, where from voles colonise adjacent fields during outbreak periods (Rodríguez-Pastor et al . 2016). Each transect was divided into 3 m sections (33 in total), with each section noting the presence or absence of one or more signs of vole activity (i.e., latrines by burrows, fresh vegetation clippings, and recent burrow excavations). The proportion of sections with signs of vole presence per transect was then used as the abundance index. The number of surveys carried out at any time varied adaptively with the perceived risk of an outbreak (according to changes in estimated abundance in previous monitoring surveys).
Analyses of travelling waves typically use some measure that can detrend from long-term temporal trends and autocorrelation, such as phase angle or log difference growth rates (Liebhold et al . 2004; Vindstadet al . 2019). As such, the response variable typically used in all models is proportional growth rate (\(r_{t}=ln\left(N_{t+1}\right)ln\left(N_{t}\right)\), where\(N\) is the abundance index at time \(t\) (Royama 1992; Berryman 2002). A benefit of using \(r_{t}\), rather than \(\ln\left(N_{t}\right)\), is that any multiplicative effects of site quality are cancelled out, provided they are constant over time. To calculate \(r_{t}\), vole abundance indices are required at the same location in subsequent time periods (i.e., and \(t+1\)). To achieve this, transects were temporally aggregated into a respective yearly quarter (e.g., January to March 2014). Transects were then spatially aggregated by using an arbitrarily chosen transect as a reference point and assigning all transects within a 5 km radius to the \(i^{\text{th}}\) centroid, with any transect only assigned to a single centroid. Once complete, the mean Julian day, X and Y UTM (Universal Transverse Mercator) were calculated for each centroid.
We chose three-month intervals and 5 km centroids as these time periods and spatial scales maximised the number of centroids with successive abundance indices, increasing the number of growth rate estimates that could be calculated. A constant of 3.03 was added to \(N\) to avoid zero entries (3.03 was the lowest non-zero value of N observed). The final dataset consisted of 3,751 observations ofrit (SFig 1).
Analysis
Bespoke models were constructed for all considered parameterisations of the travelling waves (summarised in Fig. 1 and Table 1), based on previous models of travelling waves (Lambin et al . 1998; Mosset al . 2000; Berthier et al . 2014). All of the travelling wave models contained at least three components. The first component estimated distance from either a reference planar direction or an epicentre location (Distance equation, Table 1). The next used these distances and converted them to a space-modified time variable (Space-modified time equation , Table 1), itself then used in a GAM (generalised additive models) to explain growth rates (Growth equation , Table 1). These models reflect various ways to modify space and time so that the dynamics at each location can be explained by one or two underlying cycles (see Fig. 1). The parameters defining the space-modified time variables of the travelling waves were estimated using a stochastic annealing (SANN) optimiser (Bolker 2008), using 15,000 iterations for each model. SANN initial values were determined using a direct search method to crudely characterise the parameter space. Conditional on the values of the space-modified time variables, the underlying cycles were fitted using GAMs as described below.
Three versions of a ”null” model (i.e., no travelling wave pattern) were included in the analysis and fit using generalised additive models alone. These included a true null model (N1), a model which assumed true synchrony (N2), and a final model which proposed growth rates were explained by space alone (N3) (Table 1).
All GAM components (Growth equation , Table 1) assumed a Normal distribution for random errors and included a weighting term. The weighting term was the square root of the differential in surveys from a centroid (\(\mathcal{w}_{t,\mathcal{i}}=\sqrt{\frac{n_{t,i}\times\ n_{t-1,i}}{n_{t,i}+\ n_{t-1,i}}}\), where \(\mathcal{w}_{\mathcal{i}}\) is the weights for centroid iat time t, and \(n\) is the number of surveys in time tand t-1 ). The term sought to account for observation variance caused by the adaptive vole monitoring intensity, whereby the number of transects varied over time (transects per centroid ranged from 2 to 111, with a mean of 18.5). The appropriateness of the weight term was checked by plotting model residuals against the weighted term.
