Animal movement paths display substantial complexity and variability, promoting efforts to identify universal rules and models that best describe them. Using high-resolution (≥ 10 Hz) movement from 43 vertebrate species spanning diverse taxa, body sizes, and lifestyles, we show that paths are universally composed of straight-line steps interspersed with sharp turns, echoing patterns documented in lower taxa such as bacteria. We report how vertebrate “fundamental steps” - straight travel segments between successive detected turns (with F_stepduration as the turn-to-turn interval and F_steplength as the corresponding distance when displacement is available) - and “fundamental turn angles” (F_turnangles; net changes in travel heading between successive steps) vary with species’ mass, locomotor mode, behaviour, and environment. Here, “fundamental” denotes the finest-scale step/turn events resolvable under our sampling rate and turn-detection criteria; these event-scale steps/turns are intrinsically different from the straight-line segments inferred from low-resolution position data. To explain these relationships, we posit that animals inherently move in a straight line until sensory information signals a better heading, triggering a turn. Across all species examined, animals spent the vast majority of their travel time moving in straight lines (species-level means >90%), with turns representing discrete decision points influenced by body size, locomotor mode, and ecological context. Larger animals turned less frequently, consistent with biomechanical constraints of mass and rotational inertia, while aerial species often exhibited higher turning rates driven by soaring flight demands. We further show that turns can be linked to diverse behavioural drivers, including prey pursuit, obstacle avoidance, predator evasion and exploitation of environmental energy. By explicitly quantifying turns, we clarify how distributions of step durations and turn angles interact to shape movement patterns and why different statistical models (e.g., correlated random walks, Lévy flights) emerge when lower-resolution data are analysed. Finally, we demonstrate how fundamental steps and turns can be incorporated into an agent-based modelling framework using penguins as a case study, enabling reconstruction of realistic tracks and prediction of movement responses to environmental change. Straight-line travel punctuated by decision-driven turns thus emerges as a fundamental principle of vertebrate movement, linking fine-scale movement structure, ecological context, and emergent patterns of space use.

Tag deployment details and analysis

We deployed Daily Diary (DD) tags (Wilson et al. 2008) on 43 species (15 birds, 3 fish, 3 reptiles, and 22 mammals) covering a size range of 0.3-10,000 kg to obtain data on their movement patterns. The tags contained tri-axial magnetometers, tri-axial accelerometers, and pressure sensors (Wilson et al. 2008), allowing travel headings over time to be deduced (Gunner et al. 2021a). Sampling rates ranged from 10 to 40 Hz, and derived tracks were sub-sampled to either 10 or 20 Hz for analysis.

Tag data were first processed to exclude periods immediately following the tagging process to minimize the probability of potential tagging effects; this exclusion period typically lasted a few days. Data were then examined to identify extended periods of active movement (excluding non-traveling movement behaviour), noting the travel medium: air, water, or on land. From these periods, a single continuous segment (usually between 5 and 36 h) was analyzed per individual animal. The variation in duration was entirely due to the variation in activity patterns among species. For example, pine martens were typically only continuously active for a few hours a day, whereas sharks swam continuously. The compass and acceleration data were used to calculate headings following methods described in Gunner et al. (2021a). Additionally, for select case studies, absolute animal locations over time were determined via dead-reckoning, incorporating periodic verified locations obtained through co-deployed GPS to correct for drift (Gunner et al. 2021b). In these instances, step lengths could be used to quantify movement patterns. Step lengths can be interpreted as the integral of speed over each step duration, while turn angles quantify changes in movement direction between successive steps.

