The model

At time t, the position of a badger is given by and can be modelled by a stochastic differential equation defined as

(1)

where represents the drift term, is a real matrix that represents the diffusion term, and is a two-dimensional Wiener process. The Wiener process (alternatively called Brownian motion) is widely used, typically to describe a random, but continuous, motion of atoms or particles, but here has been extended to model the movements of animals. We assume that badgers move isotropically in space, i.e. the diffusion is the same in all directions. This implies that we can write the diffusion matrix as a constant multiplied by the identity matrix, i.e., for some value we can represent the diffusion matrix as

(2)

where is the identity matrix.

Estimating diffusion of badgers

The diffusion term in Eq (1) can be estimated using the Kramers–Moyal formula, given by

(3)

Because the data are collected as a time-series with discrete intervals, it is not possible to consider the limit as in Eq (3) (i.e., infinitely small time steps). However, an estimate can still be obtained using finite difference approximations. Since we assume that badgers move isotropically in space, we wish to calculate c in Eq (2).

We estimate c as follows: given m measurements at time t_i, with , let denote the time difference between two successive GPS measurements, then

(4)

Numerical investigations show that good approximations to c and thus σ can be found despite ignoring the limit and when is not constant. Since c is always a positive number (as we take the square root), it is found that c is log-normally distributed, which is confirmed when running a Shapiro–Wilk test for normality [32].

We fitted a generalised linear mixed-effects model (GLMM) to examine the relationship between the diffusion value c and various predictors including capture year (i.e. GPS-collar monitoring year), month, and sex. The model also included a random intercept for animals-within-site to account for repeated measurements on the same individuals. The model used a normal distribution with a log link function and was fit using penalised maximum likelihood estimation (PL). Statistical tests were implemented in Matlab [33].

Koopman operator and extended dynamic mode decomposition

We utilise the Koopman operator framework to analyse the dynamic behaviour of badger populations. We refer to the badger data as a dynamical system, which changes state over time according to a fixed rule, often described by differential equations such as Eq (1). The eigenvalues and eigenfunctions of the Koopman operator provide valuable insights into the behaviour and stability of these systems. We are interested in applying the Koopman operator to badger movement data to investigate metastability, which refers to the tendency of the population to remain in a region for extended periods before transitioning to an alternative region.

We consider discrete trajectories from Eq (1) with a state space , representing the two-dimensional geographic location of a badger. The evolution of these trajectories is governed by an operator (also referred to as the flow map) associated with the underlying dynamical system. In this context, the flow map describes how a badger’s position changes over time, essentially, it models how a badger moves from one location to another. More concretely, if is the badger’s position at an initial time (e.g., the first GPS collar reading), then applying the flow map gives its estimated position after a fixed period of time τ:

where τ represents a chosen time lag. This lag time can be interpreted as the interval between GPS observation, assuming those intervals are regular and consistent.

While the flow map F describes how a badger’s position evolves over time, the Koopman operator provides a complementary perspective by describing how measurable features of the system, such as speed, evolve. That is, whilst F acts directly on positions in the state space (i.e., ), the Koopman operator acts on functions defined over the state space. We define the Koopman operator for data from a stochastic process as

(5)

where x is an arbitrary point, denotes the expected value, and is a measurable quantity of the system such as the speed of a badger. Under certain conditions, the operator can be extended from to L₂ (i.e. so that it applies to a broader set of functions), allowing inner products to be used in Galerkin-like methods (see [34] for more details). Since the operator acts on functions, is infinite dimensional and linear even if F is finite dimensional and nonlinear.

Koopman analysis tries to find key measurement functions. The eigenfunctions and the corresponding eigenvalues λ of the Koopman operator, defined by

provide detailed insights into the underlying patterns and temporal evolution of the dynamical system. The eigenvalues and the associated eigenfunctions are of particular interest in the badger movement system, as they provide crucial insights into the slow dynamics and the metastable sets. Small eigenvalues govern faster time scales, such as the daily dynamics of the badgers. The number of dominant eigenvalues corresponds to the number of metastable sets within the system. In some cases, it can be challenging to determine the number of metastable states due to no clear spectral gap, which is the barrier between the faster and slower dynamical modes. That is, given N eigenvalues , a spectral gap is defined as the first such that

(6)

for some value of . See S3 Appendix, where we plot the eigenvalues and their successive differences against i and also a different method to find the spectral gap to compare.

