Parallel Processing: From Metaphysics Towards Biophysics, Part III

In this post I will again hammer wake-up calls into my own camp, researchers in the field of individual and population wildlife ecology, including the theory of habitat selection and and animal space use. For example, I have previously claimed that the Burt legacy has hampered progress in individual home range modelling, as has standard calculus done for population dynamics of open systems (including spatially extended versions). The two classical toolboxes for space use models; based on specific postulates from statistics and standard mathematics, are hampering progress towards improved model realism. As long as there still is a strong reluctance to replace these postulates by extending the theory head-on in the direction of biophysics of memory-influenced processes it is my personal conviction that the quagmire will prevail. However, there are now rapid and promising progress from research outside the traditional community of ecologists.

These directions are pointing towards spatial models from two directions; on one hand the progress comes from physiological research on the functioning of the brain’s complex information processing. On the other hand progress comes from the field of physics dealing with models with so-called “stochastic resetting of particles”, including memory effects. I’m convinced that combining these disparate directions of research will in due course bring focus on a unification: the concept of parallel processing and the complementary theories of MRW and the Zoomer model.

Ecologists should take notice. Such a development may potentially rattle the more dogmatic framework of space use theory, for example by offering solutions to the old problems of simulating Allée effects, Taylor’s power law, pink noise spectra in population series, and long distance connectivity in population structure (complex network topology, pointing towards alternatives to classic metapopulation models).

Will brain research on mammals and recent developments in physical models of MRW-like behaviour ultimately extend towards Parallel processing-based memory execution and scale free space use in a broader context, like birds (here a hooded crow Corvus cornix) and many groups of invertebrates? Photo: AOG.

What regards brain research, professor Marianne Fyhn, Associate Professor Torkel Hafting and their research group at the faculty of Medicine, University of Oslo, are investigating processes in the brains of awake, behaving animals in order to better understand how the brain works in real time, and reveal the mechanisms underlying perception, learning and memory. Recent progress is astonishing! The same regards the Nobel laureates and Professors May-Britt Moser and Edvard Moser at the Kavli Institute and the Centre for Neural Computation, Trondheim. They are interested in the fundamental neural computations underlying cognition and behaviour. To decipher these computations, they focus on the mechanisms for mapping of local space in the mammalian cortex. Their past work includes the discovery of grid cells. Grid cells are place-modulated neurons whose firing fields define a triangular array across the entire environment in the actual nerve system. These cells are thought to form an essential part of the brain’s coordinate system for metric navigation.

I’m sensing that these breakthroughs in brain research on memory and GPS-like functions may be approaching the concept of Parallel processing in spatio-temporal navigation at a rapid pace.

What regards the concept of stochastic resetting of particles, in a recent review Evans et al. (2019) generalize this rapidly expanding field of non-random path crossing of moving particles (i.e., a core property of MRW) to an arbitrary stochastic process (e.g. Lévy flights or fractional Brownian motion) and non-Poissonian resetting (e.g. power-law waiting time distribution for intervals between resetting events). They also go on to discuss multi-particle systems as well as extended systems, such as fluctuating interfaces, under resetting. They consider memory effects, which implies resetting the process to some randomly selected previous time (like MRW). Finally the review gives an overview of recent developments and applications in the field. By the way, it also refers to some MRW papers (Gautestad and Mysterud 2005; 2006) as application of particle resetting in a biological context.*


*) However, the MRW model and the parallel processing concept were introduced already in my Dr. philos. thesis from 1998 and in Gautestad et al. (1998).


Evans, M. R., S. N. Majumdar, and G. Schehr. 2019. Stochastic Resetting and Applications. arXiv:1910.07993v1 [cond-mat.stat-mech].

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

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

Gautestad, A. O., I. Mysterud, and M. R. Pelton. 1998. Complex movement and scale-free habitat use: testing the multi-scaled home range model on black bear telemetry data. Ursus 10:219-234.


Parallel Processing: From Metaphysics Towards Biophysics, Part II

When I returned to the University of Oslo in 1990 to explore alternative pathways towards complex dispersion of populations it was natural to start out by orbiting around the Department of biology’s division that focused on population dynamical modelling. However, as become increasingly obvious was a tension that grew up between my choice of off-piste approaches, the introduction of rather unorthodox concepts and on the other side meeting a culture that focused on the classical mathematical and statistical toolbox. I simply could not find satisfactory local support for working on scale-free dispersal processes under these terms, despite what I observed as thoroughly and broadly documented instances of such paradigm-breaking behaviour in the hundreds of papers surrounding confirmation of – for example – Taylor’s power law and fractal-patterned population dispersion. The theoretical culture was stubbornly shying away from exploring a series of paradoxes that in my view were crying for an entirely novel approach for a solution. Thus, luckily in 1992 I knocked on Ivar Mysterud’s door further down the same corridor.

