Intraspecific Cohestion from Conspecific Attraction, Part II: Paradox Resolved

I briefly mentioned in Part I that the combination of spatial memory utilization and scale-free space use under the Parallel processing conjecture (PP) may lead to a fractal compliant population dispersion of intrinsic origin, given the additional condition of conspecific attraction. Below I elaborate on heterogeneous population dispersion as expected under the Paradigmatic framework (Markovian process, mean field compliance) and the contrasting PP kind of space use. In particular, one may find that two locations with different population density under the PP condition may reveal similar intensity of space use! Under the Paradigm such a result will appear paradoxical. Under the PP framework (the MRW and the Zoomer model) the paradox is resolved.

Before switching to empirical results, consider the following crucial question for ecological theory of space use. What is the driving force behind the typical pattern of a shifting mosaic of population abundance over a range of spatial scales, apparently even before the landscape structure (habitat heterogenity) is considered as a complicating factor? In other words, a population is heterogeneously scattered over space. This widespread phenomenon, widely documented in insects, was intensively debated during the last half of the 20th century. In particular, because the dispersion typically adhere to a power law of fluctuation of abundance, i.e., a scale-free phenomenon (Taylor 1986). However, despite many attempts over the years to model such complex patterns they still appear quite paradoxical, with little degree of consensus. My own attempt to drill into this phenomenon led to the idea and development of the PP concept already in the early 1990’s. I cite from my post “Simulating Populations VI: the Unrealism of Standard Models“, dated 10 March, 2018:

First, consider the condition where the individuals are living in an open environment, which is the general condition. No species is abundant everywhere, meaning that a given population is surrounded by unoccupied space […] In some cases this fringe zone may be easily understood from the perspective of unfriendly neighbourhood in habitat terms, but often the surroundings appear quite similar to the conditions inside the distribution range. For example, the over-all population may be spatially fragmented with respect to abundance; particularly along the core areas’ perimeters, with no apparent reason why small and large chunks of the intermediate areas should not be habitable. Pick any well-studied species, and ask an expert on its distributional range to explain population absence in some regions with apparently similar ecological conditions to the population’s present range. In an in my view unsatisfactory manner the way out of this dilemma (the “shoot from the hip” answer) is normally to point to some environmental factor still not revealed, or some kind of statistical chance effect. In my Zoomer model such apparently inexplicable “clumping” of a population is explained as an emergent property of conspecific attraction and scale free redistribution of some individuals. A given number of individuals cannot be everywhere all the time.

The concept of trivial and non-trivial congregation of individuals in a population below carrying capacity is illustrated in a hypothetical spatial arena, where the local carrying capacity (cc) is defined by circle size at grain scale 1/25 of arena size, and population density relative to local cc is defined by the black part of the circles. The two situations in the upper row show homogeneous arena conditions, while the lower parts show heterogeneous conditions. A trivial dispersion pattern (Paradigm-compliant) is illustrated to the left in both rows, where Poisson-variability in abundance is simplified as uniform mean density expectancy at similar local cc levels, and the expected abundance changing linearly (proportionally) with cc. Abundance below cc can – for example – be due to a population slowly recovering from a recent resource bottleneck event. A population growth rate that is slower than the rate of inter-population mixing is traditionally assumed in population dynamics models, due to their assumption of valid mean field approximations. Thus, individuals are expected to be distributed in linear proportion to local resource levels even below carrying capacity levels. A non-trivial abundance pattern (PP compliant), to the right in both rows, is defined as density levels that does not correlate proportionally with cc fluctuations, whether cc is uniformly dispersed or not.

To illustrate concept of system complexity from intraspecific cohesion, consider the sketch to the right (Gautestad and Mysterud 2006). As shown in the upper row, even in a hypothetical homogeneous habitat the process of conspecific attraction predicts heterogeneous space use (upper right). As stated above, “A given number of individuals cannot be everywhere all the time”. Individuals are congregating spatially within the limits that are set by local environmental constraints (size of the circles; upper row, right). In a heterogeneous environment (lower row, right), where these local constraints vary, this intrinsic force is mingling with the effect from local conditions*.

The sketch illustrates population dispersion at the given spatial scale of the virtual grid. Crucially, under PP a similar kind of heterogeneous dispersion with similar parameter values except for a trivial rescaling operation is expected if the observational scale is changed to another grid resolution. In other words, the dispersion is statistically scale-free, or fractal-compliant.

