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.

 

The Inner Working of Parallel Processing

The concept of scale-free animal space use becomes increasingly difficult to avoid in modeling and statistical analysis of data. The empirical support for power law distributions continue to pile up, whether the pattern appears in GPS fixes of black bear movement or in the spatial dispersion of a population of sycamore aphids. What is the general class of mechanism, if any? In my approach into this challenging and often frustrating field of research on complex systems, one particular conjecture – parallel processing (PP) – percolates the model architecture. PP requires a non-mechanistic kind of dynamics. Sounding like a contradiction in terms? To illustrate PP in a simple graph, let’s roll dice!

Please note: the following description represents novel details of the PP concept, still awaiting journal publication. Thus, if you are inspired by this extended theory of statistical mechanics to the extent that it percolates into your own work, please give credit by referring to this blog post (or my book). Thank you.

The basic challenge regards how to model a process that consists of a mixture of short term tactics and longer time (coarser scale) strategic goals. Consider that the concept of “now” for a tactical response regards a temporally finer-grained event than “now” at the time scale for executing a more strategic event, which consequently takes place within a more “stretched” time frame relative to the tactical scale.

Strategy is defined in a hierarchy theoretical manner; coarser scale strategy consequently invokes a constraint on finer scaled events (references in my book). For example, while an individual executes a strategic change of state like starting a relatively large-distance displacement (towards a goal), finer-scaled events during this execution – consider shorter time goals – are processed freely but within the top-down constraint that they should not hinder the execution of the coarser goals. Hence, the degrees of process freedom increases with the scale distance between a given fine-scaled goal and a coarser-scaled goal.

To illustrate such a PP-compliant scale range from tactics to strategy within an extended statistical-mechanical system, consider the two-dimensional graph to the right. The x-axis represents a sequence of unidirectional classic time and the y-axis represents a log2-scaled* expression of time’s orthogonal axis, “elacs” (ε) along this sequence.

The continuous x-y plane has been discretized for simpler conceptualization, and each (x,y) pair shows a die. This die represents a potential change of state of the given process at the given point in time and at the given temporal scale. An actual change of state at a given (t,ε) location is marked by a yellow die, while a white die describes an event still in process at this scale. The respective number of eyes on each die could represent a set of available states for a given system variable at this scale. To illustrate complex dynamics (over-)simplistically in terms of concepts from quantum mechanics, consider each magnitude of ε at the y-axis to represent a wave length in a kind of “complex system” wave function and each yellow die represents a “collapse” of this probability wave into a specific execution of the given event at a given point of unit time this time scale.

As the system is viewed towards coarser time scales (larger ε), the average frequency of change of state vanishes proportionally with 1/ε = 1/bz, where b is the logarithmic base and increasing z describes increasing scale level of bz. In other words, the larger the z, the more “strategic” a given event at this scale. In short, consider that each die on scale level 1 [log(b0)=1] is rolled at each time increment t=1, t=2, …, t=8; each die at level 2 [log(b1)=2] is on average rolled each second time increment, an so on.

In the illustrative example above, no events have taken place during the eight time increments at the two coarsest scales bz where z=7 (ε=128) and z=8 (ε=256). A substantial increase of the observation period would be needed to increase the probability of actually observing such coarse-scaled change of system state.

More strategic events are executed more rarely. Strategic events at a given scale bare initiated in a stochastic manner when observed from a finer time scale (smaller z), but increasingly deterministic when observed from coarser time scales. At finer scales such a strategic event may be inexplicable (thus appearing unexpectedly at a given point in time), while the causal relationship of the given process is established (visible) when the process is observed at the properly coarsened time scale. However, at each time scale there is an element of surprise factor, due influence from even coarser scale constraints and even lower frequency change of state of the system at these coarser scales. 

The unit time scale, log(b0)=1, captures the standard time axis, which is one-dimensional as long as the system can be described as non-complex. In other words, the y-axis’ dynamics do not occur, and – consequently – it makes no sense to talk about a parallel process in progress**. In this standard scale-specific framework, time is one-dimensional and describes scale-specific processes realistically. This includes the vast theories of low order Markovian processes (“mechanistic” modeling), the  mathematical theory of differential equations (calculus), and standard statistical mechanics.

For a deeper argument why a PP kind of fundamental system expansion seems necessary for a realistic description of system complexity, read my book and my previous blog posts. By the way, it should of course be considered pieces of a theoretical framework in progress.