All bespoke models reflected either radial or planar waves parameterisations. Models P, RE and RC were the simplest and included either a single planar (P), expanding radial (RE), or contracting radial (RC) travelling wave. A further suite of models assumed the presence of two spatially isolated, i.e., non-interfering, waves separated by the Duero river, with the waves being either planar (PF), expanding radial (RFE), or contracting radial (RFC). The potential for a single pattern informed by dual additive, overlapping waves was captured by allowing models to have two waves, either planar (PD), expanding radial (RDE) or contracting radial (RDC) waves. Compared to the models with two waves isolated by a physical feature, these models assumed that both waves influenced all populations in the landscape. This suite of models represents various predicted forms of travelling waves and some logical extensions to ensure a number of candidate models are considered. Given the richness of our data, the panel of models considered extends previous research that has generally used a single form or descriptive methods that could not rule out competing hypotheses. Using this approach, we can quantitively assess which description of the spatio-temporal patterns are most supported by our data. Parameterisations of each travelling wave model are included in Table 1.
The final model was chosen based on parsimony considerations using ΔAIC and the corresponding hypothesis selected over the alternatives (see supplementary material 1 for each model’s AIC values). AIC, as reported by the final GAM, was adjusted to incorporate the additional number of wave parameters as;
\begin{equation} Adjusted\ AIC=AIC+2K\nonumber \\ \end{equation}
where K is the number of wave parameters (table 1).
Confidence profiles for each parameter were determined using profiling as described in Bolker (2008). All analyses and visualisations were carried out in R version 4.0.2 (R core team 2020) using the mgcv(Wood 2011), emdbook (Bolker, 2020), ggplot2 (Wickham 2016) and patchwork (Pedersen 2020) packages. The code used for the analysis is embedded in supplementary material 1.
To determine the statistical method’s effectiveness at retrieving known parameter values, we carried out a brief simulation study, available in supplementary material 2. Model assumption checks, residual plots, and summaries of each model are included in supplementary material 3.
Results
The null models (N1, N2, and N3) were discarded through model selection (see supplementary material 1), indicating that it is unlikely that there was true synchrony (N2), or that the observed growth rates are related to static environmental conditions (N3). The relative lack of support for N2 (true synchrony) provides evidence that large scale true synchrony is not the pattern characterising our dataset.
Of the models which assumed the presence of travelling waves, RDE (dual expanding radial travelling waves) was selected, with the next most parsimonious model (dual contracting radial, RDC) having a\(\Delta AIC\ =\ 53.2\). The final model had epicentres estimated 75.2 km apart (Fig. 2, Table 2). The first was in a well-known problematic area with higher-than-average vole abundances, with recurrent and severe outbreaks (Tierra de Campos ). In contrast, the second was positioned further southeast, in an area that experiences lower than average abundances (see SFig. 2 for a Gi*cluster analysis of the 42,973 abundance indices).
Additionally, when plotting the predicted growth rates on the space-modified time variables, the possibility that the overall pattern can be decomposed into possible activator and inhibitor influences (themselves, phenomenological statistical descriptions) on vole population growth is suggested; the first, slow wave predominantly inhibited growth and was estimated to travel radially at 148 km per year, while the second, faster wave was estimate to travel radially at 835 km per year and generally promoted growth (Fig. 3). When the effects of both of these waves are visualised over true space and time, the cumulative spatio-temporal pattern becomes apparent (Video 1) and the speed at which it traverses the region is approximately 0.9 km per day (or 329 km per year, calculated by extracting the furthest south predictions where \(r_{t}>0.5\) [i.e., the wave front] at two arbitrarily chosen times, then calculating the distance between those and dividing by the difference in time).
Discussion
We find clear evidence of a self-organising spatio-temporal pattern in the population growth rates of common voles in Castilla-y-León, resulting from two travelling waves spreading radially at contrasting speeds. Further, in line with Johnson et al . (2004), we find that the pattern in CyL is best approximated as two expanding radial travelling waves. However, the waves detected here are not independent as in Johnson et al . (2004), instead acting additively as activator and inhibitor, suggesting they may be more than phenomenological descriptions of an overall pattern. The dual expanding, fast and slow radial travelling waves, suggesting activator and inhibitor dynamics respectively, are of a form not previously observed in the empirical literature but are in line with the fundamental interactions of activators (e.g., host) and inhibitors (e.g., pathogens) in population cycles. Such activation and inhibition, and their spatial diffusion are similarly believed to be the process generating synchrony (Vasseur & Fox 2009). Further, activator and inhibitor dynamics are inherently included in travelling wave simulations. As such, we find convergence between understandings of synchrony, travelling waves and population cycles.
