**Please observe**: there is an unfortunate typo at page 2 of my book. Correct © should read 2015, not 2016. Thus, please refer to the book as Gautestad (2015), which is in compliance with ISBN.

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**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 N* _{ij}* 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

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 N* _{ij}*, extrinsic influence – like immigration of individuals – is always stochastic and thus density independent with respect to N

**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, i*f 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.

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 N* _{ij}*, will at the next time t+1 either embed N-100 individuals if they all leave location

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 *k*_{i+x} takes place with probability 1/*k*_{i+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 immigration rate connects the local population to surrounding populations, but is – by necessity from the standard model design – density *independent* with respect to the dynamics in N* _{ij}*.

**) 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 will tend to drive N towards zero (extinction from standard diffusion in open environments). Individuals will “leak” from an open border zone to the surroundings.

***) 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 *k _{i}=*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.

**The standard theories on animal space use rest on some shaky behavioural assumptions, as elaborated on in my papers, in my book and here in my blog. One of these assumptions regards the assumed lack of influence of positive feedback, in particular the self-reinforcing effect that emerge when individuals are moving around with a cognitive capacity for both temporal and spatial memory utilization. The common ecological methods to study individual habitat use; like the utilization distribution (a kernel density distribution with isopleth demarcations), use/availability analysis, and so on, explicitly build on statistical theory that not only disregards such positive feedback, but in fact requires that this emergent property is not influencing the system under scrutiny.**

Unfortunately, most memory-enhanced numerical models to simulate space use are rigged to comply with negative rather than positive feedback effects. For example, the model animal successively stores its local experience with habitat attributes while traversing the environment, and it uses this insight in the sequential calculation of how long to stay in the current location and when to seek to re-visit some particularly rewarding patches (Börger *et al.* 2008, van Moorter* et al.* 2009, Spencer 2012, Fronhofer *et al.* 2013; Nabe-Nielsen *et al.* 2013). In other words, the background melody is still to maintain compliance with the marginal value theorem (Charnov 1976) and the ideal free distribution dogma, which both are *negative* feedback processes and not a self-reinforcing process that tends to counteract such a tendency.

Negative feedback (or balancing feedback) occurs when some function of the output of a system, process, or mechanism is fed back in a manner that tends to reduce the fluctuations in the output, whether caused by changes in the input or by other disturbances. Whereas positive feedback tends to lead to instability via exponential growth, oscillation or chaotic behavior, negative feedback generally promotes stability. Negative feedback tends to promote a settling to equilibrium, and reduces the effects of perturbations. Negative feedback loops in which just the right amount of correction is applied with optimum timing can be very stable, accurate, and responsive.

https://en.wikipedia.org/wiki/Negative_feedback

The above definitions follow the usual path to explain negative feedback as “good”, and positive feedback as something scary (I will return to this misconception in a later post in this series). It echoes the prevailing “Balance of nature” philosophy of ecology, which I’ve criticized at several occasions (see, for example, this post).

In a previous post, “Home Range as an Emergent Property“, I described how memory map utilization under specific ecological conditions may lead to a self-reinforcing re-visitation of previously visited locations (Gautestad and Mysterud 2006, 2010); in other words, a positive feedback mechanism*. Contemporary research on animal movement covering a wide range of taxa, scales, and ecological conditions continues to verify site fidelity as a key property of animal space use.

I use a literature search to test an assumption of the ideal models that has become widespread in habitat selection theory: that animals behave without regard for site familiarity. I find little support for such “familiarity blindness” in vertebrates.

Piper 2011, p1329.

**Obviously, in the context of spatial memory and site fidelity it should be an important theme for research to explore to what extent and under which conditions negative and positive feedback mechanisms are shaping animal space use.**

Positive feedback from site fidelity will fundamentally influence analysis of space use. For example, two patches with *a priori* similar ecological properties may end up being utilized with a disproportionate frequency due to initial chance effects regarding which patch happened to gain familiarity first**. Further, if the animal is utilizing the habitat in a multi-scaled manner (which is easy to test using my MRW-based methods), this grand error factor in a standard use/availability analysis cannot be expected to be statistically hidden by just studying the habitat at a coarser spatial resolution within the home range.