The ε-concept was introduced in my book to allow for complex dynamics within a non-Markovian physical architecture. In other words, to allow for a proper description of parallel processing the concept of time as we know it in standard modeling in my view needs to be heuristically expanded to a two-dimensional description of dynamics.

The bottom line: it works! In particular, it seems to survive the acid tests when applied on empirical data, both with respect to individual space use and population dispersion.

Environment is hereby expanded with a two-dimensional representation of dynamical time. This implies that an individual’s environment not only consists of its three-dimensional surroundings at a given point in time but also its temporal “surroundings” due to the log compliant (scale-free) scale-stretching of time. In this manner an implementation of parallel processing turns the common Markovian, mechanistically modeled framework into a special case. According to the special case of standard mechanistic dynamics a given process may be realistically represented either by a scale-specific process at a given (unit) scale or a trivial linear superposition of such processes (e.g., a composite random walk toggling between different magnitudes of the diffusion parameter for each “layer”). On the other hand, complexity arises when such a description that is based on one-dimensional time is not sufficient to reproduce the system realistically.

Observe that in a PP-system several events (change of system state) may be executed in parallel! In the illustration above, see for example the situation for t=5 where events at three time scales by chance are initiated simultaneously but at different time scales as defined by ε. Such a kind of dynamics represents a paradox within the constraint of a Markovian (mechanistic) system.

An earlier illustration of the PP framework was given here. For other examples, search this blog for “parallel processing” or read my book.

Various aspects of scaling in animal space use; from power law scaling of displacement lengths (Lévy like distribution), fractal dispersion of GPS fixes (the home range ghost model) and scale free distribution of populations (Taylor’s power law and the Zoomer model) may be natural outcomes of systems that obey the PP conjecture.

NOTE

*) The base, b, of the logarithm does not matter. Any positive integer introduces scaling of the ε-axis.

**) in a standard, mechanistic process an event describes a change of system state at a given point in space at a given point it time. No “time stretching” takes place.

 

Scrutinizing the MRW Model: Random Returns

The devil is in the details. The Multi-scaled Random Walk (MRW) model is merging scale-free habitat exploration with occasional returns to previously visited locations. Both components are by default expressed by stochastic rules. How can random returns be justified as realistic in a heterogeneous environment – where some localities are expected to experience a higher return frequency than others?

MRW regards a mixture of scale-free exploratory steps and targeted return events. The former kind of movement is modeled by a standard Lévy walk equation for random displacements with no directional bias. On the other hand, return targets are modeled as randomly chosen points in time where the animal revisits a previous location rather than performing yet another exploratory Lévy step. In other words, the animal is in this parsimonious (ground level) model formulation executing a mixture of scale-free moves and occasionally non-random self-crossing of its previous path.

Little stint, Calidris minuta, taking a nap while considering what to do next. Photo: AOG.

By default such self-crossings – strategic returns – are in the basic simulations picked randomly and at a chosen average frequency among the entire series of previous locations. Consequently, the only bias towards site preference that gradually build up is the effect from self-reinforcing space use: if a site by chance has been targeted twice in the past, it has a twice as large probability to receive yet another visit, relative to another location with only one visit. Before considering the concept of “randomness”, let’s focus on this question: how can this model design, which intrinsically reflects movement in a homogeneous environment, be considered realistic, given that the animal’s environment typically is strongly heterogeneous in both time and space?

Randomly picked return targets are easy to imagine in a homogeneous environment. However, simple statistical logic may justify such random picking also in a more realistic scenario with heterogeneous distribution of various resource patches and no-go zones. In particular, two home ranges in a homogeneous versus a heterogeneous environment may show a similar average characteristic scale of space use (CSSU) when the full set of relocations (e.g., GPS fixes) are considered, given that the average environmental conditions are similar for the two sets, respectively. In short, in this case the two data sets is expected to show similar magnitudes of the parameters in the Home range ghost equation, I(N) = cNz, where c expresses the CSSU and 1/c is the intensity of space use*.  The heterogeneity with respect to locally varying intensity of space use within a home range will surface only if one studies local or more short term conditions; for example, by splitting the data into spatial or temporal sub-sets. Influence of locally or temporally variable influence on habitat selection is then expected to be reflected in the respective subsets’ variation of 1/c and z of the Home range ghost equation at the chosen intra-home range scale of analysis.