Ivar (now Professor emeritus) was mentor of my Master’s degree on a population of tawny owl Strix aluco, Now I sat in his office again, advocating my ideas for a PhD on animal space use in broader terms and how I felt it was necessary to drill into behavioural ecology at the individual level to seek solutions to the paradoxical dispersion processes in real populations. Characteristically for Ivar, he was immediately positive to such an endeavor that was aiming at some rather unfamiliar theoretical terrain. The more off-piste, the better! He already had concluded from his own field experience that completely new approaches were probably needed. But where to find them? Not in standard text books. Neither in contemporary papers that were influencing animal movement and habitat selection those days.

Ivar Mysterud (left) and myself, displaying a novel equation that links home range utilization to to its fractal properties. From a 2010 article in the Norwegian resarch magazine Apollon. Photo: Francesco Saggio.

With Ivar’s very solid background as field ecologist in wildlife behaviour and management, he had already struggled in practical terms with analyzing animal space use complexity, in particular related to the aspects of home range behaviour. He offered me both an office and a four year scholarship to work freely (!) on whatever successively emerged as the most promising path forward. Best of all, Ivar’s and his students’ telemetry data on habitat use by free-ranging sheep Ovis aries were made available. He also ensured contact with his good friend Mike Pelton, professor at the University of Tennessee, where we were allowed to work on his lab’s extensive material on black bear Ursus americanus movement.

Within a year following the toggle from population kinetics to individual space use the first “metaphysical” property of the sheep data emerged from our analyses under the new terms (Gautestad and Mysterud 1993), including frequent and inspirational rounds on the office’s too small blackboard.

A catchy new concept emerged, “the Home range ghost” (Gautestad and Mysterud 1995).

For the initial 1993 paper the minimum convex polygon method* was used to study home range size as a function of sample size of telemetry fixes, n, including all ‘outliers’ available (more proper methods in follow-up works confirmed the same “ghost” aspects). The data were found to be satisfactorily non-auto-correlated at the given sampling intervals of several hours between successive relocations (fixes).

Then, what was the “metaphysical” Home range ghost property? For a start, consider demarcating space use from a total sample of N fixes using two protocols; (a) samples of n fixes (nmin < n <= N) that are drawn uniformly from the pool N, and (b) samples of n fixes that represent continuous series (time-close segments) from the total path of N fixes.

Ordinary theory and common sense predicted that the A(n) curves for home range area A from both methods should be quite overlapping, given that the fixes were temporally non-auto-correlated. In other words, n fixes from uniform sampling were expected to show similar A(n) as continuous sampling series. Otherwise, the latter should show smaller A for a given n. Further – again from the home range paradigm – the A(n) should be expected to flatten out towards an area asymptote for larger A.

The sheep data told us another story.

First, the asymptote of A(n) was not found (Figure 5a above). In fact, with log-log scaled axes the function satisfied a straight line with slope z about 0.5; i.e., a scale-free power law log(A) = log(c) + z*log(n). Area grew proportionally with square root of n rather than towards an asymptote, despite large N. Second, despite the non-asymptotic function, the two sampling methods “uniform over N” and “sections within N” overlapped!

A similarly strange power law pattern with z=0.5 was also found in Mike’s and his students material on black bear (Gautestad et al. 1998), which I recently also re-confirmed using latest methods of the Home range ghost theory (see this blog post). In a meta-analysis of A(N) data from many species, taxa and ecological conditions, the overall result also confirmed the same power law with z≈0.5 (Gautestad and Mysterud 1994).

In short, we were able to show scale-free space use by applying the relatively new concepts of statistical fractals (Mandelbrot 1983, Feder 1988). But what was most explosive in our results were a property of space use that resonated well with the strange, apparently “metaphysical” property of sycamore aphids with respect to “time-independent” and “scale range dependent” pattern, like seen in 1/f noise spectra and in a subsection of Taylor’s power law (see Part I of this 2-part post). This time from analysis of individual space use.


*) The MCP method, which was quite dominant in home range analyses at the time, was replaced by more robust statistical procedures in the follow-up work.


