Temporally Constrained Space Use, Part II: Approaching the Memory Challenge

In Part I three models for temporally constrained space use were summarized. Here in Part II I put them more explicitly into the context of ecology with focus on some key assumptions for the respective models. Area restricted search (ARS), Lévy walk (LW) and Continuous time random walk (CTRW) are statistical representations of disparate classes of temporally constrained space use without explicit consideration of spatial memory effects. Hence, below I reflect on a fourth model, Multi-scaled Random Walk (MRW), where site fidelity gets a different definition relative to its spatially memory-less counterparts.

A cattle egret Bubulcus ibis is foraging within a wide perimeter surrounding its breeding site. Spatial memory is utilized not only to be able to return to the nest but also to revisit favored foraging locations during a bout, based on a memory map of past experience. Photo AOG.

First, ARS is typically formulated as a composite random walk-like behaviour in statistical terms, which could be suitable for situations where a Markovian compliant (“mechanistic”) behaviour is either verified or can be reasonably assumed (memory-less  and scale-specific movement in both time and space). In this scenario the diffusion exponent can be estimated for movement bouts in different habitats and time intervals, and the result can be interpreted behavioural-ecologically. For example, the diffusion rate can be expected to be smaller i optimal patches than elsewhere. In other words, the local staying time increases due to a more jagged path.

Second, Lévy walk is a special kind of random walk. Most steps are relatively short but others may be extremely long. Sequences of short steps in-between the long ones make the overall space use appear locally constrained during these periods*). Lévy walk is characterized by a spatially memory-less statistical representation of scale-free (“hierarchical”) movement within a given spatial scale range. Beyond this range the distribution of step lengths will show increased compliance with a non-scaling, truncated Lévy walk; i.e., a composite model with exponential tail rather than a power law for the extreme part of the step length distribution. By analyzing the step length distribution within the scale-free (power law) regime using different sampling intervals one should be able to verify model compliance from stationary power exponent. A Lévy walk is statistically self-similar in space, and thus the power exponent is expected to be relatively unaffected by the sampling scheme; see Reynolds (2008). Calculating the difference in the median step length for a given sampling interval when studying subsets of the movement data under different environmental conditions brings the model into the realm of ecology [see a practical method in; for example, Gautestad (2012)].

Third, Continuous Time Random Walk (CTRW) is suitable where the animal is found to occasionally stop moving. The temporal distribution of the duration of such resting episodes can then be fitted to statistical models; for example, a power law, a negative exponential, or a mixture as in the distribution for truncated power law. The spatial distribution of step lengths is in CTRW fitted independently of the temporal distribution. Bartumeus et al. (2010) applied the CTRW framework to study “intensive versus extensive searching” (scale-free sub-diffusive versus super-diffusive search) in foraging of Balearic shearwaters Puffinus mauretanicus and Cory’s shearwaters Calonectris diomedea along the coast of Spain (Bartumeus et al. 2010). The authors  interpreted the results ecologically with weight on difference between presence and absence of local trawling activity. See Part I, where I gave a brief summary.

However, is CTRW a proper framework for these seabirds? At the end of each foraging bout they obviously utilized spatial memory to successfully return to their breeding location. CTRW assumes consistently random crossing of the movement path due to the model’s lack of spatial memory description. To me it seems illogical to assume that these birds should toggle between memory-dependent and goal-oriented returns at the end (and possibly at the start) of each trip and memory-less Brownian motion (ARS-like?) during foraging when moving in the proximity of trawlers. The same argument about conditional memory switch-off may be raised for scale-free (Lévy-like) search in the absence of trawlers.

In the context of memory-less statistical modelling of movement (the three models above), site fidelity is defined by the strength of “slow motion”, and how the distribution of local staying times is expected to vary with ecological conditions. Compare this with the alternative model Multi-scaled Random Walk, where site fidelity is defined as the strength (frequency) of targeted returns to a previous location on a path. This return frequency may be interpreted as a function of ecological conditions. Hence, MRW explicitly invokes both spatial memory and its relative strength:

Three time scales are defined: the implicit interval between successive displacements in simulations (t), the average return interval to a previous location (tret), and the observation interval on the movement path (tobs). The latter represents GPS locations in real data, and is applied to study the effect from varying ρ = tret/tobs (relative strength of site fidelity for a given tobs).
Gautestad and Mysterud (2013), p4

Note that an increasing tret for a given tobs implies weakened site fidelity, and the functional form of the step length distribution is influenced by the ρ = tret/tobs ratio. For example, a Brownian motion-like form may be found if ρ << 1, and a power law form can be expected when ρ >> 1, with truncated power law (Lévy-like) to be observed in-between. See Figure 3 in Gautestad and I. Mysterud (2005) and Figure 3 in Gautestad and A. Mysterud (2013).

If the animal in question is utilizing spatial memory a lot of confusion, paradoxes and controversy may thus appear if the same data are analyzed on the basis of erroneously applying memory-less models within different regimes of ρ!

The MRW model may thus offer interesting aspects with a potential for alternative interpretation of the results of space use analyses when put into the context of – for example – foraging shearwaters. Thanks to the three times scales for MRW as above – where the third variable, t, represents the unit (t≡1) spatiotemporal scales for exploratory moves – it should be possible to test for example Lévy walk or CTRW against MRW using real movement data. 

More on this in Part III.

NOTE

*) While temporally constrained space use in ARS regards difference in environmental forcing, the occurrence of short-step intervals of random occurrence in a Lévy walk is by default due to intrinsic behaviour.

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.

Gautestad, A. O. 2012. “Brownian motion or Lévy walk? Stepping towards an extended statistical mechanics for animal locomotion.” Journal of the Royal Society Interface 9: 2332-2340.

Gautestad, A. O., and I. Mysterud. 2005. Intrinsic scaling complexity in animal dispersion and abundance. The American Naturalist 165:44-55.

Gautestad, A. O. and A. Mysterud 2013. “The Lévy flight foraging hypothesis: forgetting about memory may lead to false verification of Brownian motion.” Movement Ecology 1: 1-18.

Reynolds, A. 2008. How many animals really do the Lévy walk? Comment. Ecology 89:2347-2351.

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