Animal Migration: Tactical Freedom During Strategic Constraint

Recent research on animal migration continue to challenge the paradigm of assuming relatively straight-line routes between start locations and respective targets, as shown in a study on blackpoll warblers Setophaga striata (Brown and Taylor 2017). The warblers had a surprisingly high degree of back-and-forth displacements during migration; apparently more than can be explained by adjusting steps to local habitat attributes along the path.  

Migration regards an endpoint on the scale continuum from short term movement bouts to long-distance seasonal displacements. Thus, one of the core challenges for more realistic models in wildlife ecology regards how to conceptualize and then formulate (in short: understand from simulations and from testing model predictions) the multi-scaled cognitive processing of environmental information and displacement decisions in animals. This insight should account for all time resolutions up to the migration scale.

From Insecta to Aves and Mammalia, individuals over a wide range of classes have been verified to relate to their environment in a multi-scaled manner. Migration is just one expression of a more general property of movement. At the opposite end we have area-restricted search (so-called ARS), which is often modelled as a composite random walk; i.e., a diffusion process with periodic toggling between fine-grained and coarse-grained movement as a function of intra- and inter-patch displacements. At a somewhat coarse scale range we have intra-home range paths during a day or week in the summer season. Seasonal drift then brings in the property of movement towards the other end of the scale continuum – migration.

Traditionally, movement over this wide temporal scale range has been modelled as “push” functions. For example, in the ARS model, an individual is toggling to another magnitude of the diffusion parameter when a local resource patch has bee exhausted, invoking a stronger directional bias of the following path sequence. In simplified terms, less preferred habitat pushes the animal around with stronger force than movement bouts in a preferred habitat. Inhospitable habitat (“barriers”) pushes the animal back. Skipping mathematical details, push-driven dynamics is easy to model in principle, thanks to the extensive theory of diffusion and advection.

However, as all wildlife ecologists know, animals are also under influence of “pull” forces. Individuals are pulled towards the summer range when migrating from the winter range. In short, the pull factor involves spatial memory, based on previous experience from the area the animal is attracted to and in complex interplay with traditional migration routes. Influence from conspecifics with better experience optimizes the process (cultural inter-generation transmission of spatio-temporal knowledge). Again skipping mathematical details, pull-driven dynamics is much more challenging to model. The field is still characterized by immature biophysics, but progress is emerging.

To summarize, over the range from fine- to coarse-scale movement, tactics (push processes at fine time resolutions) interacts with strategic goals (pull processes at coarser time resolutions) in complex and still little understood ways.

In Gautestad and Mysterud (2010) we illustrated this challenge by contrasting three hypothetical paths (see also more extensive descriptions in my book). Path A represents a correlated random walk with strong directional persistence. For example, consider the ARS model during a bout of inter-patch movement. Path B illustrates scale-free distribution of step lengths (Lévy walk), which is another form of moving around faster than the traditional standard random walk allows for. These variants belongs to the push framework.

Paths A and B both lack spatial memory, and just happen to reach a common location (upper bullet in the Figure) after a series of mechanistically executed displacements.

Path C illustrates the pull factor, resting on my parallel processing conjecture; i.e., the process involves a mixture of (a) a long-term goal to reach the upper bullet location in the Figure by using spatial memory and (b) simultaneously executing temporally finer-resolved goals “in parallel” during this displacement.

The respective fine-grained paths A, B and C are indicated by dashed lines. There is a qualitative difference between the three intermediate locations 1, 2 and 3 of Path B and path C. While the displacement from location 2 to location 3 of a Levy walk process (path B) is independent of the path’s history; i.e, reaching location 3 is independent of location 1, path C has inter-location dependence from start to end.

In short, parallel processing as conceptualized by path C violates the Markov principle. Hence, it also violates the dynamics of mechanistic modelling. Why? Because the pull-effect due to the goal to reach the upper location put constraint on how much wiggling and zig-zagging the animal can allow itself during the time interval to reach the target. Simultaneously (in parallel, thanks to the continuum of time resolutions involved in the process; see this post)  the sub-steps 1 and 2 are performed under some degree of tactical freedom, given that these tactics are not violating the strategic goal to reach the target 3 within the coarser time resolution. For example, during your drive to work in the morning you may find you have have time to depart from your usual route to buy a newspaper in the kiosk a block or two away.

