Intraspecific Cohesion From Conspecific Attraction, Part III: A Simple Test

In the real world, animals typically tend to congregate at the landscape scale, even when they show repellence at finer scales (territorial behaviour). In contrast, one of the basic assumptions of the vast majority of mathematical population models is independent space use by individuals. In other words, it is assumed that the individuals do not show conspecific attraction but adhere to “full mixing” from independent space use. A strange assumption, indeed! Using the part of the Parallel processing theory (PP) that was summarized in Part I and Part II it is a simple statistical exercise to test if a given population’s dispersion obeys the full mixing assumption (the Paradigm); or alternatively, indicating a positive feedback tendency from PP-compliant space use (the Zoomer model). 

The concept of conspecific attraction is verified among many species and taxa of animals. For example, lizards prefer to settle near conspecifics, even when unoccupied habitat is available nearby (Stamps 1987; 1988; 1991). Stamps (1991) searched the literature on territorial vertebrates but was unable to find any study in which a negative effect of the density of residents on the settlement of newcomers was demonstrated; i.e. the general assumption of habitat selection studies that prospective territory owners prefer relatively empty habitats proved to lack empirical evidence.

In contrast, in theoretical population models it is typically assumed that individuals both move and settle inter-independently. This “full mixing” assumption in models is a mathematical convenience. It is in fact a requirement, under the premise of the mean field framework, on which almost all population dynamical models adhere to. For example, to model population dynamics realistically with ordinary differential models one has to assume both full spatial mixing of individuals at the temporal scale of the analysis, and a closed system. If the system is open, one should apply partial differential equations, since this allows for an assumption of full mixing locally instead of system-wide (allowing for spatio-temporal “shifting mosaic” of local abundance) (Potts and Lewis 2019)*. The former is called a spatially implicit model, the latter is called a spatially explicit model.

The “full mixing” requirement in classic and contemporary theory of space use by populations is one of the main reasons why theoreticians and field ecologists often tend to drift apart. Logically, it does not make sense, even before considering behavioual fitness arguments! As I argued in Part I and Part II, independent movement and settlement in an open environment will over time drift a Markov-compliant population towards extinction from diffusion and Allée effects.

Empirical results show that (a) single-species populations tend to show scale-free spatial congregation, compliant with a power law (Taylor 1986), (b) empirical results continue to support Non-Markovian, spatial memory utilization by individuals, and (c) populations in general seem to adhere to the principle of conspecific attraction. References in by book and throughout this blog.

(a) Upper part: a superposition of five MRW series with stronger utilization distribution overlap than in Fig. 4 (inset). Lower part: a superposition of five series based on classic RW with homing tendency, with less spatial overlap in spatial utilization. (b) A log-transformed frequency histogram of local cell densities for MRWsuperpostions (filled circles) and RW-superpositions (open circles) shows that neither of the dispersion pattern at the population level in this case satisfy a power law.

Scientifically, the primary question is: how to perform the initial task (prior to making follow-up ecological inference) to test for inter-dependent or independent space use? In other words, how to test if local dispersion is influenced by conspecific attraction?

Consider the PP-based model to represent a specific alternative hypothesis, which should be tested against the null hypothesis given by the Paradigm (for example, a reaction-diffusion model for spatially explicit population dynamics).

The protocol can be quite simple, at least in the first-level approach of analysis of empirical data.

Again, consider results from simulations of PP, i addition to a Paradigm compliant population dispersion. In the Figure to the right you find situation both from scale-specific (Markovian compliant random walk, the Paradigm) and scale-free individual space use with extensive spatial overlap (the PP model), and where both conditions are void of intraspecific cohesion to make the scearia comparable in this respect. This condition contrasts with the scenario in Part I, where I showed how a population with PP compliant space use generated a scale-free dispersion under influence of conspecific attraction. In the present two scenaria the conspecific attraction factor is absent. Individuals use space inter-independently, and below you learn how to test statistically for this lack of conspecific attraction at the so-called “landscape scale” under two qualitatively distinct model frameworks.

The result of inter-independency is seen in the lower part of the Figure. Both situations result in a lack of fractal spatial dispersion of abundance of the pooled set of locations. The reason is that the distribution complies with a negative exponential function (semi-log linear, not shown) rather than power law compliant one (log-log linear).

