MRW and Ecology – Part IV: Metapopulations?

In light of the recent insight that individuals of a population generally seem to utilize their environment in a multi-scaled and even scale-free manner, the metapopulation concept needs a critical evaluation. Even more so, since many animals under a broad range of ecological conditions are simultaneously mixing scale-free space use with memory map-based site fidelity. In fact, both properties, multi-scaled movement and targeted return events to previous locations, undermine key assumptions of the metapopulation concept.

Levins (1969) model of “populations of populations” – termed metapopulation – rattled many corners of theoretical and applied ecology, despite previous knowledge of the concept from the groundbreaking research by Huffaker (1958) and others (Darwin, Gause, etc.). Since then, Ilkka Hanski (1999) and others have produced broad theoretical and empirical research on the metapopulation concept.

The Levins model describes a metapopulation in a spatially implicit manner, where close and more distant sub-populations are assumed to have same degree of connectivity. Later models (including Hanski’s work) made the dynamics spatially explicit. Hence, distant sub-populations are in this class of design more closely connected dynamically than more distant populations. Sub-populations (or “local” populations) are demarcated by large difference in internal individual mixing during a reproduction cycle relative to the rate of mixing with neighbouring sub-populations at this temporal scale. As a rule-of-thumb, the migration rate between neighbour populations during a reproduction cycle should be smaller than 10-15% to classify the system as a metapopulation. Simultaneously, intrinsic mixing during a cycle in a given sub-population is assumed to approximate 100%; i.e., “full spatial mixing” (spatial homogenization when averaging individual locations over a generation period).

 

According to the prevailing metapopulation concept, high rate of internal mixing in sub-populations is contrasted by substantially lower mixing rate between sub-populations. The alternative view – advocated here – is a hierarchical superposition of mixing rates if the individual-level movement is scale-free over a broad scale range. The hierarchy is indicated by three levels, with successively reduced intra- and inter-population mixing rate towards higher levels.

The spatially explicit model of a metapopulation is based on three core assumptions:

  1. The individual movement process for the population dynamics should comply with a scale-specific process; i.e., a Brownian motion-like kind of space use in statistical-mechanical terms, both within and between sub-populations. This property allows intrinsic population dynamics of sub-populations to be modelled as “homogeneous” at this temporal scale. This property is also assumed by the theory of differential and difference equations. It also allows the migration between sub-populations to be described as a classical diffusion process.
  2. Following from Point 1, more distant sub-populations are always less dynamically linked (smaller diffusion rate) than neighbour populations. In fact, dispersal between distant sub-populations may be ignored in spatially explicit models.
  3. Emigration from a given sub-population may be stochastic (random) or deterministic (e.g., density dependent emigration rate), while immigration rate is stochastic only. The latter follows logically from compliance with point 1. In other words, emigrating individuals may occasionally return, but only by chance and thus on equal terms with the other immigrants from neighbour populations. Hence, both the intrinsic and inter-population mixing process is assumed to lack spatial memory capacity for targeted returns at the individual level.

In my alternative idea for a spatially (and temporally) structured kind of population dynamics, individual movement is assumed to comply with multi-scaled random walk (MRW). Contrary to a classical Brownian motion and diffusion-like process, MRW defines both a scale-free kind of movement and a degree of targeted returns to previous locations. Thus, both emigration and immigration may be implicitly deterministic. The two perceptions of a structured population are conceptualized in the present illustrations. “Present idea” regards the prevailing metapopulation concept, and the “Alternative idea” regards population dynamics under the MRW assumptions.

The upper part of the illustration to the right shows the two classical metapopulation assumptions in a simplistic manner. Shades of blue regards strength of inter-population mixing, which basically is reaching neighbour populations only (by a rate of less than 10-15%, to satisfy a metapopulation structure) but not more distant ones. For example, inter-generation dispersal rate between next-closest sub-populations is expected to be less than (10%)*(10%) = 1%, and falls further towards zero at longer distances. The Alternative idea at the lower part describes a more leptocurtic (long-tailed) dispersal kernel – in compliance with a power law (scale-free dispersal) – rather than an exponentially declining kernel (scale-specific dispersal), as in the standard metapopulation representation. Separate arrows for immigration from one sub-population to a neighbour population of the “Present idea” part illustrates the standard diffusion principle, while the single dual-pointing arrow of the “Alternative idea” illustrates that immigration an emigration are not independent processes, due to spatial memory-dependent return events. The emergent property of targeted returns connects even distant sub-populations in a partly deterministic manner.