Despite this theoretical-empirical insight, the large majority of wildlife ecologists still tend to use classic methods resting on the negative feedback paradigm to study various aspects of space use. The rationale can be described by two end-points on a continuum: either one ignores the effect from self-reinforcing space use (assuming/hoping that the effect does not significantly influence the result), or one use these classic methods while holding one’s nose.

The latter category is accepting the prevailing methods’ basic shortcomings – either based on field experience or inspired by reading about alternative theories and methods – but the strong force from conformity in the research community is hindering bold steps out of the comfort zone. Hence, the paradigm prevails. Again I can’t resist referring to a previous post, “Why W. H. Burt is Now Hampering Progress in Modern Home Range Analysis“.

Within the prevailing modelling paradigm, implementing spatial memory utilization in combination with positive feedback-compliant site fidelity is a mathematical and statistical nightmare – if at all possible. However, as a reader of this blog you are at least fully aware of the fact that some numeric models have been developed lately, ouside the prevailing paradigm. These approaches not only account for memory map utilization but also embed the process of positive feedback in a scale-free manner (I refer to our papers and to my book for model details; see also Boyer *et al.* 2012).

NOTES

* The paper explores space use under the premise of positive feedback during superabundance of resources, in combination with negative feedback during temporal and local over-exploitation.

** In Gautestad and Mysterud (2010) I described this aspect as the distance from centre-effect; i.e., the utilization distribution falls off at the periphery of ulilized patches independently of a similar degradation of preferred habitat.

REFERENCES

Börger, L., B. Dalziel, and J. Fryxell. 2008. Are there general mechanisms of animal home range behaviour? A review and prospects for future research. Ecology Letters 11:637-650.

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.

Charnov, E. L. 1976. Optimal foraging: the marginal value theorem. Theor. Popula. Biol. 9:129-136.

Fronhofer, E. A., T. Hovestadt, and H.-J. Poethke. 2013. From random walks to informed movement. Oikos 122:857-866.

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

Gautestad, A. O., and I. Mysterud. 2010. Spatial memory, habitat auto-facilitation and the emergence of fractal home range patterns. Ecological Modelling 221:2741-2750.

Nabe-Nielsen, J., J. Tougaard, J. Teilmann, K. Lucke, and M. C. Forckhammer. 2013. How a simple adaptive foraging strategy can lead to emergent home ranges and increased food intake. Oikos 122:1307-1316.

Piper, W. H. 2011. Making habitat selection more “familiar”: a review. Behav. Ecol. Sociobiol. 65:1329-1351.

Spencer, W. D. 2012. Home ranges and the value of spatial information. Journal of Mammalogy 93:929-947.

van Moorter, B., D. Visscher, S. Benhamou, L. Börger, M. S. Boyce, and J.-M. Gaillard. 2009. Memory keeps you at home: a mechanistic model for home range emergence. Oikos 118:641-652.

**In empirical data GPS fixes are never exact positions. A “fuzziness field” will always be introduced due to uncertain geolocation. When analyzing a set of fixes in the context of multi-scaled space use, are the parameter estimates sensitive to this kind of statistical error? Simultaneously, I also explore the effect on constraining the potential home range by disallowing sallies to the outermost range of available area.**

To explore the triangulation error effect on space use analysis I have simulated Multi-scaled random walk in a homogeneous environment with N=10,000 fixes (of which the first 1,000 fixes were discarded) under two scenaria; a “sharp” location (no uncertainty polygons), and strong fuzziness. The latter introduced a random displacement to each x-y coordinate with a standard deviation (SD) of magnitude approximately equal to the system condition’s Characteristic scale of space use (CSSU). Displacements to the outermost parts of the given arena was disallowed, to study how this may influence the analyses. I then ran the following three algorithms in the MRW Simulator: (a) analysis of A(N) at the home range scale, (b) analysis of A(N) at a local scale (splitting the home range into four quadrants), and (c) analysis of the fix scatters’ fractal property.

