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.
Temporally constrained space use is a key property of animal movement. With respect to vertebrates three main statistical representations are particularly popular among modelers, based on disparate theoretical foundations. Which one should one use for analysis of a particular data set? As always in ecological research, one needs some simple protocol to distinguish between alternative model assumptions.
Animals paths are neither straight lines nor a dense dot of positions from juggling back and forth at the same spot. Typically we see a complicated combination of these two extreme patterns; some quite straightforward moves occasionally abrupted by more jagged movement. In order to infer behavioural and ecological results from space use one needs to study the data in the context of a realistic theoretical framework.
Outside the realm of temporal site fidelity; e.g., a drifting home range (Doncaster and Macdonald 1991), ecological textbooks typically explain the mixture of straight and convoluted movement bouts as Area restricted Search (ARS).
Area-restricted search. A foraging pattern in which a consumer responds to an intake of food by slowing down its movement and remaining longer in the vicinity of the most recently located food item. This behaviour causes consumers to remain longer in areas where the density of food items is high than in areas where it is low.
A Dictionary of Ecology. Encyclopedia.com. 29 Jul. 2018.
In terms of statistical models, this rather qualitative description of behaviour may be formulated in many ways. A popular one is to combine classic or correlated random walk with two distinct parameter values for intrinsic step length distribution (the λ value) in F(r) ∝ e-λr. In this manner, movement varies with the jaggedness of the path (number of turns pr. period of time) rather than the movement speed. A larger λ implies smaller step lengths on average, which tend to increase local staying time during intervals when this movement mode is active. By fine-tuning respective λ1 and λ2 and in such a superposition of two “randomly toggling modes” this so-called composite random walk can even be made to mimic the second main approach to model complicated paths, the Lévy flight model (Benhamou 2007; but see Gautestad 2013):
Lévy flights are, by construction, Markov processes. For general distributions of the step-size, satisfying the power-like condition, the distance from the origin of the random walk tends, after a large number of steps, to a stable distribution due to the generalized central limit theorem, enabling many processes to be modeled using Lévy flights.
Lévy flights (and walks) are typically thought of as producing “fat tailed” step length distributions. However, within an often observed range of the distribution (Pr(U>u = O(u)-k with 1<k<2) in real animals, one should not forget that half of the displacements in the distribution are in fact relatively short! In fact, the dominating step length bin is ultra-short moves, leading to a path that is conceptually (albeit not statistical mechanically) similar to the slow-down effect of more jagged moves during composite random walk.
Such “knots” of ultra-short moves of a Lévy path brings us to the third class of movement models, Continuous time Random Walk (CTRW). In this case the movement may be arrested for a shorter or longer period:
The step length distribution and a waiting time distribution (“resting” between steps) describe mutually independent random variables. This independence between jump lengths and waiting time to perform the next step makes the difference between CTRW and ordinary random walk (including Brownian motion).
Page 91 in: Gautestad, A. O. 2015, Animal Space Use: Memory Effects, Scaling Complexity, and Biophysical Model Coherence Indianapolis, Dog Ear Publishing.
Again, by fine-tuning the intrinsic model formulations and parameter conditions of CTRW one may in fact produce a Lévy flight-like movement pattern (Reynolds 2010; Schulz et al. 2013). Towards another extreme, one may achieve a classic random walk look-alike pattern. Conceptually, CTRW thus belongs on the left wall of the Scaling cube (x=0; y and z varies in accordance to model variant and parameter settings).
CTWR (which is closely connected to the concept of fractional diffusion) is typically invoked to explain a kind of switching – “intensive versus extensive searching” (scale-free sub-diffusive versus super-diffusive search) – that is now being found in real animal data like foraging of Balearic shearwaters Puffinus mauretanicus and Cory’s shearwaters Calonectris diomedea along the coast of Spain (Bartumeus et al. 2010).
In particular, they found that the birds in the presence of fisheries were more “restless” when taking advantage of fishery discards than in absence of trawlers, implying a higher probability of leaving a localized area pr. unit time during trawling activity. Specifically, during period of fishery discard utilization tended to show a smaller temporal scaling exponent for staying time (the temporal aspect of CTRW; larger β in their site fidelity function S(t) ∝ t–β, implying fewer events with a particularly prolonged staying time). In the spatial aspect of flight length distributions, when fisheries discard was present the birds tended to show good compliance with a negative exponential function*.
On the other hand, in the absence of trawlers a compliance with Lévy walk (truncated power law) was found. Space and time brought together, they found that the birds tended towards sub-diffusive foraging in the presence of fisheries discard (despite – somewhat counter-intuitively – a smaller local staying time within a given patch of a given spatial resolution), and super-diffusive foraging under natural conditions.
