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 my 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 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 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 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.

Slow Motion in Books on Animal Movement

Over the last years we have seen a range of interesting and important books appearing in the field of animal movement and space use. In this post I mention four of them. Unfortunately, only two of these books (disregarding my own contribution) presents any reference to animals’ capacity for spatial reorientation beyond the individual’s current perceptual field. However, all that is offered in these two titles – covering hundreds of pages with deep theory – is a couple of sentences or paragraphs. Why such a slow implementation in mainstream models with respect to this key aspect of behavioural ecology? Why such stubbornness to bridge theory to empirical knowledge by including spatial memory as an important factor that influences how animals use their habitat?

First, I was happy to discover that Turchin’s classic book “Quantitative analysis of animal movement” (Turchin 1998) has now appeared in new print, dated 2015. However, like the original version you will search in vain for any reference to animals that utilize spatial memory. Since this is not a re-write, Dr Turchin is excused. However, I had expected some update on models on movement as they have developed over the last 20 years, in particular with respect to scale-free space use (e.g., Lévy walk) and spatially explicit returns, where the latter explain an individual’s home range as an emergent property rather than some kind of mystic and pre-set constraint on area use.

Second, I have read “Stochastic foundations in movement ecology” (Mendéz et al. 2014) with great interest. However the downturn also here was the lack of any reference to animals utilizing space in a strategic manner; i.e., relating to its environment beyond the perceptual field around the present location. Several sections are included to cover advanced space use in the context of scale free movement and even continuous time random walk models (CTRW). However, again I missed descriptions of animals with spatial memory utilization. Before this field of research reach a full integration between scale-free movement and strategic, memory-influenced displacements the theory is a half-told story.

Third, I was thrilled by the title of the latest book in this field of research, “Animal Movement: Statistical Models for Telemetry Data” (Hooten et al. 2017). This book is written by researchers with a proper empirical experience with modern data sets on space use. Still, only two paragraphs over the 306 pages were dedicated to the spatial memory aspect, only referring to papers where the reader may find some research on spatial memory.

Fourth, “The Physics of Foraging” (Viswanathan et al 2011)  was very promising and I really looked forward to get a copy, since we here finally got a book that was explicitly dedicated to the interface between physics (e.g., statistical mechanics-related) and behavioural ecology. It does in fact include 12 index references to memory, but – disappointingly – all but two of them regard temporal memory (CTRW, correlated random walk and other variants) rather than spatially explicit memory as expressed by utilization of a memory map. Anyway, I appreciate that this important book does at least mention references to our own work on spatial memory!

To conclude, so far my book Animal space use is still alone on the arena what regards focusing on the integration of scale free space use and spatial map utilization. I hope it soon gets competition.

Temporally Constrained Space Use, Part III: Critique of Common Models

There is no doubt among field ecologists that animals from a broad range of taxa and over wide range of ecological conditions utilize their environment in a spatial memory-influenced manner. Spatial map utilization have now been verified also well beyond vertebrates, like dragonflies and some solitary wasps. To me at least it is thus a mystery why theoretical models that are void of influence from a memory map; for example ARS, Lévy walk and CTRW (see Part I, II), are still dominating ecological research with mostly no critical questions asked about their feasibility.

It is a fact that the memory-less mainstream models all have a premise that the data should not be influenced by map-dependent site fidelity. In other words, applying ARS, Lévy walk and CTRW models as stochastic representation of space use also implies accepting that the animal’s path is self-crossing by chance only, and not influenced by targeted returns. Such returns can be expected to seriously disrupt results on – for example – habitat selection, since self-reinforcing patch utilization (positive feedback) obviously becomes a serious issue for methods that are based on memory-less space use where revisits are statistically independent events.

Despite performing hypothesis tests on data that obviously contradicts this hidden assumption about lack of spatial memory influence, for example movement in a home range context (where the home range is an emergent property from such returns), memory-less models are applied by cultural instinct or a misconception that alternatives do not exist. “Everybody else is using these standard models, so why not me?”

This attitude obviously hinders space use-related ecological research on its path towards becoming hard science at the level we are used to find in physics, chemistry and geology; i.e., models with strong predictive power. The laid-back excuse that animal ecology is not only more complicated but also basically more complex does not hold anymore. Biophysical research, for example based on inspiration from – or developed in compliance with – my parsimonious MRW model (Song et al. 2010; Boyer et al. 2012; Boyer and Solis-Salas 2014; Mercado-Vásquez and Boyer 2018), show how even complex space use systems may now be treated analytically with success.

So far, there still exists only one book (Gautestad 2015) that is dedicated to criticizing the sloppy culture of model selection in ecological research. The statistical errors that follow from ignoring the frequently violated assumption about memory-less space use are percolating both my book and my blog*.