Extended dynamic mode decomposition (EDMD) is a data-driven method to estimate the Koopman operator and its eigenvalues, eigenfunctions, and modes. The method requires a data set of snapshot pairs, i.e. , where each pair would consist of observations of the dynamical system such that . In other words, given time series data , then is the next observation in the time series. Each observation is a GPS-derived location of a badger, numbers of which vary among sites (n), from which a finite number of data points are collected m_i (for ), and the value of m varies from animal to animal. The coordinate data for the ith badger is represented as a vector, at time t_j, where . Ideally, the time intervals between observations in the time series are equal, as EDMD assumes evenly spaced data for accurate analysis. When the time intervals are irregular, interpolation is necessary to construct a uniformly spaced time series before applying the method. We organised the GPS data in the matrices ,

(7)

(8)

In simpler terms, the Y matrix is made up of the first pair of coordinates to the penultimate coordinates for each badger, and the matrix starts from the second entry of coordinates and includes all entries up to the last.

The method also requires a dictionary of observables ψ, , where N_K (the number of basis functions) is always finite-dimensional in practice because working with infinite-dimensional dictionaries is not feasible [34]. The function space itself, however, remains infinite-dimensional. Consequently, the operator learned from the data is an approximation, specifically a Galerkin projection. Here, indicator functions are used as the set of basis functions—a mathematical term referring to foundational building blocks for approximating complex functions—in EDMD, which is equivalent to using Ulam’s method to estimate a Markov state model [35]. Other basis functions such as monomials up to a certain order could be used. However, the resulting matrices might then become numerically ill-conditioned. The optimal choice of dictionary elements depends strongly on the dynamical system and remains an open question. When defining the domain for the EDMD indicator functions, we considered a box discretisation (i.e. we superimposed a grid over the area of coordinates). For consistency, the domain was divided into a fixed number of grid cells, requiring the number of boxes along each axis to be an integer. This resulted in box dimensions that were approximately 100 m by 100 m, with minor variation across years depending on the extent of the domain.

Let be the vector-valued function

(9)

for an arbitrary value of x. Then, we define the transformed data matrices as

(10)

(11)

The minimisation problem for the transformed data matrices becomes,

(12)

where ⁺ denotes the Moore–Penrose pseudoinverse of a matrix. The matrix is a representation of the Koopman operator projected onto the subspace spanned by the chosen basis functions, i.e. a Galerkin approximation.

Finally, let be the jth eigenvector of K with the eigenvalue , then the EDMD approximation of an eigenfunction of is

(13)

Once the Koopman eigenvalues and eigenfunctions have been computed, the number of dominant eigenvalues is chosen based on the spectral gap defined in Eq (6) and the corresponding eigenfunctions are clustered using k-means clustering [36]. For example, if there are eight dominant eigenvalues, the corresponding first eight eigenfunctions form clusters that define metastable states, representing the spatial areas that badgers rarely leave.

Data collection

We accessed three sets of data from studies where collars carrying GPS devices had been fitted to badgers. One dataset derived from a study of badgers resident in Woodchester Park, Gloucestershire, England [13,23], another related to four different sites in Cornwall, England [19], and the third was from a population in County Down, Northern Ireland [20].

The Woodchester Park Study was established in 1976 to investigate TB in a wild, naturally infected badger population [37]. The study includes analysis of capture-mark-recapture data, which involves trapping and re-trapping individually marked (tattooed) badgers to infer group membership [13], and the analysis of GPS data. The study site itself covers c. of predominantly permanent pasture and woodland [12]. When designing the data collection schedule at Woodchester, a 35-minute interval was chosen as a balance between collecting sufficient GPS fixes and preserving collar battery life. The collar was programmed to collect fixes from 20:00 to 04:00 daily. Due to the collar’s internal scheduling based on a midnight-to-midnight cycle, the last fix of one day is taken at 23:30, and the first fix of the next day is taken at 00:00. While this creates a 30-minute interval between these two fixes, it is a consequence of the scheduling boundary at midnight, not an intentional deviation from the 35-minute interval.

The four sites in Cornwall each included at least two beef farms and two dairy farms and were chosen to achieve representation of varying environmental conditions [19]. Sites C2 and C4 in North Cornwall, and F2 on the South Coast of West Cornwall are dominated by pastoral landscapes with wooded valleys, and site F1 on the North coast of West Cornwall is bounded by granite cliffs and moorland. The GPS units deployed on badgers recorded position fixes every 20 minutes between 18:00 and 06:00 while the badger was active.