The Alternative idea design is termed the Zoomer model, which is explored both theoretically and by preliminary simulations in my book. A summary was presented in this post. A long-tailed dispersal kernel may embed and connect subpopulations that are separated by a substantial width of matrix habitat. Since the tail is thin (only a small part of individual displacements are reaching these distances), long-distance moves and directed returns happen with a small rate. 

The Zoomer model as an alternative to the classical metapopulation concept has far-reaching implications for population dynamical modelling and ecological interpretation. For example, the property that two distant sub-populations may sometimes be closer connected than connectivity to intermediate sub-populations due to emergence of a complex network structure at the individual level was illustrated by my interpretation of the Florida snail kite research (see this post). At the individual level, research on Fowler’s toad (see this post) and the Canadian bison (see this post) shows how distant foraging patches may be closer connected than some intermediate patches. Also this is in compliance with the Zoomer concept and in opposition to the classical metapopulation concept. In many posts I’ve shown examples of the leptocurtic distribution of an individual’s histogram of displacement lengths, covering very long distances in the tail part – potentially well into the typical scale regime of a metapopulation (for example, the lesser kestrel).

REFERENCES

Hanski, I. Metapopulation Ecology. Oxford University Press. 1999.

Huffaker, C.B. 1958. Experimental Studies on Predation: Dispersion factors and predator–prey oscillations. Hilgardia 27:83-

Levins, R. 1969. Some demographic and genetic consequences of environmental heterogeneity for biological control. Bulletin of the Entomological Society of America 15:237–240

Random Walk Should Not Imply Random Walking

Random walk is one of the most sticky concepts of movement ecology. Unfortunately, this versatile theoretical model approach to simplify complex space use under a small set of movement rules often leads to confusion and unnecessary controversy. As pointed out by any field ecologist, unless an individual is passively shuffled around in a stochastic sequence of multi-directional pull and push events, the behavioural response to local events and conditions is deterministic! An animal behaves rationally. It successively interprets and responds to environmental conditions – within limits given by its perceptive and cognitive capacity – rather than ignoring these cues like a drunken walker. Any alternative strategy would lose in the game of natural selection. Still, from a theoretical perspective an animal path may still be realistically represented by random walk – given that the randomness is based on properly specified biophysical premises and the animal adhere to these premises.

Photo: AOG

Outside our house I can study a magpie pica pica moving around, apparently randomly, until something catches its attention. An insect larva? A spider or other foraging rewards? After some activity at this patch it restarts its exploratory movement. As ecologist it is easy to describe the behaviour as ARS (area restricted search). In more general terms, the bird apparently toggles between relatively deterministic behaviour during patch exploration and more random exploratory moves in-between. If I had radio-tagged the magpie with high resolution equipment, I could use a composite random walk model (or more contemporary: a Brownian bridge formulation) derived from ARS to estimate the movement characteristics for intra- and inter-patch steps respectively, and test ecological hypotheses.

However, what if the assumptions behind the random walk equations are not fulfilled by the magpie behaviour? Now and then the magpie flies back in a relatively direct line to a previous spot for further exploration. In other words, the path is self-crossing more frequently than expected by chance. Also, the next day the magpie may be return to our lawn in a manner that indicates stronger site fidelity than expected from chance, considering all the other available gardens in the county. The magpie explores, but also returns in a goal-oriented manner, meaning that the home range concept should be invoked. Looking closer, when exploring the garden the magpie also seems to choose every next step carefully, constantly scanning its immediate surroundings, rather than changing direction and movement speed erratically. Occasional returns to a previous spot, in addition to returning repeatedly to our garden, indicates utilization of a memory map. In short, this magpie example may not fit the premises of an ARS the way it is normally modeled in movement ecology, namely as a toggling between fine- and coarser-scale random walk.

Hence, two challenges have to be addressed.

  1. What are the conditions to treat the movement as random walk when analysing the data?
  2. What are the basic prerequisites for applying the classical random walk theory for the analysis?