In this simulation scenario the intra-home range CSSU varies from large (I) to small (IV), reflecting a more intense space use in the latter section. Despite this intra-home range heterogenity of space use the average CSSU for the four sub-sets I-IV equals the CSSU that is found in the pooled set of relocations; i.e., at the home range scale. Note that calculation of CSSU from the home range ghost formula makes it independent of the number of actual relocations in each section.

Hence, even if returns during the path sampling time are skewed towards some localities (on expense of other localities) due to preference based on habitat heterogeneity, this effect will be “averaged out” when considering the path as a whole. In other words, the basic model may reflect the over-all space use condition realistically, as seen from the scale of the home range. This aspect is thoroughly verified by simulations of MRW in homogeneous vs. heterogeneous environment (Gautestad, unpublished).

First, the property of random moves is simply a reflection of a sufficiently deep “hidden layer” to allow for a statistical-mechanical representation of movement and space use. Second, by zooming into subsections (space or time) of this over-all space use picture, the environmental heterogeneity may be revealed. Thus, the basic version of the MRW model for homogeneous habitat conditions my be feasible even in a heterogeneous habitat when studying the over-all conditions.

Similarly, if one finds difference in CSSU between two sets of home range data, this reflects difference at the home range scale of space use between these two sets. Finer-grained heterogeneity is hidden from the analysis (coarse-grained away) for the sake of studying inter-home range differences rather than intra-home range heterogeneity.

NOTE

*) When calculating CSSU at different spatial scales, some trivial statistical-mechanical “tweaking” (rescaling) is performed to adjust for this difference.

The Limited Scope of Lévy Walk and the LFF Model

The Lévy flight foraging (LFF) hypothesis describes a toggling between classic Lévy flight/walk (LW) and classic Brownian motion (BM) as a function of the individual’s current resource field properties (its “environment”). Both states of motion are statistical by nature – and explicitly defined as such. The LFF describes movement as random walk in two disparate modes; scale-free LW versus scale-specific BM. However, the LFF premise of animals moving like drunken LW/BM walkers logically does not make sense unless the animal in question does not possess a capacity for spatial memory utilization or because the environment is so volatile that returning to a previous location has no fitness value with respect to optimal foraging. Under these premises of value-less spatial map utlilization the LFF hypothesis should be expected to make sense, otherwise one should expect to find better compliance with other movement-related models and hypotheses.

The theoretical model developments surrounding both the LW concept in general and the LFF hypothesis in particular have ignited new life into the interface between biology (animal movement, behavioural ecology) and physics (Lagrangian aspects of statistical mechanics, biophysics of animal space use). At last, the classic random walk – Brownian motion (BM) – got justified competition, but recall that both LW and BM are variants of random walk; respectively scale-free and scale-specific. Hundreds of papers have emerged at an increasing pace over the last 30 years, exploring optimal foraging theory and other behavioral-ecological aspects using this statistical walk approach. However, despite the increasing pile of research that has verified LW-like movement in data the results have often spurred both confusion and controversy, including alternative explanations for LW-like power law compliance of step length distribution.

Setting the LW versus pseudo-LW discussion aside, my own critique has on one side focused on premise that the two movement classes contained in LFF regard toggling between behavioural modes, and not just realizations of basically deterministic behaviour that is viewed statistical-mechanically; through “a hidden layer” (path sampling rather than continuous observation). On the other hand I have also criticized LFF from the empirically supported perspective that LW is intrinsically scale-free with respect to processing time but in practical terms scale-specific with respect to processing space.

First, when a path is sampled rather than observing the movement behaviour directly, the statistical compliance with LW, BM or other classes is an emergent property of the statistical-mechanical and intrinsic property of the actual process, regardless of the degree of deterministic or stochastic movement mode at the real-time scale.

Second, how can a process that generates a scale-free distribution of step lengths be scale-specific in the spatial domain? The simple answer is the effect of environmental influence when moving in heterogeneous space. For example, a potentially long distance displacement in progress is under the LW model expected to be easily terminated (truncated) by movement-influencing events and local conditions that successively appear within the trailing perceptual field – the interrupt distance that idealizes a specific scale – as the animal moves along. Consequently, the result is a deflation of super-long step lengths relative to expectation from absence of such interrupts. Such “truncated power law” distributions of step lengths in a non-homogeneous environment dominates the empirical results, where scale-free movement is consequently constrained to medium-range displacements*. Further down this continuum towards increased frequency of large-step interrupts we meet the BM kind of movement – the patch utilization condition of LFF – where the scale-free aspect of movement has vaporized altogether.