Consequently, both at the individual and the population level, to study PP-compliant space use where this property is verified (scale-free and memory-driven patch attraction or conspecific attraction, respectively) it is required to use PP-derived methods to infer intensity of local space use. This contrasts with the simpler expectation from standard mean field compliant modelling (the Paradigm), where a relatively straightforward positive correlation between local resource level and local abundance is expected (upper and lower left images in the Figure). As repeatedly shown elsewhere, under the Paradigm the intensity of space use is basically proportional with population density. Under the PP model, the intensity of use is less straightforward, and requires different quantification.

As an example of the statistical strength of applying PP-derived methods to study habitat selection where the animals had scale-free and memory-influenced space use, follow this link (PDF).

To summarize, one should first test whether a given space use dispersion complies with the Paradigm or the PP framework. Next, one should apply a method from respective statistical toolboxes to infer ecology, like quantifying local intensity of space use as a function of – for example – local resource level.

By applying a PP model, one may find that two locations with similar local density of individuals have different intensity of space use (indicating different strength of habitat selection). One may also find that two locations with different density may reveal similar intensity of space use! Under the Paradigm such results appear paradoxical. Under the PP framework (the MRW and the Zoomer model) the paradox is resolved.

NOTE

At the individual level and using the MRW model, a similar self-organized “clumping” emerges due to self-reinforcing patch use from targeted returns.

REFERENCES

Taylor, L. R. 1986. Synoptic dynamics, migration and the Rothamsted insect survey. J. Anim. Ecol. 55:1-38.

Intraspecific Cohesion From Conspecific Attraction, Part I: Overview

Animal survival requires some kind of intraspecific cohesion (“population glue”); an intrinsically driven tendency to counteract the diffusion effect from free dispersal. Populations of most species live in an open environment. Thus, without some kind of behavioural capacity to continuously or periodically seek and maintain contact with conspecifics only the most extreme and bizarre kind of environmental constraint would be required for the species’ long term survival. Despite the general agreement on this basic requirement, ecological models for individual movement and population dynamics have for a century maintained a different paradigm, a stubborn assumption that the animals follow the basic principles of mechanistic (Markovian) movement at the individual level and diffusion-advection laws at the population level. Fueled by empirical results and – in my view – common sense the Paradigm has from some researchers come under attack for many years, but mostly in vain. Ecological models and statistical methods are still deeply dependent on the Paradigm.

Visual, auditory and olfactory cues are often not satisfactory mechanisms for individuals to effectively return home when a sally brings them too far away to maintain sufficient population contact. Hence, under the premise of Paradigm compliant behaviour*, the result is an extensive “lost in space” risk, in particular at the fringe zones of a population. However, since field data typically reveals a capacity for targeted returns beyond sensory guidance, this property should be included in space use model designs where applicable.

What does “a capacity for targeted returns beyond sensory guidance” mean? In both the book, in papers and throughout this blog I have presented two complementary and parsimonious models for the individual and the population level respectively, the Multi-scaled random walk (MRW) and the Zoomer model. These inter-coherent approaches are based on two Paradigm-opposing assumptions; (a) individuals have a capacity for spatial memory, and (b) individuals are processing this spatio-temporal information in compliance with the Parallel processing conjecture (PP; providing a potential for scale-free habitat utilization along three axes; space, time and scale). Both PP aspects are hypothesized to be key to mimic real population dynamics realistically. For example, they open for “a capacity for targeted returns beyond sensory guidance”.

How to mimic conspecific attraction at the system resolution of individuals? The series show the accumulated spatial dispersion of relocations from seven simulations of MRW with partly inter-dependent series, by letting a uniformly sampled subset of locations, a ‘‘seed set’’, from one series become a starting point for another series. This seed set kernel, the common ‘‘return base’’, then represents a space use template, which the new series will tend to emphasize for the rest of the series. This can be interpreted as an animal that is utilizing its habitat similarly but not identically to another individual with respect to patch preferences over a range of scales. In particular, the individuals also share some common meeting points. The smaller the seed set, the more independent space use by the individuals. From Gautestad and Mysterud (2006).

Most crucially, I have presented several statistical methods that have the power to test PP-compliant space use against the Paradigm, including the challenging task to differentiate between PP and Markov-compliant execution of spatial memory. I have also presented several such tests on empirical data. On one hand, the spatial dispersion of an accumulated set of individual relocations (fixes) and on the other hand a snapshot of population dispersion of individuals produce clear statistical fingerprints using proper analyses, which can distinguish between a Paradigm- and a PP-compliant kind of behaviour.