Despite the mathematical and numerical attractiveness of composite random walk, Lévy walk and CTRW and their well-explored statistical properties, they unfortunately all lack what may be a crucial component of foraging behaviour: spatial memory; i.e., the condition x>0 of the scaling cube.
Without spatial memory, self-crossing of an individual’s path happens by chance only, not intentionally by returns to a previous location (site fidelity, whether we consider short term or long term time scales).
The memory aspect of temporally constrained space use will be explored in Part II.
*) Somewhat confusingly relative to common practice Bartumeus et al. (2010) use the exponential formula variant F(r) ∝ e-r/λ, which makes average step length proportional with λ rather than 1/λ.
Bartumeus, F., L. Giuggioli, M. Louzao, V. Bretagnolle, D. Oro, and S. A. Levin. 2010. Fishery discards impact on seabird movement patterns at regional scales. Current Biology 20:215-222.
Benhamou, S. 2007. How many animals really do the Lévy walk? Ecology 88:1962-1969.
Doncaster, C. P., and D. W. Macdonald. 1991. Drifting territoriality in the red fox Vulpes vulpes. Journal of Animal Ecology 60:423-439.
Gautestad, A. O. 2013. Animal space use: Distinguishing a two-level superposition of scale-specific walks from scale-free Lévy walk. Oikos 122:612-620.
Reynolds, A. 2010. Bridging the gulf between correlated random walks and Lévy walks: autocorrelation as a source of Lévy walk movement patterns. J. R. Soc. Interface 7: 1753–1758.
Schulz, J. H. P., A. V. Chechkin, and R. Metzler. 2013. Correlated continuous time random walks: combining scale-invariance with long-range memory for spatial and temporal dynamics. J. Phys. A: Math. Theor. 46:1-22.
Examples of positive feedback loops in population dynamics abound. Even if the majority of models are focusing on negative feedback, like the logistic growth function, non-equilibrium “boom and bust” kind of model designs have also been developed. In this post I elaborate on the particular kind of positive feedback loop that emerges from cross-scale dual-direction flow of individuals that is based on the parallel processing conjecture.
The image to the right illustrates – in simplistic terms – a spatially extended population model of standard kind (e.g., a coupled map lattice design) where each virtually demarcated local population j at spatial resolution i and at a given point in time t contains Nij individuals. No borders for local migration are assumed; i.e., the environment is open both internally and externally towards neighbouring sites.Typically, these individuals are set to be subject to a locally negative feedback loop in accordance to principles of density dependent regulation*. The larger the N the larger the probability of an increased death rate and/or and increased emigration rate from time t to t+1, eventually leading both the local and the over-all population to a steady state. This balancing** condition lasts until some change (external perturbation) is forcing the system into a renewed loop of negative feedback-driven dynamics. In a variant of this design, density regulation may be formulated to be absent until a critical local density is reached, leading to boom and bust (“catastrophic” death and emigration), which may be more or less perturbed by random immigration rate from asynchronous developments in respective surrounding Nij. More sophisticated variants abound, like inclusion of time lag responses, interactions with other trophic levels, and so on.
As previously explained in other posts, this kind of model framework depends on a premise of Markov-compliant processes at the individual level (mechanistic system), and thus also at the population level (local or global compliance with the mean field principle). In this framework intrinsic dynamics may be density dependent or not, but from the perspective of a given Nij, extrinsic influence – like immigration of individuals – is always stochastic and thus density independent with respect to Nij. In other words, the net immigration rate during a given time increment is not influenced by the state of the population in this location (i,j). You can search my blog or read my book to find descriptions and details on all these concepts.
To implement cross-location and dual-direction deterministic dynamics, multi-scaled behaviour and spatial memory needs to be introduced. My parallel processing conjecture; which spins off various testable hypotheses, creates turmoil in this standard system design for population dynamics because it explicitly introduces such system complexity. For example, positive feedback loops may emerge. Positive feedback as described below may effectively also counteracting the paradoxical Allée effect, which all “standard” population models are confronted with at the border zone of a population in an open environment**.
The dynamic driver of the complexity is the introduction of spatial memory in combination with a scale-free kind of dynamics along both the spatial and the temporal dimensions. In statistical-mechanical terms, parallel processing is incompatible with a mechanistic system. Thus, a kind of extended statistical mechanics is needed. I refer to the post where I describe the scale-extended description of a metapopulation system.
For the most extensive individual-level test of the parallel processing conjecture until now (indirectly also verifying positive feedback of space use), see our paper on statistical analysis of space use by red deer Cervus elaphus (Gautestad et al. 2013; Gautestad and Mysterud 2013). In my blog I have also provided several anecdotal examples of third party research potentially supporting the parallel processing conjecture. For the sake of system coherence, if parallel processing is verified for individual space use of a given species and under given ecological conditions, this behaviour should also be reflected in the complementary population dynamical modelling of the given species and conditions.