Gautestad, A. O. and I. Mysterud. 1993. Physical and biological mechanisms in animal movement processes. J. Appl. Ecol. 30:523-535.

Gautestad, A. O. and I. Mysterud. 1994. Fractal analysis of population ranges: methodological problems and challenges. Oikos 69:154-157.

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

Gautestad, A. O., I. Mysterud, and M. R. Pelton. 1998. Complex movement and scale-free habitat use: testing the multi-scaled home range model on black bear telemetry data. Ursus 10:219-234.

Parallel Processing: From Metaphysics Towards Biophysics, Part I

This series of posts is an attempt to briefly summarize historically the conceptual development of the Parallel processing idea (PP) as a modelling framework to better understand the complex space use by animals. The endeavor started by a couple of enlightening Eureka moments, and swiftly split into two lines of approach. The challenge was to get the grips of the deeper process behind scale-free dispersion. This fascinating behaviour would in my view have to be coherently modelled along two lines of exploration, first from the perspective of spatio-temporal population dynamics and then from individual space use. PP was born during those early periods of confusion and frustration, and it has focused (some will say haunted) my research for more than 30 years.

Spaced-out gregariousness in sycamore aphids Drepanosiphum platanoides under tree leaves in Oslo, Norway. Photo: AOG.

At the population side it all started out with a couple of frustrating years exploring mathematical models on animal population dynamics, trying to find a satisfactory road towards the very general phenomenon that animals tend to show aggregated abundance over a broad range of space and time scales, apparently driven by intrinsic behavior in addition trivial environmental forcing (Taylor 1961, 1986). However, even struggling with and tweaking of the flexibility of partial differential equations and spatially explicit coupled map lattice models did in my view not satisfy a proper reproduction of the inner working of animal dispersion processes at a very fundamental, parsimonious level. Hence, I had to put these attempts behind me, to seek an alternative approach towards a more realistic modelling framework.

Obviously to me at least, the spatio-temporal complexity had to be hidden inside the individuals’ brain, leading to the complex, scale-free statistical space use pattern as a so-called emergent property.

Around 1991 I finally stumbled into Kennedy and Crawley’s (1967) work on sycamore aphids Drepanosiphum platanoides. Here a phenomenon called “spaced-out gregariousness” was explained well by a specific behaviour at the scale of aphids in a petri dish*.

However, Taylor’s power law had shown the universality of gregarious, scale-free population abundance extending over broad scales in both space and time – and crucially – over a broad range of species, taxa and ecological conditions. How to get in touch with such a universal system behaviour in parsimonious model terms?  As a first shot after the inspiration from aphids I developed an individual-based model including inter-individual interaction that was satisfactorily able to reproduce spaced-out gregariousness at the petri dish scale. However, the pattern broke down to standard (so-called mean field) statistics when the model output was coarse-grained to broader scales. In other words, a dead end with respect to realism.

A screenshot of aphid dispersion in a petri dish, from Kennedy and Crawley (1967).

Empiric data had to set the path forward. By luck, the campus at the University of Oslo had some easily accessible alleys of sycamore trees, containing a rich abundance of aphids. Thus, during the summer of 1991 I collected data on these aphids’ distribution over spatial scales from half-leaves to groups of leaves and so on, including how the abundance changed at various time scales during the season. The results were written up in an extensive series of drafts during 1992-1994**, but was not published – in an updated version*** – until my book appeared many years later (Gautestad 2015).

Interestingly, the simulations confirmed scale-free dispersion of aphids, in line with expectation of Taylor’s empirical foundation. But I went a step further, by studying how the pattern was influenced by distance between sample units. Strangely, the result was quite insensitive to this scale aspect, meaning that a group of leaves in close proximity showed similar scaling parameters as leaves that were more spaced-out. And the dispersion was strongly heterogeneous, not uniformly distributed.

For the critical relevance of this apparently metaphysical phenomenon in biophysical terms I refer to my book and to another post on this theme at my blog. In particular, the population level model I developed, the “Zoomer model”, was able to reproduce not only power law dispersion over both space and time simultaneously, but also to link this property to another long lasting controversy in population fluctuations, called the Pink noise phenomenon in population abundance. In the period from February to April 2018 at this blog I presented a series of ten new analyses of various aspects of the Zoomer model properties.