Regarding the first question, contemporary ecological modelling of movement typically defines the random parts of an animal’s movement path as truly stochastic (rather than as a model simplification of the multitude of factors that influence true movement), in the meaning of expressing real randomness in behavioural terms. The Lévy flight foraging hypothesis is an example of this specification. The remaining parts of the path are then expressing deterministic rules, like pausing and foraging when a resource patch is encountered, or triggering of a bounce-back response when sufficiently hostile environment is encountered. In my view this stochastic/deterministic framework is counterproductive with respect to model realism, since it tends to cover up the true source of randomness.

To clarify the concept of randomness in movement models one should be explicit about the model’s biophysical assumptions. Different sets of assumptions lead to different classes of random walk. In my book I summarized these classes as eight corners of the Scaling cube. Sloppiness with respect to model premises hinders the theory of animal space use to evolve towards stronger realism.

  • Random walk (RW) in the classical sense; i.e., Brownian motion-like, regards a statistical-mechanical simplification of a series of deterministic responses to a continuous series of particle shuffling. Collision between two particles is one example of such shuffling events. In other words, during a small increment of time a passively responding particle performs a given displacement in compliance with environmental factors (“forces”) and physical laws at the given point in space and time. Until new forces are acting on the particle (e.g., new collisions), it maintains its current speed and direction. In other words, under these physical conditions the process is also Markov-compliant: regardless of which historic events that brought the particle to it current position, its next position is determined by the updated set of conditions during this increment. The next step is independent of its past steps.
  • The average distance between change of movement direction of a RW is captured by the mean free path parameter. This implies that RW is a scale-specific process, and the characteristic scale is given by the mean free path during the defined time extent.
  • Since the RW particle is responding passively, its path is truly stochastic even at the spatio-temporal resolution of the mean free path. When sampling a RW path at coarser temporal resolutions a larger average distance between successive particle locations is observed. Basically, this distance increases proportionally with the square root of the sampling interval. This and other mathematical relationships of a RW (and its complementary diffusion formulation) is predictable and coherent from a well-established statistical-mechanical theory.
  • Stepping from a physical RW particle to a biophysical representation of an individual in the context of movement ecology implies specification and realism of two assumptions: (1) the movement behaviour should be Markov compliant (i.e., scale-specific), and (2) the path should be sampled at coarser intervals than the characteristic time interval that  accompanies the mean free path (formulated in the average “movement speed” at the mean free path scale). At these coarser spatio-temporal resolutions even deterministic movement steps becomes stochastic by nature, due to lumping together the resultant displacement from a series inter-independent finer-grained steps.

    An animal is observed at position A and re-located at position B after t time units. The vector AB may be considered a RW compliant step if – and only if – the intermediate path locations (dotted circles) in totality are sufficiently independent of the respective previous displacement vectors to make the resultant vector AB random. Each of the intermediate steps may be caused by totally deterministic behaviour. Still, the sum of the sequence of more or less inter-independent displacements makes position B unpredictable from the perspective of position A. The criterion for accepting AB as a step in a RW sequence is fulfilled at temporal scale (sampling resolution) t, even it the “hidden layer” steps are more or less deterministic at finer resolutions <<t.

    In my book I refer to such observational coarse-graining as increasing the depth of the hidden layer, from a fine-resolved unit scale – where local causality of respective displacements are revealed – to a coarser resolution where deterministic (and Markov-compliant) behaviour requires a statistical-mechanical description.

Regarding the second question raised above regarding Markov compliance, see the RW criterion in the Figure to the right [as was also exemplified by “Markov the robot” in Gautestad (2013)].

However, what if the animal violates Markov compliance? In other words, what if it is responding in a non-Markovian manner, meaning that path history counts to explain present movement decisions? Is the magpie-kind of non-Markovian movement typical for animal space use, from a parsimonious model perspective, or is multi-scaled site fidelity the exception rather than the rule? These are the core questions any modeller of animal movement should ask him/herself. One should definitely not just accept old assumptions just because several generations of ecologists have done so (many with strong reluctance, though).

Instead of accepting classical RW or its trivial variants correlated RW and biased RW as a proper representation of basic movement by default, albeit while closing your nose, you should explore a broader application of other corners of the Scaling cube, each with respective sets of statistical-mechanical assumptions.

 

REFERENCE

Gautestad, A. O. 2013. Lévy meets Poisson: a statistical artifact may lead to erroneous re-categorization of Lévy walk as Brownian motion. The American Naturalist 181:440-450.