If the PP conjecture continues to gain support, it is a short step to prolong this parsimonious modelling approach to explicitly explore the mathematically challenging aspect of intraspecific cohesion from conspecific attraction – outside the comfort zone of standard mechanistic and statistical-mechanistic (Markovian) designs. The illustration to the right shows an example using the MRW design. Alternatively, the Zoomer design could be explored if one prefers a population level abstraction of space use (an “ensemble” rather than a “particle” approach in statistical-mechanical terms).

Basically, both approaches require just one extra assumption to study intraspecific cohesion. This assumption states that contact with conspecifics represent a resource for the animals, in a sense in line with other resources like food, shelter, refuges from predators, and so on (Gautestad and Mysterud 2006). However, inclusion of conspecific attraction in compliance with the PP conjecture predicts a qualitatively different kind of dynamics than you find under the Paradigm, resulting in complex space use and population dispersion. In particular, space use dispersion of individuals and local density fluctuation of the population develop an emergent property of a statistical fractal (aggregations within aggregation within…).

A log–log transformed histogram of frequency of local grid cell densities of locations from the merged set of the seven series of relocations confirm a power law distribution with slope close to the expected -1 (superimposed, best fit). The result supports a statistical fractal of space use for the population of individuals. From Gautestad and Mysterud (2006).

By applying the PP-developed methods, both the MRW and the Zoomer model find support in data covering a wide range of species and taxa. Long-lasting but generally under-communicated paradoxes that appear when the Paradigm is confronted with real data find solution under the complementary PP models.

As always in ecology, exceptions to the general rule should be expected, but so far the PP compliance seem to be very comprehensive across the animal kingdom of vertebrates, and apparently also among many invertebrates (references are given in my book and here in my blog). Again, I challenge readers to test their own animal space use use data; whether they regard GPS fixes of mammal movement or multi-scale analysis of population dispersion of insects.

Unfortunately, my book (Gautestad 2015) is still the only extensive theoretical wrap-up of the Paradigm, why it needs to be replaced as the default framework of space use ecology, and how an  alternative approach may be developed, explored and empirically tested. A broader scrutiny – involving follow-up from expertise covering a wide range of fields from wildlife management to theoretical modelling – is needed.

NOTE

*) Model examples of Paradigm compliant population dynamics are spatially implicit differential equations, spatially explicit partial differential equations and difference equations (including metapopulation models).

REFERENCES

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. 2006. Complex animal distribution and abundance from memory-dependent kinetics. Ecological Complexity 3:44-55.

 

Conservation Biology and SLOSS , Part I: Time to Challenge System Assumptions

The contrasting ideas of a single large or several small (SLOSS) habitat reserves ignited a heated debate in conservation biology (Diamond 1975; Simberloff and Abele 1982). The recent development in movement ecology – in particular the theoretical aspects of spatial memory and scale-free space use of individuals – makes time ripe to initiate a study of the SLOSS concept under this contemporary perspective. In order to produce realistic predictions community, population and individual processes need to be understood from a coherent system theory involving all levels of system abstraction. Under this premise the original SLOSS concept seems to fall apart.

A single large reserve was argued to be preferable to several smaller reserves whose total areas were equal to the larger (Diamond 1975). On the other hand, if the smaller reserves had unshared species it was possible that two smaller reserves it sum could have more species than a single large reserve of the same total area (Simberloff and Abele). Opposing the latter view, it was argued that habitat fragmentation is probably the major threat to the loss of global biological diversity (Wilcox and Murphy 1985).

However, the SLOSS concept – originating in the field of conservation biology – was understood from the perspective of dispersion of species under various landscape configurations. In my view that approach drove the debate into a dune of sand, due to over-simplification or ignorance of a community system’s lower-level dynamics.

In a traditional metapopulation system, immigration to a given local population (green area) can be concptualized as a fixed rate, representing the average number of emigrant individuals from other populations that by chance happen to reach the present population pr. unit time. In the alternative Zoomer model; the population level formulation of the individual MRW model, immigrants represent a mixture of individuals from other populations that perform exploratory moves over a wider range of scales than assumed by the classic model; i.e., scale-free movement, in addition to individuals that return in a spatially memory-dependent manner (red arrows). Such goal-oriented spatial behaviour leads to the emergence of complex population kinetics.