The Northern Ireland study area in County Down, near the town of Banbridge, covers c. of primarily pasture and arable land [38]. The resident badger population was the subject of a Test and Vaccinate or Remove (TVR) project whereby captured animals were tested for TB using the Dual-Path Platform VetTB test (DPP) and test-positives were euthanised, whilst test-negative badgers were vaccinated against TB with Bacille Calmette Guerin, fitted with GPS collars and released [38–40]. The GPS units recorded position fixes once per hour between 21:00 and 04:00.

The various study areas contained different numbers of badger social groups and individuals fitted with GPS devices over a range of time periods (Table 1). The number of social groups was estimated using field knowledge specific to each site: in Woodchester, bait-marking data were used [41]; in Northern Ireland and Cornwall, capture-recapture and GPS collaring data were applied (Table 1 shows the total number of social groups monitored throughout the total GPS-collar monitoring period). GPS data in the form of locational fixes were not captured at a uniform rate within or among years of study (Fig 1), although most fixes were collected in the latter half of the year. Most collars were deployed in the summer/autumn (the peak season for badger trapping [42]), which means that the battery life waned during the first half of the following year, which overlaps with the closed season when trapping is suspended to prevent disturbance of reproductive females and their offspring [43]. Battery longevity is dependent on the frequency with which fixes are recorded. The temporal distribution of captured badgers (split by sex) for each site is shown in S3 Fig, highlighting variability in the number of individuals monitored (i.e., with active GPS collars) per month. At their respective peaks, Northern Ireland monitored twice as many animals as Woodchester, and up to six times more than the Cornwall sites.

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Table 1. Summary of site and badger monitoring across all study sites and years used in this study.

https://doi.org/10.1371/journal.pcbi.1013372.t001

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Fig 1. Summary of distribution of GPS fixes.

Number of fixes by month (left) and by capture year for respective sites (right). See S3 Fig, highlighting variability in the number of individuals monitored per month.

https://doi.org/10.1371/journal.pcbi.1013372.g001

To account for the impact of culling and other interventions across the different study sites, it is important to consider their varying management histories. At Woodchester, badger culling began on the farmland surrounding the core of the study area in 2018 (Woodchester Park belongs to the National Trust which has a no-cull policy), meaning one might expect change to the social structure in all years from 2018 onwards due to population reductions. In Cornwall, there was no active culling during the study period for sites F1, F2 and C2. Although site F1 was in a former treatment area of the Randomised Badger Culling Trial (RBCT) [44], potentially influencing badger dynamics through residual effects from past culling. The Cornwall site C4, however, was culled from 2016 onwards [18]. In Northern Ireland, the selective removal of TB-positive badgers after the first year of the study [38] might have disrupted social cohesion, prompting increased ranging behaviour or social group reconfigurations in response to changing population density. These variations in culling and TB management strategies must be considered when interpreting EDMD results and comparing them with field observations of badger social groups, as they likely played a role in the movement patterns observed.

Data cleaning and preparation

To avoid location errors, and to be consistent with the pre-cleaned data obtained from the Cornwall sites [19], we excluded all GPS-collar locations associated with fewer than four satellites, or with dilution of precision greater than four (as detailed in Supporting Information [25]) from the Woodchester and Northern Ireland data sets (where the parameters allowed). Moreover, improbable GPS locations (e.g., those far outside the study areas, or in water bodies), were excluded from all data sets. We chose to remove data derived from one Woodchester Park badger and five individuals from the Northern Ireland study because they were clearly isolated from the remaining data (see S1 Fig). In line with previous studies where badgers are recorded to have been moving at speeds of up to [11], we excluded any data points where badgers were found to be moving faster. Additionally, any single data points where the recorded movement within a time interval appeared biologically unrealistic for badgers were removed, such as travelling a distance of greater than 1500 m in two consecutive time steps (e.g. 70 minutes). As a result of these actions, the Woodchester dataset was reduced by 19.0%, the Northern Ireland data by 13.3%, and the data from the Cornwall sites was reduced by 1.1%. See S1 Appendix for the breakdown of reasoning for data removal.

