Temporally Constrained Space Use, Part I: Three Models

Temporally constrained space use is a key property of animal movement. With respect to vertebrates three main statistical representations are particularly popular among modelers, based on disparate theoretical foundations. Which one should one use for analysis of a particular data set? As always in ecological research, one needs some simple protocol to distinguish between alternative model assumptions.

Animals paths are neither straight lines nor a dense dot of positions from juggling back and forth at the same spot. Typically we see a complicated combination of these two extreme patterns; some quite straightforward moves occasionally abrupted by more jagged movement. In order to infer behavioural and ecological results from space use one needs to study the data in the context of a realistic theoretical framework.

Outside the realm of temporal site fidelity; e.g., a drifting home range (Doncaster and Macdonald 1991),  ecological textbooks typically explain the mixture of straight and convoluted movement bouts as Area restricted Search (ARS).

Area-restricted search. A foraging pattern in which a consumer responds to an intake of food by slowing down its movement and remaining longer in the vicinity of the most recently located food item. This behaviour causes consumers to remain longer in areas where the density of food items is high than in areas where it is low.
A Dictionary of Ecology. Encyclopedia.com. 29 Jul. 2018.

In terms of statistical models, this rather qualitative description of behaviour may be formulated in many ways. A popular one is to combine classic or correlated random walk with two distinct parameter values for intrinsic step length distribution (the λ value) in F(r) ∝ e-λr. In this manner, movement varies with the jaggedness of the path (number of turns pr. period of time) rather than the movement speed. A larger λ implies smaller step lengths on average, which tend to increase local staying time during intervals when this movement mode is active. By fine-tuning respective λ1 and λ2 and  in such a superposition of two “randomly toggling modes” this so-called composite random walk can even be made to mimic the second main approach to model complicated paths, the Lévy flight model (Benhamou 2007; but see Gautestad 2013):

Lévy flights are, by construction, Markov processes. For general distributions of the step-size, satisfying the power-like condition, the distance from the origin of the random walk tends, after a large number of steps, to a stable distribution due to the generalized central limit theorem, enabling many processes to be modeled using Lévy flights.
https://en.wikipedia.org/wiki/Lévy_flight.

Lévy flights (and walks) are typically thought of as producing “fat tailed” step length distributions. However, within an often observed parameter range of the distribution (Pr(U>u = O(u)-k with 1<k<2) in real animals, one should not forget that half of the displacements in the distribution are in fact relatively short! In fact, the dominating step length bin is ultra-short moves, leading to a path that is conceptually (albeit not statistical mechanically) similar to the slow-down effect of more jagged moves during composite random walk.

Such “knots” of ultra-short moves of a Lévy path brings us to the third class of movement models, Continuous time Random Walk (CTRW). In this case the movement may be arrested for a shorter or longer period:

The step length distribution and a waiting time distribution (“resting” between steps) describe mutually independent random variables. This independence between jump lengths and waiting time to perform the next step makes the difference between CTRW and ordinary random walk (including Brownian motion).
Page 91 in: Gautestad, A. O. 2015, Animal Space Use: Memory Effects, Scaling Complexity, and Biophysical Model Coherence Indianapolis, Dog Ear Publishing.

Again, by fine-tuning the intrinsic model formulations and parameter conditions of CTRW one may in fact produce a Lévy flight-like movement pattern (Reynolds 2010; Schulz et al. 2013). Towards another extreme, one may achieve a classic random walk look-alike pattern. Conceptually, CTRW thus belongs on the left wall of the Scaling cube (x=0; y and z varies in accordance to model variant and parameter settings).

CTWR (which is closely connected to the concept of fractional diffusion) is typically invoked to explain a kind of switching – “intensive versus extensive searching” (scale-free sub-diffusive versus super-diffusive search) – that is now being found in real animal data like foraging of Balearic shearwaters Puffinus mauretanicus and Cory’s shearwaters Calonectris diomedea along the coast of Spain (Bartumeus et al. 2010).

Since I lack good pictures of shearwaters I decorate this post with a snapshot of another seabird, the sandwich tern Thalasseus sandvicensis. At least the picture was taken in the same general area of the Mediterranean. Photo: AOG.

In particular, they found that the birds in the presence of fisheries were more “restless” when taking advantage of fishery discards than in absence of trawlers, implying a higher probability of leaving a localized area pr. unit time during trawling activity. Specifically, during period of fishery discard utilization tended to show a smaller temporal scaling exponent for staying time (the temporal aspect of CTRW; larger β in their site fidelity function S(t) ∝ tβ, implying fewer events with a particularly prolonged staying time). In the spatial aspect of flight length distributions, when fisheries discard was present the birds tended to show good compliance with a negative exponential function*.

On the other hand, in the absence of trawlers a compliance with Lévy walk (truncated power law) was found. Space and time brought together, they found that the birds tended towards sub-diffusive foraging in the presence of fisheries discard (despite – somewhat counter-intuitively – a smaller local staying time within a given patch of a given spatial resolution), and super-diffusive foraging under natural conditions.

Despite the mathematical and numerical attractiveness of composite random walk, Lévy walk and CTRW and their well-explored statistical properties, they unfortunately all lack what may be a crucial component of foraging behaviour: spatial memory; i.e., the condition x>0 of the scaling cube.

Without spatial memory, self-crossing of an individual’s path happens by chance only, not intentionally by returns to a previous location (site fidelity, whether we consider short term or long term time scales).

  • May a model that implements spatial memory offer an alternative interpretation of the results presented by Bartumeus et al. 2010?
  • May this alternative hypothesis even offer a logical explanation for the apparent paradox that the birds were more restless locally when the movement simultaneously was more spatially constrained in overall terms?

The memory aspect of temporally constrained space use will be explored in Part II.

NOTE

*) Somewhat confusingly relative to common practice Bartumeus et al. (2010) use the exponential formula variant F(r) ∝ e-r/λ, which makes average step length proportional with λ rather than 1/λ.

REFERENCES

Bartumeus, F., L. Giuggioli, M. Louzao, V. Bretagnolle, D. Oro, and S. A. Levin. 2010. Fishery discards impact on seabird movement patterns at regional scales. Current Biology 20:215-222.

Benhamou, S. 2007. How many animals really do the Lévy walk? Ecology 88:1962-1969.

Doncaster, C. P., and D. W. Macdonald. 1991. Drifting territoriality in the red fox Vulpes vulpes. Journal of Animal Ecology 60:423-439.

Gautestad, A. O. 2013. Animal space use: Distinguishing a two-level superposition of scale-specific walks from scale-free Lévy walk. Oikos 122:612-620.

Reynolds, A. 2010. Bridging the gulf between correlated random walks and Lévy walks: autocorrelation as a source of Lévy walk movement patterns. J. R. Soc. Interface 7: 1753–1758.

Schulz, J. H. P., A. V. Chechkin, and R. Metzler. 2013. Correlated continuous time random walks: combining scale-invariance with long-range memory for spatial and temporal dynamics. J. Phys. A: Math. Theor. 46:1-22.