The original SLOSS debate was considering relative presence and absence of species in various settings of reserve designs. From this classic community system perspective the theory tended to ignore population dispersal effects explicitly. However, to the extent that dispersal was considered [primarily in the context of metapopulation theory; see, for example Robert (2009)] it was assumed that emigration was a deterministic process at the spatial scale of local populations while immigration was a stochastic process at this scale (random influx). This direction-dependent toggle between deterministic and stochastic population flow follows logically from the traditional premise of a diffusion-advection framework for metapopulation systems. I criticized this core assumption of metapopulation theory in this post, and in the Figure to the right I summarize the argument. My critique pinpointed the theoretical consequences of allowing for complex population kinetics (the Zoomer model). The basic metapopulation principle of slow rate of population mixing between subpopulations may in this system variant be additionally influenced by some degree of returning individuals; i.e., a deterministic component of population flow even of the immigration rate. This potential is facilitated by spatial memory and temporally multi-scaled displacements (parallel processing). In other words, system complexity.

Such complexity plays havoc with a classic metapopulation system. After occasional sallies to other habitat reserves/sub-populations, individuals may perform directed and potentially long distance returns to a previous reserve. And such returns could be a function of the respective individuals’ current conditions both at source and target reserves.

Consequently, individuals are – according to this paradigm-opposing assumption – occasionally able to transcend the matrix of intermediate environment between refuges in an energetically effective manner and with sufficiently low travel risk to make such coarse-scale moves statistically worthwhile and positive in fitness terms (see this post). This property of the Multi-scaled random walk (MRW) theory is now gaining additional anecdotal support, for example from studies on Fowler’s toads Anaxyrus fowleri (Marchand et al. 2017) and free-ranging bison Bison bison.

To be more precise, let’s first assume that we are considering a “several small” reserve system where the spatial scale (size) of these local reserves is reflecting – say – 50-60% of the median displacement distance of the respective population constituents at the time resolution of a reproductive season. If we consider day-to-day movement rather that the coarser time scale of a season, long displacements beyond this defined 50-60% limit accumulate a small part of the area under the displacement distribution (the scale-free dispersal kernel becomes more apparent at fine temporal scales). In this manner, due to the “thin” long-tale part of displacements under the premise of scale-free space use, the system complies with the basic metapopulation property of a low inter-season migration rate between subpopulations relative to the higher intra-population mixing rate at finer scales*. Also consider that the long-tail part of the displacement distribution extends to the scale of this  system’s spatial extent. Hence, individuals may occasionally displace themselves over the entire arena (exploratory sallies of various length and duration), and occasionally perform directed returns to previous locations within the same scale range.

This assumed property of scale-free and memory-influenced returns at the individual level violates the traditional metapopulation theory at the population level and – of course – the traditional SLOSS theory at the community level. In addition to the relatively high frequency mixing of individuals at the local scale a less frequent and partly memory-driven mixing takes place at coarser scales. The latter contributes to the emergence of an inter-connected network of complex migration.

The very interesting studies on snail kite Rostrhamus sociabilis plumbeus in Florida (Reichert et al. 2016; Valle et al. 2017) illustrates the profound potential this kind of system complexity may generate, with consequence also for community ecology in general and the SLOSS debate in particular. The results 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. I cite from my discussion of Valle et al.’s paper:

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. […] 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 particular, observe that even long distance returns could be a function of the respective individuals’ current conditions both at source and target reserves. Referring to the illustration above, the relative number of directed returns (deterministic behaviour, red arrows) may suddenly – at a finer temporal scale than the inter-season population change –  increase or decrease substantially. The cause of such events cannot be understood simply from local conditions, as under traditional mechanistic modelling. Coarser scale conditions both in time and space need to be studied in parallel. A single, more or less fixed immigration rate as applied in traditional metapopulation modelling does not suffice. Citing from above, “…some distant patches occasionally showed stronger modular connectivity than intermediate patches“. This is system complexity in a nutshell, whether we are considering single- or multi-species scenaria. Much work is needed to achieve a better understanding of multi-scaled behaviour under influence of spatial memory.

NOTE

*) As explained in other posts (for example, here), a scale-free displacement curve for individuals is expected to become more truncated if the time scale is coarsened, for example from short term displacements to longer sampling intervals. The reason is the increased influence from intermediate return events as observation intervals slide from the fine-grained time scale of population kinetics (observing fast mixing from individual movement) towards the coarser-grained dynamics of changing dispersion for example at the scale of a season (Gautestad 2012).

REFERENCES

Diamond, J.M. 1975. The Island Dilemma: Lessons of Modern Biogeographic Studies for the Design of Natural Reserves. Biological Conservation 7:129–146.

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

Reichert, B. E., R. J. Fletcher, 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.

Robert, A. 2009. The effects of spatially correlated perturbations and habitatconfiguration on metapopulation persistence. Oikos 118:1590-1600.