True synchrony or partial synchrony?
While we refer to the population cycle of common voles in CyL as partially synchronous, various studies have apparently demonstrated that cyclic populations, both of common voles and other cyclic species, occur synchronously. To understand this apparent contradiction, it is important to note that synchrony occurs, not as a dichotomous state but as a spectrum (Koenig 1999; Bjørnstad et al . 1999, see Fig. 1). Nevertheless, the dichotomous representation of synchrony has led to an approach whereby evidence of synchrony (notably synchrony which decays with distance) can be perceived as evidence, or lack-there-of, of true synchrony (Smith 1983; Andersson & Jonasson 1986; Erlinge et al . 1999; Huitu et al . 2003; Sundell et al . 2004; Lambinet al . 2006; Huitu et al . 2008; Fay et al . 2020;). The use of the terms “synchrony” and “asynchrony”, which implies a dichotomous state, may lead to the view that there are no nuanced forms of synchrony.
If travelling waves are ubiquitous in cyclic populations, a crucial component to detecting such nuanced forms of synchrony, overcome in the present study, is the requirement for a vast amount of data to distinguish between more subtle forms (Koenig 1999). Early descriptions of population cycle synchrony were largely limited to qualitative assessments, where populations were deemed synchronous if they peaked sometime in the same year (e.g., Andersson & Jonasson 1986). While such qualitative assessments of synchrony may reflect genuine true synchrony, they likely suffer from temporal aggregations, i.e., where population synchrony is deemed to have occurred because the same phase is experienced within the same broad period of time (see SFig. 3 for an example of where a travelling wave could be misconstrued as true synchrony using a qualitative approach). While research on synchrony has become more quantitative, some subsequent attempts to characterise synchrony have suffered from similar issues, namely, a lack of either or both spatial and temporal resolution (Koenig, 1999).
Perhaps owing to the long history of time series use in population cycle literature, many datasets which test for synchrony generally last for a long period of time (e.g., 21 years in Huitu et al . 2003). However, even in long term datasets, the temporal resolution can be severely limited. For instance, Sundell et al . (2004) used the annual breeding output of raptors in 50 km x 50 km areas across Finland, as a vole abundance index to characterise synchrony across the country. While such datasets are likely able to determine if true asynchrony or true synchrony are better supported (e.g., peaks occur in the same year), they seem ill-suited for detecting more subtle forms of synchrony as any signal of a within year spatio-temporal delay in synchrony (e.g., a travelling wave) would be obscured.
While such issues surrounding temporal resolution and aggregation may mask more subtle forms of synchrony, such as travelling waves, a lack of spatial resolution is perhaps equally detrimental. Indeed, in many instances, population synchrony has been characterised using far fewer spatial replicates than those used in this analysis (Huitu et al . 2003; Lambin et al . 2006; Huitu et al . 2008). In such cases of comparatively low spatial resolution, as with studies with a low temporal resolution, the result may be an ability to distinguish between the two extremes of synchrony but an inability to explore where a metapopulation exists on the spectrum of synchrony.
Indeed, whenever spatio-temporal datasets have been rich in both spatial and temporal resolution, the outcome appears to be the detection of travelling waves, irrespective of the method used (Lambin et al . 1998; Cummings et al . 2004; Johnson et al . 2004; Grenfellet al . 2013; Berthier et al . 2014). Such datasets tend to exist only for species with public health or economic interests, such as pest species (Bjørnstad 2001), which may, in part, explain the relatively few examples of travelling waves in the literature compared to detections of apparent true synchrony. However, if the waves captured here do represent activator-inhibitor dynamics and their dispersal, it is logical to assume that all cyclic systems are synchronised via travelling waves, which are only subsequently modified more or less by the Moran effect (Hugueny 2006).
Activator-inhibitor waves
Given the long history of using activator-inhibitor systems to model population cycles (e.g., Levin 1974), as well as the finding that trophic interactions and dispersal promote synchrony, our findings, which suggest the presence of activator and inhibitor travelling waves, provide some measure of consistency between understandings of synchrony and population cycle theory (Bjørnstad 2001; Bierman et al . 2006). Such activator-inhibitor travelling waves have previously been detected in cellular biology (Kapustina et al . 2013; Martinetet al . 2017) but not in ecology.