According to the LFF hypothesis, next time the wagtail Motacilla alba revisits this patch it is always by chance. A strategic future return; i.e., a non-random crossing of the bird’s movement path that is based on a decision outside the perceptual field surrounding this patch, requires an element of spatial memory. Photo: AOG.

However, many animals have been shown to perform scale-free movement also in heterogeneous space beyond the expected scale of potential truncation from local step length interrupts. Even during foraging such LFF breaking pattern has frequently been documented, as I have described with various references to empirical data in several posts in this blog. Some vital aspect of animal movement is obviously missing from the LW framework, and thereby also constraining the feasible applicability of the specific LFF hypothesis.

In my view there are two dis-harmonic properties of LW when confronted with empirical data of vertebrate movement; (a) the apparent ability of animals to be “back on track” after some kind of local event has interrupted a long-distance move in progress. This obviously violates LW’s intrinsic property of being a Markov compliant process. Further,  (b) according to LW properties the individual is expecting to re-visit a previous location by chance only, not as a result of a strategic return. This behavioural aspect – involving spatial memory and strategic moves – is absent from the Lévy approach. Lack of spatial memory utilization does not harmonize with the behavioural ecology of a very wide range of species and ecological conditions. Vertebrates and large groups of invertebrates have now been verified to possess a capacity for memory map utilization. Under a very broad range of environmental conditions it makes sense to logically expect a potential fitness value from returning non-accidentally to a previously utilized food patch, for example.

In particular, under the premise of spatially memory-less LW, the emergent property of “clumped” and statistical fractal-compliant aggregations of locations from self-reinforced space use (strategic returns) cannot appear. If one finds support for scale-free space use in the step length distribution in combination with support for memory utilization – “the home range ghost” (Ic√N) – it is not LW that has produced the pattern but Multi-scaled random walk (MRW). LW and MRW may under some conditions share a similar distribution of both displacement lengths and spatial distribution of relocations (power laws, with or without identical exponents), but only MRW embeds the expectation of a home range kind of space use. A home range is a verification of spatial memory utilization. Even under the condition where the ratio of returns vs. exploratory sallies under MRW is very small (leading to a larger home range for a given N; i.e., a larger c in the ghost equation), it still has a large statistical effect on how the animal is utilizing local patches with respect to self-reinforcing space use.

So far, whenever we have tested for LW vs. MRW it has been the latter that has been supported: a Lagrangian power law distribution of step lengths and an Eulerian power law compliant home range ghost equation. And the respective environments have been heterogeneous from fine towards coarse scales. Hence, a wide range of species seem to utilize their environment in a scale-free processing manner (due to the statistical power law compliance) over both time and space. This should have consequences for future tests of statistical patterns in animal movement, in particular with respect to model assumptions. New ecological methods should be implemented if memory is a factor.

I do not claim that the LW and the LFF hypothesis are erroneous – all I underscore is that these models’ assumptions with respect to absence of spatial memory influence should be scrutinized a priori and not taken as granted. I believe these models’ scope of realism will be accepted to be substantially limited if this introductory test is performed on a regular basis.

A statistical-mechanical modelling approach is needed when animal movement and space use is sampled; i.e., a hidden layer is invoked. One is studying not only the statistical aspects of the behaviour in the traditional sense. One is simultaneously also shifting the level of system abstraction from a temporally moment-to-moment analysis to a more abstract process level that appears from system coarse-graining of observational time scale (path sampling).

A statistical-mechanical theory for animal movement and space use should implement all variants of movement classes. In The scaling cube I have capture these classes by its eight corners, where the cube’s interior is expressing respective continua. From the Lagrangian perspective, spatio-temporally scale-free displacements are formulated by the MRW model’s step length distribution; from the Eulerian (spatially explicit) perspective the MRW model is expressed both by the Home range ghost formulation and by the complementary Zoomer model. The mathematical formulation of the parsimonious Zoomer model – a memory-extended statistical mechanics of ensembles of particles (individuals) under scale-free influence of conspecific attraction –  is found here.

NOTE

*) This pattern where the extreme section of the long tail part of the distribution is constrained does not per se lend support to the classic Lévy framework despite partly power law. Other models, for example the Multi-scaled random walk (MRW), also gives expectation of a truncated scale-free step length distribution over a wide range of conditions (see various posts on this blog).