Simberloff, D. S. and L. G. Abele. 1982. Refuge design and island biogeographic theory – effects of fragmentation. American Naturalist 120:41-56.

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.

Wilcox, B. A., and D. D. Murphy. 1985. Conservation strategy – effects of fragmentation on extinction. American Naturalist 125:879-887.

 

Accepting spatial Memory: Some Alternative Ecological Methods

In this post I present a guideline that summarizes how a memory-based model with an increasing pile of empirical verification covering many species – the Multi-scaled Random Walk model (MRW) – may be applied in ecological research. The methods are in part based on published papers and in part based on some of the novel methods which you find scattered throughout this blog. 

In the following, let us assume for a given data set that we have verified MRW compliance (using the standard memory-less models or alternative memory-implementing models as null hypotheses) by performing the various tests that have already been proposed in my papers, blog posts and book. Typically, a standard procedure should be to verify (a) site fidelity; i.e., presence of a home range, (b) scale-free space use by studying the step length distribution from high frequency sampling, and (c) the fractal dimension D≈1 of the spatial scatter of relocations in the resolution range between the dilution effect (very small pixels) and the space fill effect (very large pixels).

The obvious next step is to explore specific ecological hypotheses using the MRW as the model for space use. Here follows a quick tutorial:

  1. Study the Home range ghost model, I(N) = cNz, to estimate c, the individual’s characteristic scale of space use (CSSU). Search my blog for methods how to optimize grid resolution, and in particular also consider the recent breakthrough 11 months ago where I show how to estimate CSSU from auto-correlated data. Variations in CSSU quantifies difference in intensity of space use, which logically illuminates aspects of habitat selection.
  2. After proper estimation of CSSU study the power exponent, z, of the Home range ghost model. If your data lands on z≈0.5 – the default condition – you have verified that the individual has not only utilized its environment in a scale-free manner but also has put “equal weight” into relating to its habitat across the spatial scale range within its home range. On the other hand, finding 0.2<z<0.3 indicates that a model for an alternative movement class, the Markov-compliant MemRW, may be more suitable for your data. 0 < z < 0.3 indicates that the individual has concentrated its space utilization primarily towards finer resolutions, like you would expect from a Markov-compliant kind of cognitive processing. More detailed procedures should be applied to select model framework, since MemRW and MRW describes qualitatively disparate classes. Observe that the traditional KDE method is not able to discriminate between these classes; hence, typically showing weak N-dependence on area demarcation, due to its implicit assumptions that successive revisits to a local patch are independent events and that the process should be memory-less (see link above).
  3. Is the individual’s space use stationary, or is the home range drifting over time? Spatial autocorrelation in your series of fixes typically has two causes; high-frequency sampling of fixes from space use relative to a slower return frequency (ρ>>1; see my previous post) or high- or medium-frequency fix sampling under the condition of a drifting home range. Split the data into several subsets of magnitude Ns where the number of fixes (N) in each set is constant. Then study the overlap pattern of incidence I(N) at spatial resolution of CSSU (see method here). Low degree of overlap between successive subsets implies a non-stationary kind of home range. By comparing non-adjacent subsets in time one may even quantify the degree of non-stationarity (the speed by which the space use is drifting). These results can then be interpreted ecologically.
  4. What about the fractal dimension of the total set of fixes, for example by applying the box counting method? By default one expects D≈1 when z=0.5. Deviations from D=1 over specific spatial resolutions can be interpreted ecologically. For example, 1.5<D<2 at the coarsest resolutions may indicate that space use is constrained by some kind of borders. However, it could also appear from missing outlier fixes in the set (Gautestad and Mysterud 2012) or a simple statistical artifact (the space fill effect). On the other end, 1<<D can be hypothesized to emerge where the animal has concentrated its space use among a set of fine-scale patches rather than scattering is optimization more smoothly (in a statistical sense) over a wider range of scales. In Gautestad (2011) I simulated central place foraging, where i found 0.7<D<1. More sophisticated but logically simple methods can contribute to various system properties and statistical artifacts that contribute to deviation from D≈1, for example by varying the sample size of fixes as illustrated in the Figure to the right (copied from the link above).

The MRW theory also offers several other methods to study ecological and biological aspects of space use. For example, the data may reveal whether the temporal memory horizon has been constrained or unlimited (infinite memory, or remembering previous visits only over a limited, trailing time window). Temporally constrained memory will be shown by example in my next post. For more theoretical or technical details of the methods above please search this blog for the actual term, or find references in the subject index of my book.

REFERENCES

Gautestad, A. O. 2011. Memory matters: Influence from a cognitive map on animal space use. Journal of Theoretical Biology 287:26-36.