When these two distinct system conditions are viewed from the population perspective, they both lead to mean field-like system properties with respect to the (M, F) regression at the population level. In other words, even the scenario where individual space use was PP compliant, the lack of conspecific attraction masked the fractal PP property of individuals when analyzed at the population level. Under log-log transformation of frequency of cells in respective bins of grid cell abundance the regression lines were not linear; i.e., not power law compliant. Basically, to test PP compliant space use under the additional property of intraspecific cohesion from conspecific attraction one needs to verify a scale-free (log-log linear) frequency distribution of local density of individuals.

This result illustrates the interesting system property where space use at the individual level may adhere to scale-free dispersion of locations of respective individuals (analyzed separately) while space use at the population level (local abundance of the pooled sets of locations) apparently shows mean field compliance: local fluctuations of abundance are negative exponential compliant! Conclusion: the full mixing premise of the Paradigm – the mean field compliance – can easily be tested on real data.

I’d like to finish this post with an additional study:

In sum, we experimentally tested in breeding mallards two alternative and mutually exclusive hypotheses of habitat selection rules, and found more support for the conspecific attraction rule. However, taking into account the pattern of habitat distribution of breeding mallards (see references in Introduction; this study) pairs certainly use other ways of habitat assessment than mere presence of conspecifics. Some lakes had relatively stable pair numbers while others remained empty independently of experimental treatment.
Pöysä et al. 1998, p287

As I stated in Part II, “A given number of individuals cannot be everywhere all the time”. Thus, some lakes should – as a logical consequence under the premise of conspecific attraction – always be expected to be void of breeding pairs…

NOTE

*) Using the toolbox of partial differential equations and some alternatives it is shown how “diffusion-taxis” equations may show system-intrinsically driven heterogeneity of local population abundance between populations (Potts and Lewis 2019). However, this phenomenon regards interspecific cohesion (or repellence) between separate populations, not intraspecific cohesion. It is assumed that full mixing of individuals is satisfied for each of the populations at the temporal scale of analysis.

REFERENCES

Potts, J. R. and M. A. Lewis. 2019. Spatial memory and taxis-driven pattern formation in model ecosystems. arXiv:1903.05381v05382.

Pöysä, H., J. Elmberg, K. Sjöberg, and P. Nummi. 1998. Habitat selection rules in breeding mallards (Anas platyrhynchos): a test of two competing hypotheses. Oecologia 114:283-287.

Stamps, J. A. 1987. Conspecifics as cues to territory quality: a preference of juvenile lizards (Anolis aeneus) for previously used territories. The American Naturalist 129:629-642.

Stamps, J. A. 1988. Conspecific attraction and aggregation in territorial species. The American Naturalist 131:329-347.

Stamps, J. A. 1991. The effects of conspecifics on habitat selection in territorial species. Behavioral Ecology and Sociobiology 28:29-36.

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

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 II: Parallel Processing and Network Topology

Consider the following two system conditions and theoretical assumptions. First, a large nature reserve has a smaller periphery than several small ones with the same total area. Thus, all other factors equal, a smaller periphery pr. unit area of a large reserve implies a higher implicit survival rate for its embedded species. Fewer animals are “lost in space” due to accidentally diffusing out of their reserve. Consequently, a larger reserve is expected to have a proportionally larger species abundance of animals than a fragmented mosaic of smaller reserves, right? Second, again considering all other factors equal, distant reserves are functionally less connected. In other words, the population dynamics of two reserves in close proximity are assumed to be more in sync or anti-sync from intra-population dynamics than more distant ones, right? The conventional answers are obviously “yes”, but…

In my previous post I referred to empirical results on snail kite Rostrhamus sociabilis plumbeus in Florida (Reichert et al. 2016; Valle et al. 2017) that challenged the intuitive and unconditional “yes” answer to these two questions. Here I elaborate on this paradox of potential “no” answers, which may be resolved – as Reichert et al. (2016) and Valle et al. (2017) contributed to – by considering the network topology of the respective populations’ respective individuals.

Lost in space or an example of an  extremely long exploratory move? For a period of a few days in May 2017 this red-rumped swallow Cecropis daurica foraged together with a flock of local barn swallows in Ulsteinvik, Norway – a distance of about 2,500 km from the species’ ordinary habitat in the southern parts of Europe. Photo: AOG.

Species with a cognitive capacity for spatial memory may require a novel modelling approach at all level of system abstraction; individual, population and community. The “lost in space” assumption becomes less valid, because individuals may return non-randomly even after long-lasting and distant exploratory moves. Returning individuals to a specific local site – steered to some extent by strategic spatial behaviour at the landscape scale – may represent a small number of all immigration events to this site. However, the accumulated effect on local population dynamics of such deterministic and occasionally long distance returns, when summed over the larger temporal scale of a reproductive cycle, should be expected to be profound. Individuals may take advantage of the extended knowledge of both local and distant conditions. Consequently, the SLOSS concept under conservation biology may require a theoretical rethinking.