Variations in turns within the heading data were identified using the protocol defined by Potts et al. (2018). These variations ranged from an average minimum of 10 turns per hour across individuals for whale sharks to an average maximum of 207 turns per hour for the European pine marten. Briefly, the algorithm detects changes in the heading by sliding a small window across the time-series of headings and calculating the squared circular standard deviation (SCSD) within the window. Spikes in SCSD indicate turns, and candidate turns were filtered based on achieving a threshold turn angle of 30° for all species (Potts et al. 2018, Munden et al. 2021) within a species-specific time window (see Potts et al. (2018) and Appendix S1: Section S2 for details). We define a “fundamental step” as the interval between successive detected turning points (a turn-to-turn segment) during which travel heading is approximately stable; F_stepduration is the elapsed time of this interval. Where speed or dead-reckoned displacement is available, F_steplength is the distance traveled during the same interval. Each detected turning point yields a fundamental turn angle (F_turnangle), defined as the net change in travel heading between successive steps.

Agent-based modeling

We developed an agent-based model to predict the movement paths of Magellanic penguins (Spheniscus magellanicus) foraging at sea, by utilizing empirical data from GPS-corrected dead-reckoned tracks of 27 individuals, including fundamental step length (F_steplength), turn angle (F_turnangle), and associated compass heading (H). The foraging landscape was divided into equally spaced grid cells of area 10 km², enabling the calculation of unique frequency distributions (Empirical Cumulative Distribution Functions, ECDFs) for F_steplength, F_turnangle, and H within each grid cell. These distributions were further segmented by journey phase, defined as either outbound or inbound. The transition from outbound to inbound was identified by the onset of a continuous downward gradient in the penguins’ cumulative shortest distance to the colony. At the beginning of the simulation, each agent’s movement parameters (F_steplength, F_turnangle, and H) were initialized based on the outbound ECDF distributions from the grid cell corresponding to their departure point near the colony. Each agent navigated a virtual 2D spherical coordinate system. At each time step, F_steplength and F_turnangle were randomly drawn from their respective ECDFs, and the direction of each turn was selected to align with the sampled H. At each movement step, the agent was dead-reckoned using this information to recalculate its geographical position and cumulative (Haversine) distance traveled. Upon entering a new grid cell, the agent recalibrated its F_steplength, F_turnangle, and H distributions using the ECDF data specific to that grid cell. When the cumulative distance reached a user-defined proportion of the maximum distance threshold, the agent updated its grid cell ECDF information to reflect the inbound phase of the foraging trip. The simulation continued until each agent had traversed a user-defined total cumulative distance traveled.

Users can specify options, including (but not limited to) restricting the F_steplength and F_turnangle distributions to a specific quantile range, adding first-order autocorrelation to movement metrics, refining journey-phase segmentation of grid-cell ECDFs, filling empty grid cells with information from neighboring grids within a given radius or from global distribution values, and selecting from multiple heading adjustment strategies to balance turn angle and heading.

Agent-based modelling code is archived on Zenodo: https://doi.org/10.5281/zenodo.18909038 (GitHub development repo: https://github.com/Richard6195/Agent-Based_Fundamental_Steps_and_Turns_Simulation-).

References

Wilson, R. P., Shepard, E. L. C., & Liebsch, N. (2008). Prying into the intimate details of animal lives: use of a daily diary on animals. Endangered species research, 4, 123-137. https://doi.org/10.3354/esr00064

Gunner, R. M., Holton, M. D., Scantlebury, D. M., et al. (2021a). Dead-reckoning animal movements in R: a reappraisal using Gundog.Tracks. Animal Biotelemetry, 9, 23. https://doi.org/10.1186/s40317-021-00245-z

Gunner, R. M., Holton, M. D., Scantlebury, D. M., et al. (2021b). How often should dead-reckoned animal movement paths be corrected for drift? Animal Biotelemetry, 9, 43. https://doi.org/10.1186/s40317-021-00265-9

Potts, J. R., Börger, L., Scantlebury, D. M., et al. (2018). Finding turning-points in ultra-high-resolution animal movement data. Methods in Ecology and Evolution, 9, 2091–2101. https://doi.org/10.1111/2041-210X.13056

Munden, R., Börger, L., Wilson, R. P., et al. (2021). Why did the animal turn? Time-varying step selection analysis for inference between observed turning-points in high frequency data. Methods in Ecology and Evolution, 12, 921–932. https://doi.org/10.1111/2041-210X.13574