Gautestad, A. O. and I. Mysterud. 2012. The Dilution Effect and the Space Fill Effect: Seeking to Offset Statistical Artifacts When Analyzing Animal Space Use from Telemetry Fixes. Ecological Complexity 9:33-42.

Positive and Negative Feedback Part I: Individual Space use

The standard theories on animal space use rest on some shaky behavioural assumptions, as elaborated on in my papers, in my book and here in my blog. One of these assumptions regards the assumed lack of influence of positive feedback, in particular the self-reinforcing effect that emerge when individuals are moving around with a cognitive capacity for both temporal and spatial memory utilization. The common ecological methods to study individual habitat use; like the utilization distribution (a kernel density distribution with isopleth demarcations), use/availability analysis, and so on, explicitly build on statistical theory that not only disregards such positive feedback, but in fact requires that this emergent property is not influencing the system under scrutiny.

Unfortunately, most memory-enhanced numerical models to simulate space use are rigged to comply with negative rather than positive feedback effects. For example, the model animal successively stores its local experience with habitat attributes while traversing the environment, and it uses this insight in the sequential calculation of how long to stay in the current location and when to seek to re-visit some particularly rewarding patches (Börger et al. 2008, van Moorter et al. 2009, Spencer 2012, Fronhofer et al. 2013; Nabe-Nielsen et al. 2013). In other words, the background melody is still to maintain compliance with the marginal value theorem (Charnov 1976) and the ideal free distribution dogma, which both are negative feedback processes and not a self-reinforcing process that tends to counteract such a tendency.

Negative feedback (or balancing feedback) occurs when some function of the output of a system, process, or mechanism is fed back in a manner that tends to reduce the fluctuations in the output, whether caused by changes in the input or by other disturbances. Whereas positive feedback tends to lead to instability via exponential growth, oscillation or chaotic behavior, negative feedback generally promotes stability. Negative feedback tends to promote a settling to equilibrium, and reduces the effects of perturbations. Negative feedback loops in which just the right amount of correction is applied with optimum timing can be very stable, accurate, and responsive.
https://en.wikipedia.org/wiki/Negative_feedback

A common curlew Numenius arquata foraging on a field within its summer home range. Anecdotally, one may observe that a specific individual tends to revisit not only specific fields while foraging, but also specific parts of these fields. If this site fidelity is influenced by a rising tendency to prefer familiar space on expense of revisiting potentially equally rewarding patches based on previous visits, a positive feedback (self-reinforcing space use) has emerged.  This effect then interferes with the traditional ecological factors, like selection based on habitat heterogeneity, in a complex manner. Photo: AOG.

The above definitions follow the usual path to explain negative feedback as “good”, and positive feedback as something scary (I will return to this misconception in a later post in this series). It echoes the prevailing “Balance of nature” philosophy of ecology, which I’ve criticized at several occasions (see, for example, this post).

In a previous post, “Home Range as an Emergent Property“, I described how memory map utilization under specific ecological conditions may lead to a self-reinforcing re-visitation of previously visited locations (Gautestad and Mysterud 2006, 2010); in other words, a positive feedback mechanism*. Contemporary research on animal movement covering a wide range of taxa, scales, and ecological conditions continues to verify site fidelity as a key property of animal space use.

I use a literature search to test an assumption of the ideal models that has become widespread in habitat selection theory: that animals behave without regard for site familiarity. I find little support for such “familiarity blindness” in vertebrates.
Piper 2011, p1329.

Obviously, in the context of spatial memory and site fidelity it should be an important theme for research to explore to what extent and under which conditions negative and positive feedback mechanisms are shaping animal space use.

Positive feedback from site fidelity will fundamentally influence analysis of space use. For example, two patches with a priori similar ecological properties may end up being utilized with a disproportionate frequency due to initial chance effects regarding which patch happened to gain familiarity first**. Further, if the animal is utilizing the habitat in a multi-scaled manner (which is easy to test using my MRW-based methods), this grand error factor in a standard use/availability analysis cannot be expected to be statistically hidden by just studying the habitat at a coarser spatial resolution within the home range.

Despite this theoretical-empirical insight, the large majority of wildlife ecologists still tend to use classic methods resting on the negative feedback paradigm to study various aspects of space use. The rationale can be described by two end-points on a continuum: either one ignores the effect from self-reinforcing space use (assuming/hoping that the effect does not significantly influence the result), or one use these classic methods while holding one’s nose.