As far as I know this theoretical linking between two disparate aspects of  complex population dynamics – Taylor’s power law and reddened (“1/f”) spectra in time series and spatial transects – were by the Zoomer model for the first time directly reproduced in simulations as showing two sides of the same statistical elephant!

By this a priori counter-intuitive empirical result from a biophysical perspective, the first seed for the development of the PP concept was also sown. According to the PP postulate, animals are moulding fine- and coarser scale behaviour into the actual process of animal movement and redistribution. In other words, PP regards how tactics and more strategic decisions play out cognitively in a temporally parallel fashion. Crucially, its mathematical foundation describes how this property can be tested against the standard framework of statistical mechanics (again, I refer to previous papers, book and this blog).

Next steps in the endeavor: (a) switching towards the actual behaviour of animal space use at the individual level, and (b) broadening the empirical field to involve more species, taxa and ecological conditions. In short, an individual-based model for animal space use was needed, and it should from a system perspective be coherent with the novel structure of my Zoomer model for populations. The Multi-scaled random walk model (MRW) was in the pipeline! More on this in Part II.


*) For a simple introduction to the sycamore aphids’ behaviour, see this blog post by Ray Cannon.

**) I thank professor Stuart L. Pimm, who’s lab at the University of Tennessee I visited during the spring 1994, for his kind attempts to make my rather convoluted presentation both simpler and more accessible for the general ecologist. Unfortunately, the complexity of the model due to its quite unfamiliar biophysical concepts made it necessary to shelf the publication until a later time.

***) I’m forever thankful to professor Peter Yodzis (1943–2005, see obituary) for our open-minded and inspirational discussions in 1996 on the mathematics of my new multi-scaled population dynamical model, the Zoomer model, during a workshop on “Scale Dependency of Ecological Patterns and Processes”, Tovetorp Research Station, University of Stockholm, Sweden.


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

Kennedy, J. S. and L. Crawley. 1967. Spaced-out gregariousness in sycamore aphids Drepanosiphum platanoides (Schrank) (Hemiptera, Callaphididae). J. Anim. Ecol. 36:147-170.

Taylor, L. R. 1961. Aggregation, variance and the mean. Nature 189:732-735.

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


Non-Mechanistic Dynamics: a Simple Illustration

In my previous post I summarized my critique of mechanistic modelling when applied on animal movement. Simply stated, the Markovian design on which mechanistic models depend is in my view incompatible with a realistic representation of memory-influenced and scale-free space use. Below I illustrate the alternative approach, non-mechanistic dynamics, by a simple Figure. As conceptually described by the Scaling cube, an extra system dimension based on relative scale (“hierarchical scaling” of the dynamics), resolves the apparent paradox of non-mechanistic dynamics.

I cite from my first post on the Scaling cube (December 25, 2015):

The scaling cube brings these directions of research together under a coherent biophysics framework. It also forces upon us a need to differentiate between mechanistic dynamics (the M-floor) and non-mechanistic dynamics (the PP-ceiling).

As a supplement to my book presentation I have published a series of posts on this theme, where its unfamiliar nature has been revealed in a piece-wise manner. In particular, you got a “rolling dice” perception of how non-mechanistic dynamics pays out as a special universality class:

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.
From “The Inner Working of Parallel Processing” (blog post, February 8, 2019).


How to visualize non-mechanistic dynamics? Consider the output from a simple representation of a simulated animal path during a given time interval. The movement rules are constant and deterministic (but executed over a temporal scale range). Then consider repeating the simulation four times, marked by numbers in the image to the right. In compliance with classic design principles all repeat runs of the dynamics should be expected to show identical path progression over the habitat during the given interval. However, under non-mechanistic dynamics even a deterministic progression is expected to show occasional “surprise” moves (red line in run no 4).

Simply stated, what appears to be a surprising and random move from one time resolution (fine-scale “now”) may appear deterministic and quite rational from a coarser time resolution, in compliance with the rules for the simulation at these scales (coarser scale “now”).

Such unfamiliar and unconventional kind of model behaviour is expected to appear due to the fact that the path from the simulation is logged at a specific temporal scale (unit size time increments) while the dynamics are executed over a scale range, including coarser scales that the logged scale. The dynamics from parallel processing at coarser temporal scales will by necessity appear as stochastic surprise moves from the perspective of finer time scales.

For example, with reference to the “throwing dice” analogy (where number of eyes determine a specific state at the respective scale levels), most moves are executed at unit scale while some are executed at coarser scales. The latter may be formulated as deterministic moves at respective temporal resolutions, but will appear randomly; i.e., as a surprise, when observed from finer scales. The dice represent discrete-scale description of levels, which of course should be considered a continuous range in a real system.