The coordinates of the cleaned data were converted from the longitude/latitude (Woodchester and Northern Ireland) or British National Grid (Cornwall) into UTM (Universal Transverse Mercator) coordinates, so the distances between coordinates could be measured in meters. To preserve confidentiality of the location of sites (and to be consistent), the coordinates were shifted to a (0,0) base. For the estimation of the diffusion metric for badgers, the data can be used as presented. However, to account for the large distances between the last location of one night-time monitoring period and the first location of the next, during which time badgers were likely inactive in their underground setts, these intervals were excluded from the calculation. Furthermore, since the limit in Eq (3) cannot be taken, once the individual diffusion metric had been estimated, it was decided to exclude any individual’s measurements derived from ten or fewer data points, as an insufficient number of points hinders the accurate estimation of the diffusion metric.

Continuous and evenly spaced time series data are essential for EDMD, as the method assumes uniform time intervals between observations. However, GPS data for badgers often features irregular intervals, necessitating interpolation to create consistent time steps. These inconsistencies can arise due to undetected signals (e.g., when badgers are underground or beneath dense vegetation), timing issues with GPS devices (e.g., a 70-second window to connect with satellites), or large gaps caused by GPS collars losing power, which can result in days or months without data until the collar is replaced. To manage these gaps, new ID numbers were assigned to separate series of GPS fixes, ensuring continuity in the data. For instance, at Woodchester, GPS data were available from 66 unique badgers, but this was increased to 108 collar monitoring periods after accounting for interruptions. Recorded times (e.g., 20:01) were adjusted to the nearest pre-programmed time point (e.g., 20:00) to standardise the timing of locations. Notably, the time discrepancy reflects the duration between the start of GPS location calculation and the time of recording, which can vary across sites. At Woodchester Park, likely influenced by its high proportion of woodland, of badgers did not have their positions recorded at the designated time, with most locations logged within one minute of the intended capture time, but some taking up to three minutes. In contrast, only 5% of the badgers at the Cornwall sites failed to register their location within one minute. Most badgers in the Northern Ireland study area registered a location fix within sixty seconds of the appointed hourly time. Those that did not (see [45]) were excluded (2.5%) to facilitate interpolation. Although badgers are known to travel along complex paths (see [31]), linear interpolation was used to create evenly spaced intervals, closely matching the original time steps. This approach minimises the impact of interpolation on the data matrices used in EDMD.

For example, when locations were recorded every 35 minutes, such as in Woodchester, and a factor of 35 minutes is not chosen, we observe cutting corners and missing the actual recorded data point, resulting in calculating a shorter distance travelled by the badger. This underestimation is expected in any sampling of locations from continuous time, as linear interpolation inherently misses finer-scale movements and underestimates the true distance travelled. We found choosing a two-minute interpolation interval produced only small differences between consecutive positions, which were too subtle for EDMD to reliably detect underlying movement dynamics. At the same time, this high-frequency interpolation introduced artificial detail by generating multiple inferred locations between actual observations, thereby making more assumptions about the badger’s movement than were supported by the data. Using a 35-minute interval caused EDMD to fail in identifying non-trivial dynamics because the large time gap between points missed finer-scale transitions, oversimplified the system’s behaviour, and reduced the temporal resolution needed to capture complex, non-linear dynamics. For the Woodchester data, we found the optimal interval to be 17.5 minutes because it only assumes one location of the badger between observations, and EDMD was able to identify meaningful dynamics. For the Cornwall datasets, which comprised GPS fixes at 20-minute intervals, this was also found to be optimal. In contrast, the Northern Ireland dataset contained fixes recorded at hourly intervals, necessitating interpolation to create data points every 20 minutes for consistency with the Cornwall datasets. It is important to note that this approach means approximately two-thirds of the Northern Ireland dataset consists of interpolated values, which should be considered when interpreting the results.

Badger diffusion

The results of the GLMM analysis investigating badger diffusion patterns are presented in Tables 2 and 3 (see S2 Appendix for univariate analyses and S2 Fig for the data inputted into the model). The model included two fixed predictors: month and sex. Their respective estimates, standard errors, and confidence intervals are reported in Table 2. The model also allows for random slopes to account for variation in capture years across sites.