Distance in space may to a smaller or larger degree be de-coupled from distance in network-topological terms. Supported by the Florida snail kite example, the difference in local population turnover within virtual sections of a given area may be expected to be less spatially correlated, and the functional effect from a variable distance between specific reserves becomes blurred. For example, if the red-rumped swallow this particular week in May (image above) just performed an occasional sally – albeit an extremely long one – followed by a memory-driven return to its original habitat, the two localities would have been drawn closer together in terms of network topological distance.

As a complementary view of this network theoretical aspect, the “anti-dispersal effect” from intraspecific cohesion (a tendency to counteract free dispersal and effectively gluing a population together, as modelled by my Zoomer model) cannot be mimicked within a traditional, Markovian/mechanistic framework. Without spatial memory at the individual level, there is no or little glue to keep the population together in the long run. Individuals are easily lost in space, whether they – for example – happen to disperse into the surrounding matrix environment from large or small reserves. On the other hand, a capacity for targeted, memory-driven returns obviously improves survival and thus fitness, whether the context is some large or several small reserves. 

Recall from the Zoomer model for complex population dynamics that the influx of individuals to a given locality during a given interval may on average be equally distributed between short- and long distance immigrants (Gautestad 2015). This model property may sound counter-intuitive and paradoxical under the the traditional model architectures, unless one considers the premise of scale-free population flow during “zooming”. Statistically, a given individual in a distant location has a small probability of turning up at the given local site*. However, when summed over all individuals within this coarser spatial scale the migration rate becomes equal in magnitude to the probability from influx from more closely located individuals. Since number of individuals within a larger periphery are more numerous, their summed contribution to immigration to a finer-scale locality becomes equal to the immigration that originated from individuals more close-by. That is, given that individuals have shown equal weight to fine- and coarse-scale movement and space use on average (the default condition of both the MRW and the Zoomer models).

Targeted, goal-driven moves by individuals in the Zoomer model design drive the emergence of complex network topology of inter-population migration. Closely linked nodes in the network may be less dependent on physical distance between nodes (like reserves in a SLOSS context). This aspect, which has now found empirical support by studying multi-scaled individual movement’s effect on population redistribution, was illustrated by the snail kite papers that were reviewed in my previous post.

In other words, individuals, populations and communities in distant patches may be functionally connected in a manner that is inexplicable in classic metapopulation terms. A multi-scale kind of system analysis is necessary to understand both population dynamics and community ecology. Without it, paradoxes in local changes of population abundance may prevail and the predictive power of individual- and population level models in conservation ecology will remain low.

NOTE

*) Under classic models such a probability for visitors from a distant site is practically zero, due to the premise of a diffusion compliant migration; i.e., an exponentially declining distribution of individual displacement lengths. On the other hand, under the Zoomer model this exponential equation is replaced by a power law tail, which is thin but very long (up to the cut-off scale).

REFERENCES

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

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.

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.

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.

 

Parallel Processing – How to Verify It

In my previous post I contrasted the qualitative difference between animal space use under parallel processing (PP) and the standard, mechanistic approach. In this post I take the illustration one step further by illustrating how PP – in contrast to the mechanistic approach – allows for the simultaneous execution of responses and goals at different time scales. This architecture is substantially different from the traditional mechanistic models, which are locked into a serial processing kind of dynamics. This crucial difference in modelling dynamics allows for a simple statistical test to differentiate between true scale-free movement and look-alike variants; for example, composite random walk that is fine-tuned towards producing apparently scale-free movement.

First, recall that I make a clear distinction between a mechanistic model and a dynamic model. The former is a special case of a dynamic model, which is broader in scope by including true scale-free processing; i.e., PP. In my previous post I rolled dice to explain the difference.

In the traditional framework there is no need to distinguish between a mechanistic and a dynamic evolution, simply because in this special case of dynamics time pr. definition is one-dimensional. On the other hand, in the PP framework time is generally two-dimensional to allow for parallel execution of a process (for example, movement) at different scales at any moment in time.

Ignoring this biophysical distinction has over the years produced a lot of unnecessary confusion and misinterpretation with respect to the Multi-scaled random walk model (MRW), which is dynamic but non-mechanistic. The distinction apparently sounds paradoxical in the standard modelling world, but not in the PP world. I say it again: MRW is non-mechanistic, non-mechanistic, non-mechanistic – but still dynamic!