The Lesser Kestrel: Natal Dispersal In Compliance With The MRW Model

The Multi-scaled random walk (MRW) model defines a specific dispersal kernel for animal movement; a power law, which is qualitatively different from standard theory (a negative exponential function). Alcaide et al. (2009) analyzed long-term ringing programmes of the lesser kestrel Falco naumanni in Western Europe, and showed results from re-encounters of 1308 marked individuals in Spain. They found that most first-time breeders settled within 10 km from their natal colony (i.e., a strong philopatric tendency), with a negative association between natal dispersal and geographical distance. While Alcaide et al. (2009) were mainly concerned with gene flow and population effects, here I take a deeper look at their dispersal data and find strong support for MRW-compliant behaviour in the natal dispersal data. Indirectly, this pattern at the individual level also supports the MRW-analogue at the population level, the Zoomer model (Gautestad 2015).

I allow myself to copy their Figure 1, showing the natal dispersal distances:

Fig. 1. Frequency distribution of natal dispersal distances of lesser kestrels in the Guadalquivir Valley (SW Spain, N = 321 individuals, black bars; Negro et al. 1997) and in the Ebro Valley (NE Spain, N = 961, white bars; Serrano et al. 2003).

 

To visualize the difference between the expected dispersal kernel from MRW and from standard theory I here present the data above with log-scaled axes:

 

Under this transformation, compliance with a power law should resemble a straight regression line, with a slope that is defined by the power exponent. Such log-log linearity of a power law contrasts with a log-log transformed negative exponential function, which becomes convex. Interestingly, the two subsets of natal dispersal distances show strong compliance with a power law (R2=0.90 and R2=0.96, respectively), while the best-fitting negative exponential does not match the pattern that well (R2=0.60; dotted line).

Quite remarkably, even the power exponents (β=-2.02 and β=-2.00) show up very close to the standard MRW expectancy of β=-2  (Footnote 1). This particular magnitude of β is – according to MRW theory – expected from scale-free space use where the individual on average during the sampling period has put equal effort into utilizing its environment over the given scale range (in this case, from a spatial grain resolution of 10 km to an extent resolution of 440 km).

The discovery of natal dispersal data as summarized by Alcaide et al. (2009) allows me – for the first time – to study empirical model compliance in a species at relatively coarse temporal scales; i.e., over the interval from birth to first breeding the following year or two. Previous resolutions for MRW tests have typically been at temporal resolution of a few hours (GPS relocation data). Simultaneously, the good fit to power exponent β=-2, even at this coarse temporal scale, translates to β’=-1 in area terms rather than distances (Gautestad and Mysterud 2005). I recycle an illustration of this population kinetic aspect, which was also shown in this post and in my book:

 

The grey-shaded inset represents the classic dispersal kernel, expected from standard random walk at the individual level and diffusion at the population level; i.e., a negative exponential. The other elements in the illustration regard MRW (scale free power law, see also Footnote 2).

In particular, observe for the F(L) movement kernel that the coloured rectangle area of each log-scaled interval (bin) for squared distance, L2; representing “effort” by the individual to relate to respective spatial resolutions of their environment, is of similar magnitude when F=(L2)-1 = 1/L2. The area of each of the rectangles is the same. In other words; in a two-dimensional arena, an individual is then utilizing a k times larger landscape resolution 1/k times as frequently. In a population context (the Zoomer model, switching from a Lagrangian to the complementary Eulerian system perspective) – since a k times larger arena is expected to embed k times more individuals in average terms – when β=-2 the population is utilizing the landscape with equal intensity over the given scale range (Gautestad 2015, p122-132).

Footnote 1: what about the Lévy flight/walk model, which also predicts a scale-free and thus a log-log linear dispersal kernel? With respect to the lesser kestrel, as well as all other bird species, spatial memory is part of their cognitive capacity. A home range, which requires directed returns to previous locations, is exemplifying this utilization. MRW regards a combination of scale-free space use and site fidelity. Lévy flight only regards the former.

Footnote 2: With respect to lesser kestrel’s natal dispersal, the data represents the displacement distribution of many individuals (called an ensemble in statistical mechanics) rather than the distribution of a set of displacements for a given individual. Thus, the power law curve reflects these individuals’ pooled tendency for scale free space use during natal dispersal. When establishing their respective home ranges with centre of activity at the chosen breeding site, it would have been interesting to see whether the median displacement length (and β) for the following 1-2 year period deviated from natal dispersal at the same temporal resolution.