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Table 2. Fixed effect estimates from the final generalised linear mixed-effects model (GLMM) with a normal distribution and log link.

https://doi.org/10.1371/journal.pcbi.1013372.t002

Alternative model structures were explored, including configurations with both temporal variables (month and capture year) modelled as random slopes. While these models maintained a good fit (), their performance declined relative to the final model, as indicated by higher AIC values and less interpretable variance structures. Model comparison statistics, including AIC and , are provided in S1 Table. Two models appeared to perform best: the model presented in Tables 2 and 3, which includes random slopes for capture year only, and the model without capture year (the last model in S1 Table). While the first model outperforms the second model in terms of AIC (80.819 vs. 152.82), the second model outperforms the first model in terms of BIC (238.66 vs. 231.74). Hence, the fixed effects estimates for the second model have also been reported, in S2 Table, where it can be observed that the estimates are similar to those observed in Table 2. The decision to choose the first model was supported by the fact that capture years differed across sites due to varying management regimes, introducing meaningful heterogeneity best captured with random slopes on year. Diagnostic plots for residuals of the final model are shown in S4 Fig.

Monthly variation in diffusion rate was evident, with reduced movement observed from June to December relative to January. Notably, movement increased in February, with diffusion rates approximately 11% higher than in January (, p<.001), suggesting an early seasonal shift in activity. Sex was also a significant predictor of diffusion rate. Female badgers exhibited a reduction in movement compared to males, with an estimated 14% lower rate (, p<.001).

Random effects associated with capture year at the site level (Table 3) indicated modest variation in site-specific intercepts (SD = 0.068), but substantial variation in slopes across years (SD range = 0.197–0.261). This suggests that temporal trends differ across sites, some may exhibit increasing diffusion rates over time, while others may remain stable or decline. Additionally, we observe a strong correlation between capture years.

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Table 3. Random effects for Capture Year at the Site level. Standard deviations and correlations.

https://doi.org/10.1371/journal.pcbi.1013372.t003

The random intercept variance for animals within sites was 0.229, reflecting individual-level heterogeneity in movement. The residual standard deviation ( = 5.730) indicates that, despite the model’s complexity, unexplained variability in diffusion rates remains, likely attributable to unmeasured environmental or behavioural factors.

Extended dynamic mode decomposition

We employed EDMD to analyse the drift term in the stochastic differential equation describing badger movement behaviour. By extracting dominant modes and eigenvalues from observed trajectories, EDMD effectively approximated the drift component, which represents the average or expected rate of change in the system’s state.

Fig 2 displays the calculated metastable clusters for Woodchester across the full dataset and the individual years. Each panel shows badger location, with individual location fixes represented as circles, colour-coded according to their assigned metastable cluster. The purple lines represent the convex hulls of these metastable clusters, outlining the spatial extent of each cluster (with the area estimated next to it). The orange lines represent the convex hulls of social group home ranges derived directly from the raw movement data. A convex hull is the smallest convex polygon that contains all the points in the given set (calculated here using the convhull function in MATLAB). In this study, convex hulls provide a simple and interpretable way to define the spatial extent of badger activity within each cluster or social group. For further examples from the other sites, see S5 Fig.

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Fig 2. The calculated metastable clusters with areas of clusters for Woodchester.

The purple lines are the convex hull of the metastable clusters, the orange lines are the convex hull of the social group movement (home ranges) based on the data.

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Table 4 summarises the number of social groups identified at each site alongside the groupings calculated using EDMD. It also presents the mean home range area of the social groups and the mean area of the metastable clusters, both estimated from the area of their respective convex hulls. The amalgamated multi-year clustering provides a broad overview of the long-term trends in movement behaviour, where it can be observed that EDMD predicted fewer clusters than those observed via capture–mark–recapture. This pattern was most evident in Northern Ireland, where EDMD predicted 15 metastable clusters where 45 social groups had been identified by field observations, capture/recapture and GPS tracking.

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Table 4. Summary of badger social groups at each site and capture year. For each site, the number of observed social groups (left column) is shown alongside the number of clusters identified using extended dynamic mode decomposition (EDMD; right column). Bracketed values represent the mean home range area or cluster area in (standard deviation).

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Although some clusters appeared to be stable for periods of time, the number of metastable states can be seen to vary per year. For example, in years two to five at Woodchester (Fig 2), a cluster persisted in the central north of the study area (covering , , , and , respectively). Yet, during those same years, the estimated number of clusters fluctuated between one and two in the northeast. Similarly, EDMD identified four metastable clusters in the west of the study area throughout the capture period, but this reduced to one large cluster in the fourth year. Similar observations can be made in other sites, for example, in the Cornwall site C4 (S5 Fig), both the first and second capture year present similar clustering. When there is only one social group present in the data for the year (C2 and C4 during capture year three and two, respectively) then EDMD rightly picks up only one metastable state.