The following image shows the sharp fix set and the strongest fuzzyness condition.

By visual inspection is is easy to spot the effect from the spatial error (SD = 1182 length units, upper row to the right). However, the respective A(N) analyses at the home range scale generated a CSSU estimate that was only 10% larger in the fuzzy set (linear scale). When superimposing cells of CSSU magnitude onto each fix, the home ranges appear quite similar in overall appearance and size. This was to be expected, since fuzziness influences fine-resolution space use only.

Visually, both home range areas appear somewhat constrained with respect to range, due to the condition to disallow displacements to the peripheral parts of the defined arena (influencing less than 1% of the displacements).

**A(N) analysis (the Home range ghost)**. The two conditions appeared quite similar in plots of log[*I*(N)], where *I* is the number of fix-embedding pixels at optimized pixel resolution, as described in previous posts.

However, for the fuzzy series there is a more pronounced break-point with a transition towards exponent ∼0.5 in sample size of magnitude larger than log_{2}(N) ≈ 3. This break-point “lifted” the regression line somewhat for the fuzzy series, leading to a slightly larger intercept with the y-axis when interpolating towards log(N) = 0. This difference between the two conditions with respect to the y-intercept, Δlog(*c*) from the home range ghost formula log[*I*(N)] = log(*c*) + z*log(N), also defines the difference in CSSU when comparing sharp and fuzzy data sets. recall that CSSU ≡ *c*.

The spatial constraint on extreme displacements relative to the respective home ranges’ centres apparently did not influence these results.

**I have also superimposed local CSSU-analysis for respective four quadrants of the two home ranges. When area extent for analysis is constrained in this manner; i.e., spatial extent is reduced to 1:4 in area terms (1:2 linearly) for each local sub-set of fixes, respective (N,I) plot needs to be adjusted by a factor which compensates for the difference in scale range. **

Since the present MRW conditions were run under fractal dimension D=1, each local log_{2}[(N,I)] plot is rescaled to log_{2}[(N,*I*)] + log_{2}[Δ_{D}(grain/extent)] = log_{2}[(N,*I*)] + 1 when Δ regards the relative change of scale under condition D=1. After this rescaling the over-all CSSU and the local CSSU are overlapping, as shown by the regression lines in the A(N) analyses above. Overlapping CSSU implies that the four quadrants had similar space use conditions, which is true in this simplified case.

**Fractal analysis**. The Figure below shows the magnitude of incidence *I* as a function of relative spatial resolution *k* (the “box counting method”), spanning the scale range from *k*=1 (the entire arena, linear scale 40,000 units) and down to *k = *40,000/(2^{12}) = 9.8 units, linear scale*.

Starting from the coarsest resolution, *k*=1, the log-log regression obviously shows *I *= 1. At resolution *k*=1:2 and *k*=1:4 (4 and 16 grid cells, respectively), I = 4 and I = 4. In other words, all boxes contain fixes at *k*=1/2, apparently satisfying a two-dimensional object, and at k=1:4 some empty cells (12 of 16) are peeled away as empty space from the periphery of the fix scatter at this finer resolution.

This coarse-scale pattern is a consequence of the defined space use constraint. Disallowing “occasional sallies” outside the core home range obviously influences the number of non-empty boxes relative to all boxes available at the coarse resolutions 1<*k*<4^{-1}.

However, at progressively finer resolutions – below the “space fill effect” range transcending down to ca *k*=1:32 – the true fractal nature of the scatter of fixes begin to appear due to log-log linearity, confirming a statistical fractal with a stable dimension D≈1.1 over the resolution range 2^{-9} < *k* < 2^{-5} (showing log-log slope of -1.1). At finer resolutions, the dilution effect flattens further expansion of *I*. The D=1.1 scale range is close to expectation from the simulation conditions D_{true}=1, while the deviations above and below this range are trivial statistical artifacts from space filling and dilution.