First, consider multi-scale movement in the comfort zone of mechanistic models. You may also call it serial processing, or Markov compliant. In the image to the right we see a (one-dimensional) time progression over a time span t=1,….,8 of a series where unit time scale per definition equals one (ε = b0=1; see my previous post). Some sequences are processed at a coarser scale than unit scale; for example, during the interval from t=2 to t=5 the animal “related to” its environment in a particularly coarse-scaled manner relative to unit time. Consider an area-restricted search (ARS) scenario, where the unit-scale moves (light blue events) regard temporally more high-frequency search within a local food patch and more coarse-scaled moves regard temporally toggling into a mode of more inter-patch movement. Consider that the animal during this time temporarily switched to a behavioural mode whereby environmental input is less direction- and speed-influencing (as seen from the unit scale) than during intra-patch search.

Within a mechanistic framework, processing at different scales (temporal resolutions) cannot take place simultaneously. The process needs to toggle (Gautestad 2011).

Mechanistically, the ARS scenario is often parsimoniously modelled by a composite, correlated random walk. By fine-tuning the model parameters an the relative frequencies of toggling it has been shown how such a pattern may even produce approximately scale-free distribution of displacements; i.e, Lévy-like movement (Benhamou 2007). Such statistical similarity between two distinct dynamical classes has produced much fuzz in the field of animal movement research.

Next, contrast the Lévy look-alike model above with a true scale-free process to the right. Due to the dynamics being executed over a continuum of temporal scales, we get a hierarchical structure of events. Thanks to the extra ε axis, there is no intrinsic paradox – as in a mechanistic system – due to a mixture of simultaneous events at different resolutions. Again, I refer to my previous “rolling dice” description. Despite a potential for fine-tuning the composite random walk model to look statistically scale-free, this mechanistic variant and the dynamically scale-free Lévy walk belongs to different corners of the Scaling cube.

Finally, how to distinguish a PP compliant kind of scale-free dynamics from the look-alike process? Coarse-grain the time series and see if the scale-free property persists or not (Gautestad 2013)!

Simulation of a two-level Brownian motion model was performed under four conditions of ratio lambda between the scale parameter of the respective levels, lambda2/lambda1, where frequency of execution t2/t1 = 10 under all conditions. For each condition of lambda the simulated series were sampled at three time scales (lags, tobs); every step, sampling 1:10 and sampling 1:100. Original series lengths were increased proportionally in order to maintain the same sample size under each sampling scheme (20 000 steps). A double-log scatter plot (logarithmic base 2) of step length frequency, log(F), as a function of binned step length, log(L), was then made for each of the four parameter conditions and each of the three sampling schemes. (a) The result from lambda = 4 shows a linear regression slope and thus power law compliance over some part of the tail part of the distribution, with slope b = 2.9; i.e. the transition zone between Lévy walk (1 < b < 3) and Brownian motion (b >= 3 and increasing with increasing L, leading to steeper slope). At coarser time scales tobs = 10 and tobs = 100 the pattern is transformed to a generic-looking Brownian motion with exponential tail, which becomes linear in a semi-log plot: the inset shows the pattern from tobs = 100. (b) The results from lambda = 8.

Both the step length distribution (above) and the visual inspection of the path at different temporal scales reveal the true nature of the model: a look-alike scale-free and pseudo-Lévy pattern when the data are studied at unit scale where the fine-tuning of the parameters were performed, but shape-shifting towards the standard random walk at coarser scales. A true PP-compliant process would have maintained the Lévy pattern even at different sampling scales (Gautestad 2012).

Simulated paths of two-scale Brownian motion where 1000 steps are collected at time intervals 1:1, 1:10 and 1:100 relative to unit scale for the simulation, with lambda2/lambda1 = 15. The pattern shifts gradually from Lévy walk-like towards Brownian motion-like with increasing temporal scale relative to the execution scale (t = 1) for the simulations. Since the number of observations is kept constant the spatial extent of the path is increasing with increasing interval.

By the way, the PP conjecture also extends to the MRW-complementary population dynamical expression of animal space use, the Zoomer model. This property can be clearly seen in the Zoomer model’s mathematical expression.

 

REFERENCES

Benhamou, S. 2007. How many animals really do the Lévy walk? Ecology 88:1962-1969.

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

Gautestad, A. O. 2013. Animal space use: Distinguishing a two-level superposition of scale-specific walks from scale-free Lévy walk. Oikos 122:612-620.