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.