Our results are, to our knowledge, the first instance where a single spatio-temporal pattern of population cycles has been decomposed into constituent parts, which we propose represent the influences of activator and inhibitor on population vole growth. Microcosm experiments investigating the effects of dispersal and trophic interactions (and the Moran effect) found that the synchronising effect of dispersal in the presence of predation led to greater synchrony in population cycles of protists (Vasseur & Fox 2009), suggesting that the two waves here may partly represent the synchronising effects of dispersal of voles, dispersal of inhibitors (possibly pathogens or predators) and the interactions between them. Indeed, a potential candidate agent for an inhibitor, pathogens, are known to spread via travelling waves (Grenfellet al . 2001; Cummings et al . 2004).
The presence of two epicentres is in line with previous research on travelling waves (Johnson et al . 2004), though the finding that final cumulative pattern is dependent on both epicentres, with apparently distinct roles (i.e., activation and inhibition of growth rates, Fig. 3) are new to the field. The positioning of the epicentres, estimated as distinct locations with no overlap in the 95% CI, may provide some support for the interpretation of activator and inhibitor dynamics. The estimated location of the inhibitor epicentre is in an area with higher-than-average abundances of voles (Tierra de Campos , see SFig. 2). This region is known locally to practitioners for recurrently experiencing severe outbreaks, which may be related to farming practices which are more suitable for voles (Roos et al . 2019). Such a location would present an area consistent with understandings of where travelling waves of diseases initiate, as pathogen travelling waves have been found to originate in areas of high density (Grenfell et al . 2001; Cummings et al . 2004). If so, this epicentre may represent the starting location for the outward spread of pathogens because of infected dispersing individuals which serve to inhibit growth rates of voles. A testable hypothesis would be that this region experiences a higher proportion of infected individuals compared to a regional average. Indeed, two pathogens, Tularemia and bartonella, are known to occur in a density dependent relationship with vole densities in Tierra de Campos (Rodríguez-Pastor et al . 2017). A consequence of being reliant on dispersers for the propagation of the disease, in combination with various delaying processes (e.g., latency to infection), is that we would expect that the speed of the inhibitor to be slower than the activator speed, which we observe (Table 2).
Conversely, the activator epicentre was located in a lower-than-average abundance region (see SFig. 2). We propose that this may be due to a slight adjustment to the epicentre hypothesis as described in Johnsonet al . (2006). The epicentre hypothesis posits that emigration between close suitable habitats causes travelling waves. We consider that our epicentre meets these requirements in all but “suitable habitat” (i.e. lower-than-average densities). However, given the high reproductive capacity of common voles, we would assume they are able to produce as many offspring in a “less-suitable habitat” as elsewhere in the region, but that most of these offspring become emigrants. In this light, the core understanding of the epicentre hypothesis is maintained, where an epicentre is a location producing many emigrants, but altering it to take into account the reproductive ability of common voles which we do not believe has influential spatial variation. Evidence of this would come from finding a higher-than-average proportion of dispersers at this location.
The speed of the inhibitor wave was estimated at 147 km per year, while the activator was estimated at 835 km per year, which appear to be middle-ground speed estimates amongst empirical travelling wave literature (which vary from a minimum of 7-8 km per year [Berthieret al . 2014] to a maximal 1,776 km per year [Cummingset al . 2006]). Differences in speed offers some confirmation with simulations (Johnson et al . 2006), where differences in activator-inhibitor dispersal abilities was found to result in radial travelling waves. We propose that pathogen (i.e., a possible inhibitor) diffusion would be dependent on, host dispersal, mode of transmission, latency to infection, and so forth, all possible means to impart a delay in the spread to adjoining populations. Conversely, we believe that the fast speed of the activator wave may reflect the relative ease at which voles are able to disperse (i.e., habitat connectivity, where CyL is criss-crossed by field margins) or the effectiveness of a dispersal event (related to density).
Our modelling has demonstrated evidence for both activator and inhibitor influences on population growth rates in voles. Further work is required to establish the processes underlying these influences, and to collect sufficient large-scale data on other ecological systems to establish whether these too are underpinned by activator and inhibitor influences.