REFERENCES

Alcaide, M., D. Serrano, J. L. Tella and J. J. Negro. 2009. Strong philopatry derived from capture–recapture records does not lead to fine-scale genetic differentiation in lesser kestrels. Journal of Animal Ecology 78:468–475.

Gautestad, A. O. 2015. Modelling parallel processing. pp114-148 inAnimal Space Use: Memory Effects, Scaling Complexity, and Biophysical Model Coherence. Dog Ear Publishing, Indianapolis. 298pp.

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

The Florida Snail Kite: Linking Individual MRW to Population Kinetics?

The theoretical framework of the traditional space use models (“the Paradigm”) shows strong mathematical coherence between individual movement and its population level representation. Basically (in its simplest and most parsimonious form), standard random walk representing individual statistics is compliant with standard diffusion representing population statistics. Further, the trunk of the toolbox of statistical methods in space use ecology is also resting on assumptions from the Paradigm. However, the Paradigm and its large family of sophisticated sub-models is now under increasing attack from many directions – directly or indirectly – due to the growing pool of empirical results that cast doubt on its common, core assumptions. For example, recent analyses of the snail kite Rostrhamus sociabilis plumbeus in Florida indicate that individuals of this dietary specialist show a surprising capacity to rapidly adapting to changing conditions over a large range of spatial scales from localized home ranges to state-wide network of  snail-rich wetland patches (Valle et al. 2017) . In this post I allow myself to speculate on a potential for space use compliance between snail kite in Florida and a specific Paradigm challenger – the dual MRW model and Zoomer model (the parallel processing framework for the individual and population level, respectively) – that is advocated in my book and here on my blog.

Snail kite, adult male. Photo by Andreas Trepte (www.photo-natur.net).

What in my view is particularly thrilling about the snail kite analyses is the authors’ quite nontraditional approach to apply network analysis to study intra-population flow of individuals from the perspective of clustering individuals based
on the locations they visit instead of clustering locations based on their connectivity to other locations (Fletcher et al. 2013, 2015; Reichert et al. 2016; Valle et al. 2017). As shown in one of their papers, network theoretical methods applied in this manner may lead to surprising results (Valle et al. 2017). What would you say if the population you are monitoring shows a relatively sudden directional drift of many individuals towards areas far away (hundreds of kilometers) relative to the apparently still suitable locations (local wetlands) where these individuals for years have shown strong site fidelity and thus spend most of their time? Adult snail kites from this sub-species rarely depart from their local wetland during the breeding season (ca 13% probability), and annual departure rate is not larger than ca 40-60%. Still, changing conditions appearing many wetland patches away (e.g., the distance between southern and northern Florida) was observed to influence local emigration rate quite suddenly; to be described below. Something like ecology’s analogy to quantum entanglement; i.e, spooky action over large distances (yes, joking)?

In a previous post I described animal space use as a combination of push and pull; mixture of tactics and strategy, under a postulate of parallel processing. The latter – a simultaneous utilization of the environment over a range of resolutions – was described as spatio-temporally multi-scaled memory utilization. Under this assumption, individuals could gradually accumulate environmental overview quite effectively, and thus build an intrinsic potential to react more swiftly to environmental change over large distances in comparison to groups that behave in accordance to more classic assumptions; i.e., the Paradigm. For example; under multi-scaled space use, if distant patches show improvement with respect to key resources, a functional response driven by spatial memory and parallel processing may represent a net pull effect; i.e., expressed as a net directed emigration rate relative to the local habitat with more constant conditions.

Consequently, the actual “force” driving long-distance pull in a population could be explained as the coarse-scale experience that emerges from a low frequency of “occasional sallies” by an individual outside its normal day of life of habitat explorations.

Under the Paradigm; i.e., the classical home range theory, such occasional sallies have historically been treated as a statistical nuisance, creating all kinds of challenges and creative workarounds for “proper” (Paradigm-compliant) home range demarcations at the local scale where the individual spends 90-99% of its time. Technically, while the Paradigm predicts compliance with a negative exponential distribution of step lengths of an individual (consequently, swiftly running out of steam for longer displacements during a given period), a scale-free kind of space use assumes compliance with a power law distribution (more short and – crucially – more superlong displacements than expected under the Paradigm).