The crucial difference between mechanistic and non-mechanistic kind of stochastic moves is thus buried in the process itself; is the move a result of a Markovian rule that involves some influence of randomness in the decision making about what to do next, or is the move the result of a more strategic decision? To clarify this key issue one has to apply specific statistical methods that have the potential to test for parallel processing. The acid test of parallel processing is performed by comparing some basic statistics from simulations of non-mechanistic dynamics with the similar statistics from real data. My papers, book and blog provide many results of such tests, spanning many aspects of the space use behaviour.

On the Paradox of Mechanistic Movement Models

Mechanistic design is still dominating animal space use modelling. As the readers of my papers, book and blog have understood I’m very critical to this framework. In particular, because both mechanics and statistical mechanics – due to their dependence on Markovian dynamics – under their present formulations seem to be unable to implement memory-influenced movement in a realistic manner. Thus, paradoxes abound. Unfortunately most theoreticians in movement ecology either don’t care or don’t know how to approach this issue. In this post I seek to pinpoint the most basic challenge, and how it may be potentially resolved by exploring a qualitatively new direction of modelling.

Consider the standard, simplified illustration of animal foraging, which takes up much of an individual’s focus during a day. At each time increment the behaviour adheres to rules under the mathematical framework of a low order Markovian process. A similar diagram could have been shown for other behavioural modes; like looking for a mate, seeking a shelter for resting, and so on. In short, mechanistic behaviour describes rules, which may be executed deterministically, stochastically or as a mixture. The key point is that the process in model terms is described at a specific temporal resolution. In other words, each “Start” in the illustration to the right regards execution of behaviour during the current time increment; i.e., at the current moment at the given time scale.

A Markov chain is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. In probability theory and related fields, a Markov process, named after the Russian mathematician Andrey Markov, is a stochastic process that satisfies the Markov property (sometimes characterized as “memorylessness”). Roughly speaking, a process satisfies the Markov property if one can make predictions for the future of the process based solely on its present state just as well as one could knowing the process’s full history, hence independently from such history, that is, conditional on the present state of the system, its future and past states are independent.

Animals including at least mammals, birds, reptiles, amphibians and fish (apparently also many species of invertebrates), have a cognitive capacity both to orient themselves in space and to relate to a larger or lesser degree to past experiences from visited space. For example, a home range is an emergent property of this capacity, where the animal under specific biological and ecological circumstances prefers to revisit some of its previously visited locations more frequently than by chance.

There have been many proposals for modelling spatial memory, based on the standard principles of the mechanistic (Markovian) framework. For example, the animal could cognitively store its experience from successive locations along its path with respect to local food attributes and the location of these experiences (memory map utilization). Along a trailing time window; i.e., assuming a high order rather than low order Markovian design, older experiences beyond a given memory capacity is lost by computational necessity.

However, as I’ve documented over a large range of simulation examples where theory and empirical results are compared, the space use pattern as seen in large sets of individual relocations (e.g., in telemetry/GPS-based studies) do not seem to comply with the basic statistical properties of mechanistic/Markov behaviour. Animals seem to have a capacity for long term and spatially explicit memory utilization* well beyond the time scale of any high order Markov model. Simple statistical analyses from spin-offs under the alternative statistical-mechanical model designs reveal that these animals also relate to their environment in a scale-free manner, rather than the intrinsically scale-specific constraint of processing  that by necessity follows from the standard Markovian designs.

The alternative approach to modelling animal space use – MRW and the Zoomer model – is advocated my papers, my book and on this blog. These contributions also provide several simple methods to test model compliance – Markovian vs. non-Markovian processing – using your own data.

It’s of course up to each ecologist to ignore these warning signs for the realism of standard dynamic modelling of animal movement and space use, but be aware that an increasing number of physicists are now drawn towards this field of research. In an upcoming post I’ll summarize some of these recent developments, which in large part seem to have been initially inspired by my MRW simulations that began appearing in papers almost 15 years ago**.


*) Biological and ecological conditions may constrain the utilization to a more narrow range, for example when or where the environment is relatively unpredictable. In that case the value of old information i rapidly diminishing as a function of time.

**) In fact, the first simulations of MRW were published already in my PhD thesis back in 1998.


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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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


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


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

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

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

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

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

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