The most interesting pattern regards the finest resolution range, where the fuzzy set of fixes somewhat unexpectedly follows a similar log(*k,I*) pattern as the non-fuzzy set. However, the slight difference in D, which increases to D =1.17, may be caused by the fuzziness (proportionally stronger space-fill effect as resolution is increased).

**To conclude, if the magnitude of position uncertainty does not supersede the individual’s CSSU scale under the actual conditions, the A(N) analyses of MRW compliant space use does not seem to be seriously influenced by location fuzziness. The “fix scrambling” error is in most part subdued towards finer scales than the CSSU.**

However, the story doesn’t end here. I have superimposed a dotted red line onto the Figure above. Overlapping with the D=1 section of grid resolutions, the line is extrapolated towards the intersection with log(*I*) ≈ 0 at a scale that is 2^{1.5} = 2.8 times larger (linear) scale than the actual arena for the present analyses. In other words, in absence of area constraint and step length constraint (and disregarding step length constraint due to limited movement speed of the individual) one should expect the actual set of fixes to “fill up” the missing incidence over the coarse scale range, leading to D≈1 for the entire range from the dilution effect range towards coarser scales.

**I have also marked the CSSU scale as the midpoint of the red line. A resolution of log _{2}(k)=-5.5 is in fact very close to the CSSU estimate from the A(N) method (k=1:50 of actual arena using the A(N) method, versus 1:35 of the arena according to the fractal analysis). This alternative method to estimate CSSU was first published in this blog post. **

A preliminary development towards this approach was explored both theoretically and empirically in Gautestad and Mysterud (2012).

NOTE

*) This analysis of N= 8,000 fixes, spanning box counting of 1, 2, 4, 16, 32, … 16.8 million grid cells at respective scales, took ca 10 hours pr. series in the MRW Simulator, using a laptop computer. I suppose this enormous number cracking it would be outside the practical range of a similar algorithm in R.

REFERENCES

Gautestad, A. O., and I. Mysterud. 2012. The Dilution Effect and the Space Fill Effect: Seeking to Offset Statistical Artifacts When Analyzing Animal Space Use from Telemetry Fixes. Ecological Complexity **9**:33-42.

**To understand populations’ space use one needs to understand the individual’s space use. To understand the individuals’ space use one needs to acknowledge the profound influence of spatio-temporal memory capacity combined with multi-scale landscape utilization, which continues to be empirically verified at a high pace in a surprisingly wide range of taxa. Complex space use has wide-ranging consequences for the traditional way of thinking when it comes to formulate these processes in models. In a nutshell, the old and hard-dying belief in the balance of nature needs a serious re-formulation, since complexity implies “strange” fluctuations of abundance over space, time and scale. A fresh perspective is needed with respect to inter-species interactions (community ecology) and environmental challenges from habitat destruction, fragmentation and chemical attacks. We need to address the challenge by rethinking also the very basic level of how we perceive an ecosystem’s constituents: how we assume individuals, populations and communities to relate to their surroundings in terms of statistical mechanics.**

Stuart L. Pimm summarizes this Grand Ecological Challenge well in his book *The Balance of Nature?* (1991). Here he illustrates the need to rethink old perceptions linked to the implicit balancing principle of carrying capacity*, and he stresses the importance of understanding limits to how far population properties like resilience and resistance may be stretched before cascading effects appear. In particular, he advocates the need to extend the perspective from short-series local-scale population dynamics to long-term and broad scale community dynamics. In this regard, his book is as timely today as it was 27 years ago. However, in my view the challenge goes even deeper than the need to extending spatio-temporal scales and the web of species interactions.

My own approach towards the Grand Ecological Challenge started with similar thoughts and concerns as raised by Pimm**. However, as I gradually drifted from being a field ecologist towards actually attempting to model parsimonious population systems I found the theoretical toolbox to be void of key instruments to build *realistic* dynamics. In fact, the current methods were in many respects even seriously misleading, due to what I considered some key dissonant model assumptions.