MRW implements a combination of scale-free space use with memory-dependent, occasional returns to previous sites in accordance to the parallel processing conjecture. The average return interval tret to a previously visited location relative to the sampling interval tobsρ = tret/tobs, will lead to different analytical results a a function of ρ.

This important ratio defines how the observed distribution of step lengths is a function of  memory-influenced movement that complies with the MRW formulation: a mixture of scale-free exploratory steps and occasional returns to a previous location. I cite from Part II:

If the animal in question is utilizing spatial memory a lot of confusion, paradoxes and controversy may thus appear if the same data are analyzed on the basis of erroneously applying memory-less models within different regimes of ρ!

For example, an decreasing tret for a given tobs implies stronger site fidelity. The variable observer effect that is expressed by tobs becomes apparent within a quite wide transition range around tobs ≈ tret. For example, a Brownian motion-like form of the step length distribution may erroneously be found if ρ << 1, and a power law form can be expected when ρ >> 1, with truncated power law to be observed in-between. However, power law compliance may arise both in scale-free but spatially memory-less behaviour (Lévy walk) and MRW when ρ >> 1. Recall that MRW implies a combination of spatially memory-influenced and Lévy walk-like kind of movement in statistical terms.

The step length distributions to the right (Gautestad and I. Mysterud 2005)  illustrates from MRW-simulated data the effect on changing the ratio ρ >> 1 towards ρ < 1 apparently makes the step length distribution shape-shifting from a power law (apparently Lévy) to a negative exponential (apparently Brownian). This paradoxical pattern appears simply from changing sampling frequency of a given series of successive relocations.  As observation frequency becomes larger than the return frequency the paradox appears from comparing the expectation from erroneous of model based on the memory-less space use assumption; i.e., Brownian motion vs. Lévy walk. 

The Figure to the right (Gautestad and A. Mysterud 2013) illustrates the same transition more graphically. The hump (blue colour)  that is observed for for ρ = 10 towards the extreme tail of the distribution, leading to a hump-like “hockey stick” pattern, becomes almost invisible at ρ = 100 Appendix 1 in Gautestad and A. Mysterud, 2013; see also Gautestad 2012). This gradual appearance/disappearance of the hockey stick as a function of ρ >> 1 illustrates the pseudo-LW aspect of MRW. By the way, such a “hump” on the tail part of a power law distribution has in fact been found and commented in several analyses of empirical data. Citing from Gautestad and A. Mysterud (2013):

It is interesting that one of the main issues raised in this  respect regards the “problematic” occasional over-representation of very long step lengths even relative to an ideal Lévy walk distribution, invoking the term “Lévy walk-like” search (Sims and Humphries 2012; Sims et al. 2012). This “hump” in the long tail part of the distribution has been hypothesized to emerge from some kind of environmental forcing (Sims and Humphries 2012). However, here we have shown (Figure 3) that a similar hump – called a hockey stick – is in fact expected by default if MRW-compliant data are analysed within a specific range of the ratio between return events and observation interval.
Gautestad and Mysterud 2013, p14.

The take-home message from these two examples is stressing the importance of testing for spatial memory before choosing which statistical model(s) to apply for a specific analysis.

NOTE

*) In my research I also criticize memory-implementing models where spatial utilization beyond the individual’s current perceptual field builds on a mechanistic (Markov-compliant) kind of information processing. See, for example, this post. Consequently, in the Scaling cube, these Markov models are located in the lower right corner (MemRW), in contrast to the “parallel processing”-based MRW, which you find in the upper right corner. In Gautestad et al. (2013) we tested these alternative model classes on red deer Cervus elaphus, and found strong support for the MRW framework. The red deer moved both in compliance with a scale-free space utilization, in parallel with site fidelity from targeted returns in a manner which supported parallel processing. Additional research has also given support to to MRW lately; for example see Merkle et al. (2014), who tested a set of contemporary hypotheses on memory-influenced movement in free-ranging bison Bison bison and found support for a central premise of MRW in the summer ranges of this species.

REFERENCES

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

Boyer, D. and C. Solis-Salas. 2014. Random walks with preferential relocations to places visited in the past and their application to biology. arXiv 1403.6069v1:1-5.

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

Gautestad, A. O., L. E. Loe and I. Mysterud. 2013. Inferring spatial memory and spatiotemporal scaling from GPS data: comparing red deer Cervus elaphus movements with simulation models. Journal of Animal Ecology 82:572-586.

Mercado-Vásquez, G. and D. Boyer. 2018. Lotka-Volterra systems with stochastic resetting. arXiv:cond-mat.stat-mech:1809.03975v03971.