If the “buy the Newspaper” is a temporally fine-scale goal (short term impulse) relative to “reach my job in time” (your main task for the day), the two events – according to the parallel processing conjecture – are executed at different time scales. Under this framework the resultant vector for movement at a given instant in time cannot be understood as a superposition (sum) of various influences at that moment, regardless of reductionistic details (number of model terms) and their relative weights. Thus, the process is non-differentiable and non-mechanistic; ie., not Markovian. Hence, in my book I differentiate between mechanistic and non-mechanistic kinds of dynamics. Parallel processing regards an extension of the traditional statistical-mechanical theory, which rests on the superposition principle.

Back to bird movement. Brown and Taylor’s study on blackpoll warblers (Brown and Taylor 2017) illustrate some potential pull constraints during passerine migration:

… adult birds migrate using a ‘navigational map’ learned during their first migrations, allowing them to correct for displacements by using alternate migratory paths.

… Based on traditional views of migration, we anticipated that movements at regional scales would be highly variable, but still generally oriented towards the migratory goal [5]. Instead, only 13 out of 75 non-ambiguous movements were classified as ‘migratory’. The total amount of time spent in the region decreased, and the likelihood of making a migratory movement increased, as the season progressed, …

… Surprisingly, many individuals moved in directions oriented away from the migratory goal (‘indirect movement pattern’). These indirect movement patterns occurred throughout the migratory season, regardless of age group and natal origin (n = 62). Indirect movement patterns were highly variable in both their extent and path tortuosity. We suggest that they are either an extension of ‘landscape-scale stopover movements’ [ref] or a more complete representation of ‘reverse migration’, a phenomenon recorded in 10–50% of individuals each evening during migration [refs]. Regardless, their function is unknown.

… The higher prevalence of indirect movements patterns in more experienced individuals suggests that these movements are not accidental, and thus may confer a selective advantage that has either been learned by adults, or that is too energetically costly for less efficient hatch-years.

… Astonishingly, the upper range of cumulative distances flown during indirect movement patterns exceeded 1000 km, almost half the total distance that individuals must fly when they embark on their final migratory flight across the Atlantic Ocean.
Brown and Taylor (2017).

Thanks to Brown and Taylor’s study and a wide range of supporting results that have accumulated on this topic over the years it is an increasing pressure on ecological theorists to come up with a more realistic framework to model multi-scaled movement; i.e., explicitly formulating the combination of push and pull (tactics and strategy) in a meaningful manner.


Brown, J. M., and P. D. Taylor. 2017. Migratory blackpoll warblers (Setophaga
striata) make regional-scale movements that are not oriented toward their migratory goal during fall. Movement Ecology 5:1-13.

Gautestad, A. O., and I. Mysterud. 2010. The home range fractal: from random walk to memory dependent space use. Ecological Complexity 7:458-470.


Fowler’s Toads: the MRW Model Gains Additional Support

Over the years our Multi-scaled Random Walk (MRW) model has been empirically supported by various anecdotal observations and pilot tests, but also more extensive results, in particular our multi-faceted statistical analysis of a large database of red deer Cervus elaphus movement (Gautestad et al. 2013). The MRW model has now also been scrutinized by a team working on data from a non-mammalian species, Fowler’s toads Anaxyrus fowleri (Marchand et al. 2017).

In its generic form, the MRW represents a very basic statistical-mechanical description of animal space use, implementing complex space use from spatial and temporal memory utilization. It illustrates a statistically scale-free distribution of displacements (you may call it Lévy-like) in combination with a given frequency of return events to previously visited locations. By changing the values of a small set of parameters the model can be put into the context of specific biological-ecological settings. Conversely, by fitting the model to real space use data (like series of GPS fixes), ecological inference can be performed by interpreting these parameter values within the given universality class.

The MRW model has already been described in a range of blog posts, for example here. The model represents one of the eight corners  (universality classes) of the Scaling cube, which was first described in Gautestad (2015). As each successive revisit to a location increases its effective weight for future return steps, the MRW model allows home range patterns to emerge without the need to specify an ad hoc homing process*.

As stated by Marchand et al. (2017), the MRW – as do the other seven corners of the Scaling cube, represents a parsimonious model; i.e., a more powerful concept than a simple or generic model per se:

“Parsimonious models are simple models with great explanatory predictive power. They explain data with a minimum number of parameters, or predictor variables. The idea behind parsimonious models stems from Occam’s razor, or “the law of briefness” (sometimes called lex parsimoniae in Latin). The law states that you should use no more “things” than necessary; In the case of parsimonious models, those “things” are parameters. Parsimonious models have optimal parsimony, or just the right amount of predictors needed to explain the model well.”