In my book (Gautestad 2015), and here in my subsequent blog, I have summarized how – for example – individual-based modelling generally rests on a very unrealistic perception of site fidelity (March 23, 2017: “Why W. H. Burt is Now Hampering Progress in Modern Home Range Analysis“). I have also found it necessary to start from scratch when attempting to build what I consider a more realistic framework for population dynamics (November 8, 2017: “MRW and Ecology – Part IV: Metapopulations?“), for the time being culminating with my recent series of post on “Simulating Populations” (part I-X).

I guess the main take-home message from the present post is:

- Without a realistic
*understanding*;*i.e.*, modelling power, of individual dispersion over space, time and scale it will be futile to build a theoretical framework with deep explanatory and predictive value with respect to population dynamics and population ecology. In other words, some basic aspects of*system complexity at the “particle level”*needs to be resolved. - Since we in this respect typically are considering either the accumulation of space use locations during a time interval (e.g., a series of GPS fixes) or a population’s dispersion over space and how it changes over time, we need a proper formulation of the statistical mechanics of these processes.In other words, when simplifying extremely complicated systems into a manageable set of smaller set of variables, parameters and key interactions, we have to invoke the hidden layer.
- With a realistic set of basic assumptions in this respect, the modelling framework will in due course be ready to be applied on issues related to the Grand Ecological Challenge – as so excellently summarized by Pimm in 1991. In other words, before we can have any hope of a detailed prediction of a local or regional faith of a given species or community of species under a given set of circumstances, we need to build models that are void of the classical system assumptions that have cemented the belief in the so-called balance of nature.

NOTES

*) The need to rethink the concept of carrying capacity and accompanying “balance” (density dependent regulation) should be obvious from the simulations of the Zoomer model. Here a concept of carrying capacity (called CC) is introduced at a local scale only, where – logically – the crunch from overcrowding is felt by the individuals. By coarse-graining to a larger pixel than this finest system resolution we get a mosaic of local population densities where each pixel contains a heterogeneous collection of intra-pixel (local) CC-levels. If “*standard*” population dynamic principles applies, the population change when averaging the responses over a large number of pixels with similar density should be the same whether one considers the density at the coarser pixel or the average density of the embedded finer-grained sub-pixels. This mathematical simplification follows from the mean field principle. In other words, the sum equals the parts. On the other hand, if the principle of *multi-scaled* dynamics applies, two pixels at the coarser scale containing a similar average population density may respond differently during the next time increment due to inter-scale influence. At any given resolution the dynamics is as a function not only of the intra-pixel heterogeneity within the two pixels but also of their respective neighbourhood densities; i.e., the condition at an even coarser scale. The latter is obviously not compliant with the mean field principle, and thus requires a novel kind of population dynamical modelling.

**) In the early days I was particularly inspired by Strong *et al.* (1984), O’Neill *et al.* (1986) and L. R. Taylor; for example, Taylor (1986).

REFERENCES

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

O’Neill, R. V., D. L. DeAngelis, J. B. Wade, and T. F. H. Allen. 1986. *A Hierarchical Concept of Ecosystems*. Monographs in Population Biology. Princeton, Princeton University Press.

Pimm, S. L. 1991, *The balance of nature? Ecological issues in the conservation of species and communities*. Chicago, The University of Chicago Press.

Strong, D.E., Simberloff, D., Abele, L.G. & Thistle, A.B. (eds). 1984. *Ecological Communities: Conceptual Issues and the Evidence. *Princeton,Princeton University Press.

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

**In Part IX a standard Coupled map lattice model was shown to be able to display a 1/f power spectrum, by careful tuning of one of the conditions for population mixing. On the other hand, I also showed that the Zoomer model under similar general conditions showed a higher resilience with respect to the 1/f property. In the present post I explore this statistical resilience of the Zoomer model further, by stressing the population system towards other corners of extreme conditions. However, I start by elaborating on this pressing question: Why this recurrent focus on 1/f noise? To stimulate your curiosity I also reproduce from my book two analyses of empirical data; the sycamore aphid Drepanosiphum platanoides and the leaf miner Leucoptera meyricki. **

First, a brief bird’s view of complex dynamics. I have repeatedly given partly answers to the question “why focusing on 1/f noise?”, but here I seek to give you a broader perspective, starting with a citation from Scholarpedia:

1/f fluctuations are widely found in nature. During 80 years since the first observation by Johnson (1925), long-memory processes with long-term correlations and 1/f

^{α}(with 0.5 ≲ α ≲ 1.5) behavior of power spectra at low frequencies f have been observed in physics, technology, biology, astrophysics, geophysics, economics, psychology, language and even music…

…1/f noise can not be obtained by the simple procedure of integration or of differentiation of such convenient signals. Moreover, there are no simple, even linear stochastic differential equations generating signals with 1/f noise. The widespread occurrence of signals exhibiting such behavior suggests that a generic mathematical explanation might exist. Except for some formal mathematical descriptions like fractional Brownian motion (half-integral of a white noise signal), however, no generally recognized physical explanation of 1/f noise has been proposed. Consequently, the ubiquity of 1/f noise is one of the oldest puzzles of contemporary physics and science in general.

Ward and Greenwood (2007)

As scientists – whether we are ecologists or working in other fields – we are of course curious about exploring this hard nut to crack, in particular since natural populations (paradoxically, if judged by expectation from standard theory) tend to show 1/f noise in their spatial variation and temporal fluctuations. Here are two examples from my book:

Both series show close agreement with 1/f noise; and their respective derivatives trivially show similar compliance with f noise (power changing proportionally with frequency, shown as dashed plots). The first example is a spatial transect of sycamore aphids (my own data), and the second example shows my power spectrogram analysis based on a time series collected from Bigger and Tapley (1969) of the coffee leaf miner *Leucoptera meyricki*. Both data sets and methods are described in detail in my book.

**The burning question then arises: what kind of spatially extended population model may reproduce 1/f noise over space, time and scale? Within the current paradigm the coupled map lattice (CML) approach fails, since it by design cannot reproduce long distance and far back influence on a present location’s dynamics*.The standard calculus approach (ordinary and partial differential equations) also fails, since models of this design rests on the mean field approximation. In short a novel approach is needed. As previously stated, for the time being and to my knowledge the Zoomer model is the only candidate standing the test when confronting simulations with real data properties with respect to statistical mechanics.**

In a series of posts I have now in a step-by step manner presented the Zoomer model, and compared it with the paradigmatic approach in the field, the coupled map lattice design (CML). In Part IX the CML model was shown to be more sensitive to the population’s response to local overcrowding (the CC level). Only 100% redistribution of individuals where CC was temporally exceeded led to a 1/f pattern. The Zoomer model showed 1/f, whether 40% or 100% of individuals emigrated. In other words, strong statistical resilience in this regard. However, the best way to test a model’s intrinsic dynamics is to stress it to its limit. Thus, below I test the effect from changing the redistribution behaviour of the individuals that emigrate from the “CC overshoot” localities.

So far, these individuals have been set to redistribute themselves to other cells in “swarms” of X% each of all swarmers at the actual point in time. For example, if two cells have an overshoot event at this point in time with a total number of dispersers of – say – 10,000 individuals, these dispersers settle in 5% of the other cells (of 1024 cells available) in swarms of ≈ 195 individuals pr. receiving cell, chosen randomly.

Below I explore two variants of this behaviour; (1) the dispersers settle randomly and individually among the available cells, and (2) the dispersers settle in only 1% of the cells. The former implies a more smooth dispersal kernel, and the latter a more collective kind of large swarms settling in just a few cells.

In short, the power spectrum of the time series of the smooth dispersal condition shows white noise (1/f^{0}), while the extremely clumped dispersal shows more like red noise (1/f^{1.8}).**

To conclude, the 1/f property in the Zoomer model seems to be critically sensitive to the behaviour of the dispersing individuals during local crowding (boom/bust) events. Whether the bust leads to local meltdown (0% remaining) or not (60% remaining), the 1/f pattern was robust. However, how the emigrating individual re-settled themselves (independently or in a contagious manner) determined 1/f compliance. An intermediate level of contagion was in best compliance with 1/f, and thereby with the empirical results shown above. Intermediate contagion implies intermediate level of population disturbance frequency***.