Merkle, J. A., D. Fortin, and J. M. Morales. 2014. A memory-based foraging tactic reveals an adaptive mechanism for restricted space use. Ecology Letters Doi: 10.1111/ele.12294.

Sims, D. W. and N. E. Humphries. 2012. Lévy flight search patterns of marine predators not questioned: a reply to Edwards et al. ArXiv 1210.2288: [q-bio.PE].

Sims D. W, N. E. Humphries, R. W. Bradford and B. D. Bruce. 2012. Lévy flight and Brownian search patterns of a free-ranging predator reflect different prey field characteristics. Journal of Animal Ecology 81:432-442.

Song, C., T. Koren, P. Wang, and A.-L. Barabási. 2010. Modelling the scaling properties of human mobility. Nature Physics 6:818-823.

Positive and Negative Feedback Part II: Populations

Examples of positive feedback loops in population dynamics abound. Even if the majority of models are focusing on negative feedback, like the logistic growth function, non-equilibrium “boom and bust” kind of model designs have also been developed. In this post I elaborate on the particular kind of positive feedback loop that emerges from cross-scale dual-direction flow of individuals that is based on the parallel processing conjecture.

The image to the right illustrates – in simplistic terms – a spatially extended population model of standard kind (e.g., a coupled map lattice design) where each virtually demarcated local population j at spatial resolution i and at a given point in time t contains Nij individuals. No borders for local migration are assumed; i.e., the environment is open both internally and externally towards neighbouring sites.Typically, these individuals are set to be subject to a locally negative feedback loop in accordance to principles of density dependent regulation*. The larger the N the larger the probability of an increased death rate and/or and increased emigration rate from time t to t+1, eventually leading both the local and the over-all population to a steady state. This balancing** condition lasts until some change (external perturbation) is forcing the system into a renewed loop of negative feedback-driven dynamics. In a variant of this design, density regulation may be formulated to be absent until a critical local density is reached, leading to boom and bust (“catastrophic” death and emigration), which may be more or less perturbed by random immigration rate from asynchronous developments in respective surrounding Nij. More sophisticated variants abound, like inclusion of time lag responses, interactions with other trophic levels, and so on.

As previously explained in other posts, this kind of model framework depends on a premise of Markov-compliant processes at the individual level (mechanistic system), and thus also at the population level (local or global compliance with the mean field principle). In this framework intrinsic dynamics may be density dependent or not, but from the perspective of a given Nij, extrinsic influence – like immigration of individuals – is always stochastic and thus density independent with respect to Nij.  In other words, the net immigration rate during a given time increment is not influenced by the state of the population in this location (i,j). You can search my blog or read my book to find descriptions and details on all these concepts.

To implement cross-location and dual-direction deterministic dynamics, multi-scaled behaviour and spatial memory needs to be introduced. My parallel processing conjecture; which spins off various testable hypotheses, creates turmoil in this standard system design for population dynamics because it explicitly introduces such system complexity. For example, positive feedback loops may emerge. Positive feedback as described below may effectively also counteracting the paradoxical Allée effect, which all “standard” population models are confronted with at the border zone of a population in an open environment**.

The dynamic driver of the complexity is the introduction of spatial memory in combination with a scale-free kind of dynamics along both the spatial and the temporal dimensions. In statistical-mechanical terms, parallel processing is incompatible with a mechanistic system. Thus, a kind of extended statistical mechanics is needed. I refer to the post where I describe the scale-extended description of a metapopulation system.

For the most extensive individual-level test of the parallel processing conjecture until now (indirectly also verifying positive feedback of space use), see our paper on statistical analysis of space use by red deer Cervus elaphus (Gautestad et al. 2013; Gautestad and Mysterud 2013). In my blog I have also provided several anecdotal examples of third party research potentially supporting the parallel processing conjecture. For the sake of system coherence, if parallel processing is verified for individual space use of a given species and under given ecological conditions, this behaviour should also be reflected in the complementary population dynamical modelling of the given species and conditions.

Extending the standard population model. As explained in a range of blog posts, my Zoomer model represents a population level system design that is coherent with the individual-level space use process (in parsimonious terms), as formulated by the Multi-scaled random walk model. In my previous post I described the latter in the context of positive feedback from individual-level site fidelity. Below I illustrate positive feedback also at the population level, where site fidelity get boosted by conspecific attraction. In other words, conspecifics become part of the individuals’ resource mapping at coarser scales, as it is allowed for by spatial memory. Consequently, a potential for dual-direction deterministic flow of individuals is introduced (see above). Conspecific attraction is assumed to be gradually developed by individual experience of conspecifics’ whereabouts during exploratory moves.

