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.

Patch Use by Bison in Canada: Memory and Habitat Exploration

Merkle et al. (2014; 2017) have developed and tested a sophisticated memory-including patch selection model for free-ranging bison (Bison bison). Some of the empirical results add support to the Multi-scaled Random Walk (MRW) model (Gautestad and Mysterud 1995, 2005; Gautestad 2015), while other aspects require further details of the bison data to evaluate the potential for full compliance.  

In Merkle et al.’s simulation model; which is described by both a Lagrangian and a quasi-Eulerian variant (integro-difference equation, leading to similar results), animals use spatial and attribute memory to choose food patches based on three components: whether or not they have previously visited them, their reference point of patch profitability derived from recent foraging experience, and their memory of the profitability of each previously visited patch.

First, the results confirm home range as an emergent property of spatial memory. Animals returning to previously visited patches with a higher probability than path crossing by chance leads to constrained space use even without a central place attraction force or borderline repellence. In compliance with both MRW and other memory-implementing models (e.g., van Moorter 2009; Nabe-Nielsen et al. 2013) bison re-visits its memorized patches to a high degree.

Second, bison differentiates (to varying degree) between experienced patch profitability and the quality of the current patch when choosing which patch to return to. While the basic MRW formulation treats all previously visited locations along a path as equal (and thus let returns happen randomly), this “homogeneous environment” simplification is trivially extended with better realism by defining a heterogeneous habitat for the model execution (e.g., Gautestad and Mysterud 2013; see image below). In short, since patches with high profitability can be assumed to be positively correlated with longer local staying time, the MRW model’s Characteristic Scale of Space Use (CSSU) parameter can be defined to vary accordingly. Hence, the local path density (number of locations pr. unit area) in superior patches will tend to be higher than average density. Consequently, even when the animal returns randomly and with equal relative weight to any previous location along the total path, the target location for such return events will tend to be over-represented in high quality patches. Further, when averaging over all visited patches, the basic MRW model is assumed to represent the average CSSU for the given space use extent. Thus, when studying over-all space use properties rather than (for example) intra-home range patch selection, the basic MRW model with homogeneous environment (constant CSSU) satisfies the Occam’s razor principle.

Third, bison chose recently visited meadows as strongly as meadows visited farther into the past. In other words, memory decay was small. This is also a basic property of MRW. However, it could trivially be extended by allowing for varying degree of memory decay through time-dependent return probability to previously visited locations (Gautestad and I. Mysterud 2006; Gautestad and A. Mysterud 2013).

Fourth, Merkle et al. (2017) argue well for space use where targeted (memory-based) returns should be expected to be combined with random exploration. “…a certain amount of random patch use is necessary to avoid frequent returns to relatively poor-quality patches, or avoid being caught in a relatively poor quality area of the landscape” (p185). Bison data confirm this mixture of targeted returns and exploratory bouts, in compliance also with the MRW model.

Four scenarios under MRW and infinite memory horizon, where frequency of return steps is increasing by a factor of 10 per display from left towards right. The embedded red line in the lower part of each panel shows the spatial scale (for example, 1 km) for the respective arenas. For a given accumulated movement length (total path), MRW with a smaller return interval on average covers a smaller area than under condition of a larger return interval (smaller frequency of return). Thus, local density; locations pr. standardized spatial unit embedding at least one location, also increases accordingly. From Gautestad and Mysterud (2013).

What is missing from the bison results? Unfortunately, in order to test if bison confirms MRW-like space use relative to other memory-based models the data needs to be analysed for other properties. For example, the basic MRW model has two main components; (1) occasional return to previously visited patches, and (2) exploratory steps, by default defined by a (truncated) Levy flight step length function. This particular algorithm is applied to mimic scale-free space use; i.e., Multi-scaled random walk. Merkle et al. have studied patch-to-patch distances, but this method obscures a test for degree of scale-free exploratory steps. By defining patch-to-patch movement they do for obvious reasons also find an approximately negative exponential distribution tail of such lengths (Weibull function with shape parameter close to 1). Since a step is defined as the distance from the source meadow to the first encountered next meadow patch, this length distribution should a priori be expected to be a negative exponential in the tail part; i.e., apparently reflecting a scale-specific process. For example, long distance steps could be subjectively terminated if the real target was further out relative to the first-encountered patch. In Gautestad (2013) I illustrate some pitfalls leading to false verification of scale-specific space use from real scale-free movement.

Further, by defining patch-to-patch steps (meadow-to-meadow), any subsequent intra-patch exploratory steps up to next patch departure will obscure analysis of the frequency of return steps relative to exploratory steps. MRW assumes a relatively small return frequency relative to exploratory steps (the ρ ratio; see the illustration above) while Merkle et al. (2014) find ca 2/3 of steps targeting familiar patches relative to reaching previously unexplored meadow. In the bison analysis the actual definition of intra-patch movement may unintentionally have inflated this ratio substantially, since most Levy flight-like steps are relatively short. This means that estimating ρ requires more data details and another approach.

Hence, the very interesting analyses presented by Merkle et al. (2014; 2017) unfortunately does not cast light on any potentially scale-free property of bison movement. Future work could for example use the step length function studied from the original GPS series of fixes. By sub-sampling these series at different time resolutions (e.g., 3 hours, 10 hours, etc.) and spatial sub-sections (to study variations of the local CSSU estimates, for example, by differentiating between steps inside and outside of meadows) the MRW compliance – and applicability as a tool for ecological inference – could be illuminated. It would also be interesting to see degree of model compliance and parameter differences between the winter and summer season.

REFERENCES

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.

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

Gautestad, A. O. and I. Mysterud. 1995. The home range ghost. Oikos 74:195-204.

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

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

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

Merkle, J. A., D. Fortin and J. M. Morales. 2014. A memory-based foraging tactic reveals an adaptive mechanism for restricted space use. Ecology Letters 17:924–931.

Merkle, J. A., J. R. Potts and D. Fortin. 2017. Energy benefits and emergent space use patterns of an empirically parameterized model of memory-based patch selection. Oikos 126:185–195

Nabe-Nielsen, J., J. Tougaard, J. Teilmann, K. Lucke and M. C. Forckhammer. 2013. How a simple adaptive foraging strategy can lead to emergent home ranges and increased food intake. Oikos 122:1307-1316.

van Moorter, B., D. Visscher, S. Benhamou, L. Börger, M. S. Boyce and J.-M. Gaillard. 2009. Memory keeps you at home: a mechanistic model for home range emergence. Oikos 118:641-652.