This conclusion also fits well with the general model property of zooming, which reflects conspecific attraction in over-all terms. The “balanced” and intermediate kind of zooming follows prom the generic condition that the strength of zooming is distributed evenly over that actual range of spatial scales (*k*^{–b} zoomers to *k ^{b}* larger grid area, with

Interestingly, the spatial pattern log(M,V) for the white noise condition shows – as expected – a similarly rattled pattern, with slope *b ≈ *1 and log(a) >> 0. However, in the contageous dispersal scenario, the spatially self-similar property *b ≈ *2 and log(a) ≈ 0 is maintained. Hence, the spatial pattern is still fractal-compliant with similar properties as for other levels of contagion during re-distribution events, despite the more classic “random walk-like” time series of fluctuations.

**To conclude, by studying a population’s variation of abundance over space, time and scale it should be possible to analyze a wide range of key ecological signatures from the data series’ statistical properties; for example to what extent the population under the given environmental conditions adheres to a multi-scaled and scale-free kind of intrinsically driven population dynamics/kinetics, and how the population responds to local crowding events. Hopefully, this statistical-mechanical approach may lead to more realistic theory and thus better predictive power. Bringing population dynamical modelling and some basic empirical properties of real populations in closer agreement is long overdue in this important field of ecology.**

Crucially, these basic analyses should also be able to cast light on to what extent the dynamics are compliant with expectations from a classical modeling regime; i.e., a Markov- and mean field compliant kind of statistical mechanics, or the alternative framework based on the parallel processing conjecture for individual space use (the MRW model and its population level version, the Zoomer model). This challenge is of course the first obstacle that has to be passed. Hopefully the two insect examples above will inspire others to perform broader and deeper tests – for example, based on my recent series of blog posts.

REFERENCES

Bigger, M., and R. G. Tapley. 1969. Prediction of outbreaks of coffee leaf-miners on

Kilimanjaro. Bulletin of Entomological Research 58:601-618.

Ward, L. M. and P. E. Greenwood. 2007. 1/f noise. Scholarpedia 2(12):1537.

NOTES

*) The 1/f pattern from a CML example in Part IX was a 1/f look-alike due to some tweaking of conditions, but it failed when other statistical aspects were scrutinized.

**) As in previous scenaria, the time scale is set to be smaller than the population’s net growth rate of 0-1%, giving a focus of the higher rate population mixing processes; diffusion 1% for the CML examples, zooming 5% in the Zoomer examples (distributed with 1% pr. spatial scale), and a stochastic magnitude of intrinsic re-distribution of individuals from local CC-linked events under both platforms.

***) By intermediate frequency of disturbance from intrinsic reshuffling of individuals I mean on one side that a given locality’s frequency of disruptions by swarms of dispersing individuals from other localities depends on swarm size. Larger swarms mean fewer locations where they settle. On one hand, large swarms (stronger contagion among emigrants) appear more rarely at a given locality in statistical terms but will have stronger local impact when a swarm arrives. On the other hand, an additional disruptive event from such an influx may happen within the next time increment if the sum of existing and arriving individuals surpass the local CC level. One also has to consider that a higher local abundance due to arriving immigrants may change a declining local abundance to an increasing one due to the zooming process of conspecific attraction.The abundance level as seen from a broader neighbourhood scale may increase sufficiently to toggle this neighbourhood (including the actual cell receiving the immigrants) from being a net supplier; *i.e*., subject to net local population decline, to becoming net receiver of individuals in the time ahead.

This perspective again underscores the importance of studying population dynamics not only over space and time, but also over respective scale axes and in a Parallel processing manner. Hence, a Markovian-compliant model framework is not sufficient to understand complex dynamics/kinetics, including the 1/f property.

