The Hidden Layer

Focusing on the statistical pattern of space use without acknowledging the biophysical model for the process will create much confusion and unnecessary controversy. Ecologists are now forced to get a better grip on concepts from statistical mechanics than earlier generations. For example, to understand the transformation from data on actual behaviour to pattern analysis of space use, the concept of the hidden layer represents the first gate to pass.

Research on animal movement and space use has always had a central place in ecology. However, as more field data, better computers and more sophisticated statistical methods have become available, some old dogma have come under attack. Specific theoretical aspects of this quest for improved model realism have emerged from the rapidly growing cooperation between biologists and physicists in the emerging field of macro-level biophysics. The so-called Lévy flight foraging hypothesis is one example. And, of course, I can’t resist mentioning the MRW theory.

A booted eagle Hieraaetus pennatus is triggering a flock of spotless starlings Sturnus unicolor to show swarming behaviour. Malaga river delta, December 2017. Photo: AOG.

In 1985 Charles Krebs described ecology as the scientific study of the interactions that determine the distribution and abundance of organisms. In an ethological context animal space use is studied on two levels – tactical and strategic. The tactical level regards understanding individual biology and behavioural ecology on a moment-to-moment temporal scale. Strategic space use adds an extra layer of complexity to the tactical behaviour. In a simplistic manner we may refer to this layer as the animal’s state at a given moment in time; for example whether it is hungry or not (e.g., in hunting mode). Strategy also involves processing of memory-based goals. Strategies executed at coarser time scales than tactics. Some of the interaction between tactics and strategy may then – under specific conditions (see below) – be transformed to dynamic models at the tactical level; so-called mechanistic models, which consists of a set of executable rules covering respective cognitive and environmental conditions. Validating the model dynamics and resulting statistical patterns against real animal data then rates the degree of model realism. For example; realistic, tactical models have been developed to cast light on the “clumping behaviour” (dense swarming) of flock of birds that are threatened by a raptor.

The myriad of rules that influence animal movement makes detailed modelling an impossible task, and would anyway only lead to a descriptive picture with no value to ecological hypothesis testing. In fact, the signature of successful modelling is simplification. Thus, only specific aspects of the reference individual’s behaviour can be included and scrutinized.

The present post addresses one particular aspect of system simplification; coarse-graining the temporal scale. This approach implies a qualitative change of how the space use system is observed and analyzed. Actually, temporal coarse-graining is forced upon us when studying animal space use from sampling an individual’s successive displacements as a series of locations (fixes) during a given period of time. During each inter-fix interval the observed displacement regards the resultant vector from a myriad of intermediate and unobserved events. What has happened to the moment-to-moment kind of behavioural ecology? It has become buried below the hidden layer.

At the surface of this hidden layer you lose sight of behavioural details (like raptor response and swarming rules) but you gain access to an alternative perspective of movement and space use. Alternative statistical descriptors are emerging at this temporally coarser scale, following the laws of statistical mechanics. What is analyzed above the hidden layer is the over-all pattern from many displacements events that are aggregated into a spatial scatter of fixes.

For example, you may coarse-grain both the temporal and spatial system dimensions, and study the aggregated distribution of fixes at the spatial scale of virtual grid cells (pixels) and temporal scale of the fix sampling period. The spatio-temporal variations in intensity of space use within the actual space-time extents then allows for modelling and hypothesis testing, but now using statistical-mechanical descriptors of space use intensity. These descriptors are either not valid below the hidden layer (e.g., the information content of local density of fixes) or they have an alternative interpretation (e.g., movement as a “step” versus movement as a resultant vector for a given interval and location). Both levels of analysis require large sets of input to allow for statistical treatment.

Why is the hidden layer concept and the statistical-mechanical approach more important to relate to today than in earlier decennia? The short answer is the realization – seeded by better and more extensive data – that animal space use involves more than a couple of universality classes of movement (see this post). In fact, in my book, papers and blog posts I have detailed eight classes, most of which are unfamiliar to you.

