Table 1.
Time series and dates of GPS fixes, number of sheep and red deer used in this study.
Fig 1.
Analytical design to extract species interaction from time series of positional data.
Table 2.
Methods of constructing scalar time series from positional data of species using three data sets (Data 1, 2, 3, Fig 1) constructed using a modified Couzin model, each with different embedded intra- and inter-species interactions to test the following hypotheses Hi.
D: deer; S: sheep; H1. D → S: deer drives sheep; H2. S → D: sheep drives deer; H3. D ↛ S, S ↛ D: no interaction between species; ✔: results support hypothesis; ✖: results do not support hypothesis. Timespan: number of hours within spatial data area pooled for analysis (GPS fix acquisition time = 1 h). k: number of nearest neighbours in the analysis.
Fig 2.
Zones of repulsion, orientation and attraction for an animal within a two dimensions Couzin model. The individual is at position xi travelling in direction θi at speed v. The zone of repulsion is the region (in dark grey) of points within rr, the zone of orientation is the region (in light grey) of points between rr and ro of xi and the zone of attraction is points between ro and ra of xi.
Table 3.
Out-degree of summarized networks inferred using different methods of constructing scalar time series from positional data of individual animals, using three data sets (Data 1, 2, 3) constructed using a modified Couzin model, each with different embedded inter-species interactions (intra-species interaction is present and assumed and so not tested for) to test the following three interaction hypotheses Hi.
D: deer; S: sheep; H1. D↔S: deer drives sheep and sheep drives deer; H2. D→S: deer drives sheep; H3. D ↮ S: no interaction between species. Interaction is assessed by the connectedness of the inferred interaction networks, in particular, the out-degree of intra-species network edges to inter-species networks edges. Bold font used if hypothesis supported: successful detection of the interaction. Timespan: number of hours within spatial data area pooled for analysis (GPS fix acquisition time = 1 h). k: number of nearest neighbours in the analysis, either all other animals, or all animals of same species.
Fig 3.
Simulated data example network.
Inferred network for the Voronoi tesselation area transformation applied to test Data 2. Triangles denote vertices corresponding to deer and circles denote those vertices corresponding to sheep. Directed edges represent tail vertex Granger causing head vertex. In test Data 2 there should be no sheep → deer edges; only nine have been incorrectly inferred (red arrows). Although the exact relationship matrix for Data 2 has not been replicated, the overall propensity of inter and intra interactions have been captured, as reflected by the relative average out-degree of the different connection types (Table 3).
Fig 4.
Convex hull area time series for sheep using time intervals of 1 h, 2 h and 3 h. Starting time 04:00 h on 20th of October 2006. Multi-hour intervals provides a greater area and are smoother than convex hulls calculated at 1 h interval.
Fig 5.
Time series of mean 5th nearest neighbour distance for sheep and deer. First 300 h, starting time 15:00 h on 5th of October 2006.
Fig 6.
Autocorrelation function of the 5h CHA over 100 hours/lag for the five data segments of 2008 in Table 1, showing the evident daily 24 h cycle.
Fig 7.
Cross-correlation for the average distance to the k = 2 nearest neighbour transformation methods for lags between -100 and 100 h (data segment: 2006-S1). A significant cross-correlation for positive lag would indicate sheep influencing deer (sheep → deer), while for a negative lag it would indicate deer influencing sheep (deer → sheep). The bounds of the 95% confidence region are plotted horizontally in yellow, calculated non-parametrically by randomizing the order of the sheep time series and calculating the correlation 10000 times.
Table 4.
Granger causality test p-values (for order of model from q = 1 to q = 10) for 1h, 3 h and 5 h convex hull area.
Significant values are bold highlighted. See Table 2 for arrow acronyms.
Table 5.
Detected species interactions across data segments for the various time series transformations methods using Granger causality.
Interaction was judged to be detected if the p-values calculated were significant for at least three model orders (for q = 1 to 10) of the Granger causality tests. A → B means A Granger causes B, A ↔ B means symmetric interaction,—means less than three model order were significant hence no interaction detected.
Fig 8.
A visualization of the summarized network produced using conditional Granger causality to create edges using the distance to the k = 4 nearest neighbour of the same species time series for each individual animal within the 2006 (S1-S3) data set. Deer are represented by triangles, and sheep are represented as circles.
Fig 9.
The average degree for different types of inferred interaction in summarized networks of the 2006 (S1-S3) data set created using the k nearest neighbour transformation methods and conditional granger causality tests. The typically higher value for intra-species interaction suggests this is the stronger driver for animal behaviour.
Fig 10.
MDNN network shortest path length.
The average geodesic (shortest) path length for different types of inferred interaction in summarized networks of the 2006 (S1-S3) data set created using the k nearest neighbour transformation methods and conditional granger causality tests. Analogous to degree the typically lower value for intra-species interaction suggests this is the stronger driver for animal behaviour.
Table 6.
Suggested use of time series transformation methods in relation to the type of interactions aimed and the quality of the data available.