Valle et al. (2017) studied the Florida snail kite within its total distributional range over the years 1997-2013. Then, in 2005 a natural experiment unintentionally appeared. This year an exotic snail species Pomacea maculata appeared in Lake Tohopekaliga in the north of Florida, and subsequently began spreading throughout many of the northern wetlands. These exotic snails have become an important novel food resource for the snail kite population, as a supplement to the kite’s traditional and previously almost exclusive food source, the native Florida apple snail Pomacea paludosa.

With respect to snail kite, a meso-scale functional response then commenced in 2005. Even relatively sedentary adults in the south reacted by showing a rapid increase in net migration towards the northern wetlands, some 260 km apart!

When comparing the frequency with which different groups visited each site (i.e., visitation rate) before the exotic snail invasion (1997–2004) to the next time period when the only invaded site was TOHO (2005–2009), we find a substantial increase for TOHO and a significant decline for WCA3A. This is particularly noteworthy because WCA3A is one of our southernmost sites while TOHO is one of the northernmost sites, revealing a substantial geographic shift in how the landscape is used by these individuals.
Valle et al. 2017, p5

In my view it is not the distance as such that is that main point here (the snail kite can easily traverse long distances in s short period of time), but the fact that the natural experiment provided by the exotic snail showed how some distant patches occasionally showed stronger modular connectivity than intermediate patches. This property of space use is in direct violation of key assumptions of – for example – metapopulation theory (one of the branches of the Paradigm), where spatially close subpopulations cannot be more weakly connected than more distant subpopulations that are separated by intermediate ones.

In Levins’ (1969) original model it is implicitly assumed that all patches are equally connected with respect to migration rate; i.e., regardless of distance, but even this design does not embed a potential for distant patches to be dynamically stronger connected than closer ones.

Thus, strong network modularity over the meso-scale range in Florida may be indicative of a true multi-scaled space use process, involving complex spatio-temporal memory utilization with respect to patch choice by the individual kites. Hence, the Paradigm is challenged by the snail kite results.

… the lack of spatial structure identified in seasonal movements (distance related or otherwise) and results from the partial Mantel tests support previous findings that distance alone is not an adequate predictor of structure in annual dispersal of snail kites (Fletcher et al. 2015). Rather, our findings emphasize the importance of accounting for self-organized population structure, which can arise for several reasons, such as intraspecific cohesion (Gautestad & Mysterud 2006) [e.g. conspecific attraction (Fletcher 2009)], matrix resistance, or natal habitat preference. Network modularity may be a reliable approach for identifying the spatial scales relevant for understanding these processes.

Reichert et al. 2016, p1569

By the way, a glimpse into my own application of network analysis in another context can be found in this post.

 

REFERENCES

Fletcher, R.J. 2009. Does attraction to conspecifics explain the patch-size
effect? An experimental test. Oikos 118:1139–1147.

Fletcher R. J. Jr, A. Revell, B. E. Reichert, W. M. Kitchens, J. D. Dixon and J. D. Austin. 2013. Network modularity reveals critical scales for connectivity in ecology and evolution. Nature Communications 4 (2572):1-7.

Fletcher R. J. Jr, E. P. Robertson, R. C. Wilcox, B. E. Reichert, J. D. Austin and W. M. Kitchens. 2015. Affinity for natal environments by dispersers impacts reproduction and explains geographical structure of a highly mobile bird. Proc. R. Soc. B 282 (2015.1545):1-7.

Gautestad, A.O. and I. Mysterud. 2006. Complex animal distribution and
abundance from memory-dependent kinetics. Ecological Complexity 3:44–55.

Levins R. 1969. Some Demographic and Genetic Consequences of Environmental Heterogeneity for Biological Control. Bulletin of the Entomological Society of America 15: 237-240.

Reichert, B. E., R. J. Fletcher Jr, C. E. Cattau and W. M. Kitchens. 2016. Consistent scaling of population structure across landscapes despite intraspecific variation in movement and connectivity. Journal of Animal Ecology 85:1563–1573.

Valle, D., S. Cvetojevic, E. P. Robertson, B. E. Reichert, H. H. Hochmair and R. J. Fletcher. 2017. Individual Movement Strategies Revealed through Novel Clustering
of Emergent Movement Patterns. Scientific Reports 7 (44052):1-12.

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.

REFERENCES

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.”
http://www.statisticshowto.com/parsimonious-model/

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.

REFERENCES

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.