To understand space use that is influenced by spatial memory and scale-free movement, statistical mechanical modelling is a prerequisite for realistic representation of such complex systems, unless you limit your perspective to a short-term behavioural bout within a very localized arena. In other words, “a single piece of a jigsaw-puzzle of space use dynamics”. For example, if you zoom closely into a small segment of a circle you observe an approximately straight line. Take a step outwards, and you are facing the qualitatively different geometry – the mathematics of a curve and finally a full circle. Stubbornly staying within the linear framework when analyzing more extensive objects than what you observe at fine scales will force you into a corner filled with paradoxes.

Fine-grained and coarse-grained analyses of animal space use are complementary approaches to the same system.

 

MRW and Ecology- Part VII: Testing Habitat Familiarity

Consider having a series of GPS fixes, and you wonder if the individual was utilizing familiar space during your observation period – or started building site familiarity around the time when you started collecting data. Simulation studies of Multi-scaled random walk (MRW) shows how you may cast light on this important ecological aspect of space use.

First, you should of course test for compliance with the MRW assumptions, (a) site fidelity with no “distance penalty” on return events, (b) scale-free space use over the spatial range that is covered by your data, and (c) uniform space utilization on average over this scale range. One single test in the MRW Simulator, the A(N) regression, cast light on all these aspects. First, you seek to optimize pixel resolution for the analysis (estimating the Characteristic scale of space use, CSSU). Next, if you find “Home range ghost” compliance; i.e., incidence I expands proportionally with square root of sample size of fixes, your data supports (a) spatial memory utilization with no distance penalty due to sub-diffusive and non-asymptotic area expansion, (b) scale-free space use due to linearity of the log[I(N)] scatter plot, and (c) equal inter-scale weight of space use due to slope ≈ 0.5.

Supposing your data confirmed MRW, how to test for time-dependent strength of habitat familiarity? Consider the following simulation example, mimicking space use during a season and under constant environmental conditions.

The red dots show log(N,I) for various sample sizes up to the total set of 11,000 fixes. Each dot represents the average I for respective N of the two methods continuous sampling and frequency sampling (counteracting autocorrelation effect; see a previous post). However, analyzing the first 1,000 fixes separately (black dots) consistently revealed a more sloppy space use in terms of aggregated incidence at a given N, relative to the total season. The next 1,000 fixes, however, was compliant with the total series both with respect to slope and y-intercept (CSSU) (green dots).

The reason for the discrepancy in space use during the initial period of fix sampling* was in the present scenario the actual simulation condition; site familiarity was set to develop “from scratch” simultaneously with the onset of fix collection. I define strength of site familiarity as proportional with the total path length from which the model animal collects a previous location to return to**. In the start of the sampling period, the underlying path is short in comparison to the total path that was traversed during the total season, and – crucially – return steps targeted previous locations from the actual simulation period only, and not locations prior to to this start time. In other words, the animal was assumed to settle down in the area at the point in time when the simulation commenced.

To conclude, if your data shows CSSU and slope of similar magnitude in the early and later phase of data collection, you sampled an individual with a well-established memory map of its environment during the entire observation period. The implicit assumption for this conclusion is of course that the environmental conditions was constant during the entire sampling period, including the initial phase. Using empirical rather than synthetic data means that additional tests would have to be performed to cast light on this aspect.

NOTE

*) The presentation above reflects the pixel resolution that was optimized for the total series. The first 1,000 fixes showed a more coarse-grained space use, reflected in a 50% larger CSSU scale (not shown: optimal pixel size was 50% larger for this part of the series) despite constant movement speed and return rate for the entire simulation period. In this scenario a larger CSSU [coarser optimal pixel for the A(N) analysis] signals a less mature habitat utilization in the home range’s early phase. The CSSU was temporarily inflated during build-up of site familiarity, but – somewhat paradoxically – the accumulated number of fix-embedding grid cells (incidence) for a given N at this scale was smaller. These two effects, reflecting degree of habitat familiarity during home range establishment, should be considered a transient effect.

**) Two definitions should be specified:

  • I define strength of site familiarity as proportional with the total path length from which the model animal collects a previous location to return to.
  • I define strength of site fidelity as proportional with the return frequency.

Both definitions rest on the assumptions of no distance penalty on return targets and no time penalty on returns; i.e., infinite spatio-temporal memory horizon relative to the actual sampling period.

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.