In the Zoomer model , some percentage of the individuals are redistributing themselves over a scale range during each time increment. Emigration (“zooming out”) is marked by dotted arrows, and immigration (“zooming in”) is shown as continuous-line arrows. Numbers refer to scale level of the neighbourhood of a given locality. This neighbourhood scales logarithmically; i.e., in a scale-free manner, in compliance with exploratory moves in the individual-level Multi-scaled random walk model. Zooming in depends on spatial memory by the individuals, and introduces a potential for the emergence of positive feedback at the population level.

First, consider the zooming process, whereby a given rate, z, of individuals (for example, z=5% on average at a chosen time resolution Δt) at a “unit” reference scale (k=i) are redistributing themselves over a scale range beyond this unit scale***. During a given Δt consider that 100 individuals become zoomers from the specific location marked by the white circle. In parallel with the zooming out-process the model describes a zooming in-process with a similar strength. The latter redistributes the zoomers in accordance to scale-free immigration of individuals under conspecific attraction.Thus, number of individuals (N) at this location j at scale i, marked as Nij, will at the next time t+1 either embed N-100 individuals if they all leave location j and end up somewhere in the neighbourhood of j, or the new number will be N -100 + an influx of immigrants, where these immigrants come from the neighbourhood at scale i (those returning home again), scale i+1 (immigration from locations nearby), i+2 (from an even more distant neighbourhood), etc.

In the ideal model variant of zooming we are thus assuming a scale-free redistribution of individuals during zooming, with zooming to a neighbourhood at scale ki+x takes place with probability 1/ki+x (Gautestad and Mysterud 2005). Under this condition, zoomers to successively coarser scales become “diluted” over proportionally larger neighbourhood area, the maximum number of immigrants in this example is 100 + N’, where N’ is the average number of zoomers pr. location at unit scale k=i within the coarsest defined system scale k=i(max) for zooming surrounding location j at scale i.

As a consequence of this kind of scale-free emigration of zoomers, the population system demonstrates zooming with equal weight of individual redistribution from scale to scale over the defined scale range (Lévy-like in this respect, with scaling exponent β≈2; see Gautestad and Mysterud 2005). By studying the distribution of step lengths, this “equal weight” hypothesis may be tested, when combinded with othe rstatistical fingerprints (in particular, verifying memory-dependent site fidelity; see Gautestad and Mysterud 2013).

Putting this parsimonious Zoomer model with its system variables and parameters into a specific ecological context implies a huge and basically unexplored potential for ecological inference under condition of scale-free space use in combination with site fidelity.

Positive feedback in the Zoomer model. As shown in my series of simulations of the Zoomer model a few posts ago, a positive feedback loop emerges from locations with relatively high abundance of individuals having a relatively larger chance of received a net influx of zoomers during the next increment, and vice versa for locations with low abundance. The positive feedback emerges from the conspecific attraction process, linking the dynamics at different scales together in a parallel processing manner.

This positive feedback loop from conspecific attraction also counteracts extinction from a potential Allée effect (see this post and this post), which have traditionally been understood and formulated from the standard population paradigm. The Zoomer model represents an alternative description of a process that effectively counteracts this effect.

NOTES

*) The migration rates connects the local population to surrounding populations. Immigration is – by necessity from the standard model design – density independent with respect to the dynamics in Nij.

**) Since the process is assumed to obey a Markovian and the mean field principles (standard, mechanistic process), the arena and population system must either be assumed to be infinitely large or the total set of local populations has to be assumed to be demarcated by some kind of physical border. Otherwise, net emigration and increased death rate in the border zone will tend to drive N towards zero  in open environments (extinction from standard diffusion in combination with local N drifting below critical density where Allée kicks in). Individuals will “leak” from an open border zone to the surroundings where N is lower.

***) The unit temporal scale for a population system should be considered coarser than the unit scale at the individual level, since the actual scale range under scrutiny typically is larger for population systems. In particular, to find the temporal scale where for example 5% of the local population can be expected to be moving past the inter-cell borders of a given unit spatial grid resolution ki=1, one should be expected to find Δt substantially larger than Δt at the individual level.

Consider that the difference in Δt is a function of the difference of the area of short-range versus long range displacements under the step length curve for individual displacements, where the ∼5% long-step tail of this area represents the relative unit time in comparison to the rest of the distribution (thereby defined as intra-cell moves). Since this area is a fraction of the area for the remaining 95% of the displacements, the difference in Δt should scale accordingly.

REFERENCES

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

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

Gautestad, A. O., L. E. Loe, and A. Mysterud. 2013. Inferring spatial memory and spatiotemporal scaling from GPS data: comparing red deer Cervus elaphus movements with simulation models. Journal of Animal Ecology 82:572-586.