The latter category is accepting the prevailing methods’ basic shortcomings – either based on field experience or inspired by reading about alternative theories and methods – but the strong force from conformity in the research community is hindering bold steps out of the comfort zone. Hence, the paradigm prevails. Again I can’t resist referring to a previous post, “Why W. H. Burt is Now Hampering Progress in Modern Home Range Analysis“.

Within the prevailing modelling paradigm, implementing spatial memory utilization in combination with positive feedback-compliant site fidelity is a mathematical and statistical nightmare – if at all possible. However, as a reader of this blog you are at least fully aware of the fact that some numeric models have been developed lately, ouside the prevailing paradigm. These approaches not only account for memory map utilization but also embed the process of positive feedback in a scale-free manner (I refer to our papers and to my book for model details; see also Boyer et al. 2012).

 

NOTES

* The paper explores space use under the premise of positive feedback during superabundance of resources, in combination with negative feedback during temporal and local over-exploitation.

** In Gautestad and Mysterud (2010) I described this aspect as the distance from centre-effect; i.e., the utilization distribution falls off at the periphery of ulilized patches independently of a similar degradation of preferred habitat.

 

REFERENCES

Börger, L., B. Dalziel, and J. Fryxell. 2008. Are there general mechanisms of animal home range behaviour? A review and prospects for future research. Ecology Letters 11:637-650.

Boyer, D., M. C. Crofoot, and P. D. Walsh. 2012. Non-random walks in monkeys and humans. Journal of the Royal Society Interface 9:842-847.

Charnov, E. L. 1976. Optimal foraging: the marginal value theorem. Theor. Popula. Biol. 9:129-136.

Fronhofer, E. A., T. Hovestadt, and H.-J. Poethke. 2013. From random walks to informed movement. Oikos 122:857-866.

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 I. Mysterud. 2010. Spatial memory, habitat auto-facilitation and the emergence of fractal home range patterns. Ecological Modelling 221:2741-2750.

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.

Piper, W. H. 2011. Making habitat selection more “familiar”: a review. Behav. Ecol. Sociobiol. 65:1329-1351.

Spencer, W. D. 2012. Home ranges and the value of spatial information. Journal of Mammalogy 93:929-947.

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.

Analytical Sensitivity to Fuzzy Fix Coordinates

In empirical data GPS fixes are never exact positions. A “fuzziness field” will always be introduced due to uncertain geolocation. When analyzing a set of fixes in the context of multi-scaled space use, are the parameter estimates sensitive to this kind of statistical error? Simultaneously, I also explore the effect on constraining the potential home range by disallowing sallies to the outermost range of available area.

To explore the triangulation error effect on space use analysis I have simulated Multi-scaled random walk in a homogeneous environment with N=10,000 fixes (of which the first 1,000 fixes were discarded) under two scenaria; a “sharp” location (no uncertainty polygons), and strong fuzziness. The latter introduced a random displacement to each x-y coordinate with a standard deviation (SD) of magnitude approximately equal to the system condition’s Characteristic scale of space use (CSSU). Displacements to the outermost parts of the given arena was disallowed, to study how this may influence the analyses. I then ran the following three algorithms in the MRW Simulator: (a) analysis of A(N) at the home range scale, (b) analysis of A(N) at a local scale (splitting the home range into four quadrants), and (c) analysis of the fix scatters’ fractal property.

The following image shows the sharp fix set and the strongest fuzzyness condition.

By visual inspection is is easy to spot the effect from the spatial error (SD = 1182 length units, upper row to the right). However, the respective A(N) analyses at the home range scale generated a CSSU estimate that was only 10% larger in the fuzzy set (linear scale). When superimposing cells of CSSU magnitude onto each fix, the home ranges appear quite similar in overall appearance and size. This was to be expected, since fuzziness influences fine-resolution space use only.

Visually, both home range areas appear somewhat constrained with respect to range, due to the condition to disallow displacements to the peripheral parts of the defined arena (influencing less than 1% of the displacements).

A(N) analysis (the Home range ghost). The two conditions appeared quite similar in plots of log[I(N)], where I is the number of fix-embedding pixels at optimized pixel resolution, as described in previous posts.

However, for the fuzzy series there is a more pronounced break-point with a transition towards exponent ∼0.5 in sample size of magnitude larger than log2(N) ≈ 3. This break-point “lifted” the regression line somewhat for the fuzzy series, leading to a slightly larger intercept with the y-axis when interpolating towards log(N) = 0. This difference between the two conditions with respect to the y-intercept, Δlog(c) from the home range ghost formula log[I(N)] = log(c) + z*log(N), also defines the difference in CSSU when comparing sharp and fuzzy data sets. recall that CSSU ≡ c.