**In the previous post I presented a time series of the Zoomer model, verifying a 1/f (pink) noise signature. For the sake of comparison I here present a series from scale-specific dynamics (a Coupled map lattice model) using a comparable system setup. Interestingly, an apparent similarity with some statistical aspects of the Zoomer result is appearing, given specific simulation conditions. “Fine tuning”, if you wish. These critical aspects may reveal important insight in complex dynamics of spatially extended systems, with respect to whether individual (“particle”) mixing is driven by spatial memory and inter-individual congregation (multi-scaled zooming) or standard mixing (CML model with classical, single-scaled diffusion at fine resolutions).**

The present CML model conditions are the same as for the Zoomer version with 5-scale zooming of 1% zoomers pr. scale level and time increment, except for replacing zooming with 1% diffusion between neighbour cells at unit scale. Recall from previous posts in this series – and from the older post where the Zoomer model was described in mathematical terms – that zooming has a component of intraspecific cohesion. In contrast, diffusion is a memory-less (Markov compliant) kind of re-distribution of individuals.

The spatial density snapshot (upper left) shows the smooth surface expected from even a weak magnitude of memory-less inter-cell mixing. Also observe the intercept log(*a*) << 0 in the log(M,V) plot.

**By the way, such a smooth surface is what modelers of population dynamics tend to love, since it adheres to the **mean field** property. On the other hand, empirical data typically shows a very rugged surface over a wide range of resolutions (even in a relatively homogeneous environment), statistically in close compliance with what the Zoomer model generates.**

At this particular instance in time of the CML simulation above one also sees the effect from a local cell overshooting its carrying capacity (CC) and redistributing its individuals as small swarms among some of the other cells (seen as temporary “warts” on the density surface). The log(M,V) plot shows a slope close to *b *= 2 and log(*a*) << 0, as in previous examples of scale-specific CML dynamics. I have previously pointed out that log(a) << 0 (in combination with *b ≈ *2) is a hallmark of scale-specific rather than scale-free population dispersion. It does not comply with the “CV = 1” law for scale-free dynamics (*b ≈ *2, log(*a*) ≈ 0). Now and then some larger number of simultaneous local population breakdown happen, and – as was shown also for the Zoomer condition – this led to a temporary change to *b≈*1 and log(a) >> 0.

The actual time series (above) appears qualitatively different from the Zoomer equivalent (see Part IIX), being less rugged. Despite this, the correlogram from a sampled series at frequency 1:80 does not reveal any significant difference to the Zoomer model result. However, the log(M_{t},V) plot is both more clustered and with a generally smaller magnitude of log(V) than was found in the Zoomer example.

Surprisingly, the power spectrogram (log2-transformed) satisfies pink noise; 1/f^{μ} with μ ≈ 1), and is thus similar to the zoomer example:

However, if local population survival when density overshoots CC is increased from 0% to 60%, the spectrum is more flattened (more like white noise). In comparison, a similar change to the CC overshoot behaviour for the Zoomer model did not change the spectrum away from 1/f. Both spectra are shown below.

What regards the other statistical aspects of the present zoomer example with CC survival 60%, these are reproduced below:

Despite a more “noisy” time series than the 0% local survival condition (see image above for 60%, and compare with Part IIX for 0%), which is also reflected in spatial log(M,V) with *b* ≈ 1.5 and log(*a*) >> 0 dominating most of the time (during more silent intervals the CV=1 law applies), the temporal log(M,V) plot shows larger log(V) for a given log(M), relative to the CML example. Thus, despite the narrow range of log(M) the level of log(V) is at least close to the line for b ≈ 2.

**In summary, the traditional population dynamical modelling approach to apply scale-specific coupled map lattice models may to some degree mimic scale-free pattern generation with respect to the power spectrum by setting CC overshoot survival low (in this instance, survival = 0%, where “survival” implies remaining individuals in the given cell while most of the emigrants survive but redistribute themselves to other cells). On the other hand, the Zoomer model shows stronger statistical resilience over a broader range of aspects, including both the power spectrum and the CV ≈ 1 property.**

In a follow-up post I will reveal further theoretical details, by linking the Zoomer model to so-called* self-organized criticality* in statistical mechanics of complex systems. Step-by-step I’m approaching a shift towards ecological application of the Zoomer model. However; by necessity, theory first.