Being a parsimonious model – like Brownian motion (represented by, for example, classic random walk) – MRW should of course be subject to particularly strong scrutinizing tests. After all, MRW challenges the traditional approach to interpreting, modelling and analysing animal movement at a very fundamental level. However, as emphasized in this post (and others) the MRW model and its complementary statistical-mechanical framework has only recently gained stronger spotlight, both empirically and theoretically. For mathematical/physical study of the model and variants, see for example Boyer et al. (2016).

With respect to Fowler’s toads, the interesting analysis by Marchand et al. (2017) use Approximate Bayesian computation (ABC) to estimate the parameters of three variants of MRW, including the scale and shape of a Lévy-stable distribution of movement steps and the probability of returning to a known refuge rather than establishing a new one.

Specifically, Marchand et al. (2017) compare the relative fit of three versions of the return step:

  1. toads return to a randomly selected previous refuge, independent of distance;
  2. they return to the nearest refuge from their current location; or
  3. the probability of return to any previous refuge is a decreasing function of the distance to that refuge.

The authors hypothesize that either of the last two models would provide a better fit if minimizing energy expenditure were the primary factor determining refuge choice.

The assumption that toads returning to a previous refuge choose one at random may seem unrealistic. Yet it fits the data better than two alternative models we tested, where the probability of return and/or the choice of refuge were distance-dependent.
Marchand et al. 2017, p 68.

In other words, they conclude that MRW in its generic form explained the data better than the more “biologically intuitive” variants.

Why do Fowler’s toads, red deer and other tested species apparently show little degree of distance-dependent returns when executing site fidelity? Marchand et al. (2017) do not speculate on the potential parsimonious aspect of this counter-intuitive result, but in my book I do:

“For example, consider a thread of a given length representing a movement path during a given time span from one location of the animal to the next, and this interval
on average contains one or more returns to a previously visited site. If you stretch the thread to a unidirectional line, you get a large displacement during this time interval. If you curl it, the net start-to-end displacement is small. Anyway, the energy expenditure is similar since the thread length is similar. Both return steps (expressing site fidelity) and scale-free exploratory steps—the two main components of a MRW—contribute to curling of the path, which will vary in strength from one interval to the next.”
Gautestad (2015), p272.

All three hypothetical paths with respective start- and end points (red dots, representing displacement during a given time interval Δt) have approximately the same stretched-out distance as the vertical reference line to the left.

When site fidelity is influencing space use, and unless the path show a strong uni-directional drift during the total sampling period, the animal’s path is jagged in a manner that typically is not reflected in detail at the temporal scale of data sampling or analysis. In my book I refer to this aspect as a consequence of the “hidden layer” (un-observed details) of a given movement path.

In other words, when animal displacements are collected at frequency 1/Δt and Δt (the actual time resolution for successive displacement collection) is substantially larger than what would be needed to reveal the fine-resolved path, one should – a priori – not expect to find support for Model 2 and 3 above. The MWR model is explicitly defined to represent a statistical-mechanical representation of movement; i.e., Δt should in fact be sufficiently large (the hidden layer should be deep) to ensure conditions for analysis at this temporally coarse-grained level of space use abstraction.

When it comes to energy expenditure, the MRW model rests on the conjecture that a given amount of kinetic energy pr. unit time Δt may be distributed over a range of spatio-temporal scales during this interval: many small displacements, some intermediate ones, and occasionally some very large ones in a scale-free manner (the Parallel processing postulate). This property leads to a Lévy-like step length distribution.

* In fact, the site fidelity effect under MRW does not depend on this additive, self-reinforcing use of previously visited sites even if it represents an intrinsic property of the model’s generic formulation. However, absence/presence of this positive feedback property provides a testable aspect of its home range representation.


Boyer, D, M. R. Evans, and S. N. Majumdar. 2016. Long time scaling behaviour for diffusion with resetting and memory. Cond. mat.stat-mech. (arXiv:1611.06743v1).

Gautestad, A. O. 2015. Animal Space Use: Memory Effects, Scaling Complexity, and Biophysical Model Coherence. Dog Ear Publishing, Indianapolis.

Gautestad, A. O., L. E. Loe, and A. Mysterud. 2013. Inferring spatial memory and spatiotemporal scaling from GPS data: comparing red deer Cervus elaphus movements with simulation models. Journal of Animal Ecology 82:572-586.

Marchand, P, M. Boenke and D. M. Green. 2017. A stochastic movement model reproduces patterns of site fidelity and long-distance dispersal in a population of Fowler’s toads (Anaxyrus fowleri). Ecological Modelling 360:63–69.