The spatial constraint on extreme displacements relative to the respective home ranges’ centres apparently did not influence these results.

I have also superimposed local CSSU-analysis for respective four quadrants of the two home ranges. When area extent for analysis is constrained in this manner; i.e., spatial extent is reduced to 1:4 in area terms (1:2 linearly) for each local sub-set of fixes, respective (N,I) plot needs to be adjusted by a factor which compensates for the difference in scale range.

Since the present MRW conditions were run under fractal dimension D=1, each local log2[(N,I)] plot is rescaled to log2[(N,I)] + log2D(grain/extent)] = log2[(N,I)] + 1 when Δ regards the relative change of scale under condition D=1. After this rescaling the over-all CSSU and the local CSSU are overlapping, as shown by the regression lines in the A(N) analyses above. Overlapping CSSU implies that the four quadrants had similar space use conditions, which is true in this simplified case.

Fractal analysis. The Figure below shows the magnitude of incidence I as a function of relative spatial resolution k (the “box counting method”), spanning the scale range from k=1 (the entire arena, linear scale 40,000 units) and down to k = 40,000/(212) = 9.8 units, linear scale*.

Starting from the coarsest resolution, k=1, the log-log regression obviously shows = 1. At resolution k=1:2 and k=1:4 (4 and 16 grid cells, respectively), I = 4 and I = 4. In other words, all boxes contain fixes at k=1/2, apparently satisfying a two-dimensional object, and at k=1:4 some empty cells (12 of 16) are peeled away as empty space from the periphery of the fix scatter at this finer resolution.

This coarse-scale pattern is a consequence of the defined space use constraint. Disallowing “occasional sallies” outside the core home range obviously influences the number of non-empty boxes relative to all boxes available at the coarse resolutions 1<k<4-1.

However, at progressively finer resolutions – below the “space fill effect” range transcending down to ca k=1:32 – the true fractal nature of the scatter of fixes begin to appear due to log-log linearity, confirming a statistical fractal with a stable dimension D≈1.1 over the resolution range 2-9 < k < 2-5  (showing log-log slope of -1.1). At finer resolutions, the dilution effect flattens further expansion of I. The D=1.1 scale range is close to expectation from the simulation conditions Dtrue=1, while the deviations above and below this range are trivial statistical artifacts from space filling and dilution.

The most interesting pattern regards the finest resolution range, where the fuzzy set of fixes somewhat unexpectedly follows a similar log(k,I) pattern as the non-fuzzy set. However, the slight difference in D, which increases to D =1.17, may be caused by the fuzziness (proportionally stronger space-fill effect as resolution is increased).

To conclude, if the magnitude of position uncertainty does not supersede the individual’s CSSU scale under the actual conditions, the A(N) analyses of MRW compliant space use does not seem to be seriously influenced by location fuzziness. The “fix scrambling” error is in most part subdued towards finer scales than the CSSU.

However, the story doesn’t end here. I have superimposed a dotted red line onto the Figure above. Overlapping with the D=1 section of grid resolutions, the line is extrapolated towards the intersection with log(I) ≈ 0 at a scale that is 21.5 = 2.8 times larger (linear) scale than the actual arena for the present analyses. In other words, in absence of area constraint and step length constraint (and disregarding step length constraint due to limited movement speed of the individual) one should expect the actual set of fixes to “fill up” the missing incidence over the coarse scale range, leading to D≈1 for the entire range from the dilution effect range towards coarser scales.

I have also marked the CSSU scale as the midpoint of the red line. A resolution of log2(k)=-5.5 is in fact very close to the CSSU estimate from the A(N) method (k=1:50 of actual arena using the A(N) method, versus 1:35 of the arena according to the fractal analysis). This alternative method to estimate CSSU was first published in this blog post.

A preliminary development towards this approach was explored both theoretically and empirically in Gautestad and Mysterud (2012).

NOTE

*) This analysis of N= 8,000 fixes, spanning box counting of 1, 2, 4, 16, 32, … 16.8 million grid cells at respective scales, took ca 10 hours pr. series in the MRW Simulator, using a laptop computer. I suppose this enormous number cracking it would be outside the practical range of a similar algorithm in R.

REFERENCES

Gautestad, A. O., and I. Mysterud. 2012. The Dilution Effect and the Space Fill Effect: Seeking to Offset Statistical Artifacts When Analyzing Animal Space Use from Telemetry Fixes. Ecological Complexity 9:33-42.