Extending the standard population model. As explained in a range of blog posts, my Zoomer model represents a population level system design that is coherent with the individual-level space use process (in parsimonious terms), as formulated by the Multi-scaled random walk model. In my previous post I described the latter in the context of positive feedback from individual-level site fidelity. Below I illustrate positive feedback also at the population level, where site fidelity get boosted by conspecific attraction. In other words, conspecifics become part of the individuals’ resource mapping at coarser scales, as it is allowed for by spatial memory. Consequently, a potential for dual-direction deterministic flow of individuals is introduced (see above). Conspecific attraction is assumed to be gradually developed by individual experience of conspecifics’ whereabouts during exploratory moves.
First, consider the zooming process, whereby a given rate, z, of individuals (for example, z=5% on average at a chosen time resolution Δt) at a “unit” reference scale (k=i) are redistributing themselves over a scale range beyond this unit scale***. During a given Δt consider that 100 individuals become zoomers from the specific location marked by the white circle. In parallel with the zooming out-process the model describes a zooming in-process with a similar strength. The latter redistributes the zoomers in accordance to scale-free immigration of individuals under conspecific attraction.Thus, number of individuals (N) at this location j at scale i, marked as Nij, will at the next time t+1 either embed N-100 individuals if they all leave location j and end up somewhere in the neighbourhood of j, or the new number will be N -100 + an influx of immigrants, where these immigrants come from the neighbourhood at scale i (those returning home again), scale i+1 (immigration from locations nearby), i+2 (from an even more distant neighbourhood), etc.
In the ideal model variant of zooming we are thus assuming a scale-free redistribution of individuals during zooming, with zooming to a neighbourhood at scale ki+x takes place with probability 1/ki+x (Gautestad and Mysterud 2005). Under this condition, zoomers to successively coarser scales become “diluted” over proportionally larger neighbourhood area, the maximum number of immigrants in this example is 100 + N’, where N’ is the average number of zoomers pr. location at unit scale k=i within the coarsest defined system scale k=i(max) for zooming surrounding location j at scale i.
As a consequence of this kind of scale-free emigration of zoomers, the population system demonstrates zooming with equal weight of individual redistribution from scale to scale over the defined scale range (Lévy-like in this respect, with scaling exponent β≈2; see Gautestad and Mysterud 2005). By studying the distribution of step lengths, this “equal weight” hypothesis may be tested, when combinded with othe rstatistical fingerprints (in particular, verifying memory-dependent site fidelity; see Gautestad and Mysterud 2013).
Putting this parsimonious Zoomer model with its system variables and parameters into a specific ecological context implies a huge and basically unexplored potential for ecological inference under condition of scale-free space use in combination with site fidelity.
Positive feedback in the Zoomer model. As shown in my series of simulations of the Zoomer model a few posts ago, a positive feedback loop emerges from locations with relatively high abundance of individuals having a relatively larger chance of received a net influx of zoomers during the next increment, and vice versa for locations with low abundance. The positive feedback emerges from the conspecific attraction process, linking the dynamics at different scales together in a parallel processing manner.
This positive feedback loop from conspecific attraction also counteracts extinction from a potential Allée effect (see this post and this post), which have traditionally been understood and formulated from the standard population paradigm. The Zoomer model represents an alternative description of a process that effectively counteracts this effect.
*) The migration rates connects the local population to surrounding populations. Immigration is – by necessity from the standard model design – density independent with respect to the dynamics in Nij.
**) Since the process is assumed to obey a Markovian and the mean field principles (standard, mechanistic process), the arena and population system must either be assumed to be infinitely large or the total set of local populations has to be assumed to be demarcated by some kind of physical border. Otherwise, net emigration and increased death rate in the border zone will tend to drive N towards zero in open environments (extinction from standard diffusion in combination with local N drifting below critical density where Allée kicks in). Individuals will “leak” from an open border zone to the surroundings where N is lower.
***) The unit temporal scale for a population system should be considered coarser than the unit scale at the individual level, since the actual scale range under scrutiny typically is larger for population systems. In particular, to find the temporal scale where for example 5% of the local population can be expected to be moving past the inter-cell borders of a given unit spatial grid resolution ki=1, one should be expected to find Δt substantially larger than Δt at the individual level.
Consider that the difference in Δt is a function of the difference of the area of short-range versus long range displacements under the step length curve for individual displacements, where the ∼5% long-step tail of this area represents the relative unit time in comparison to the rest of the distribution (thereby defined as intra-cell moves). Since this area is a fraction of the area for the remaining 95% of the displacements, the difference in Δt should scale accordingly.
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.
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).
* 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.
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 log2(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 log2[(N,I)] plot is rescaled to log2[(N,I)] + log2[ΔD(grain/extent)] = log2[(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/(212) = 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 Dtrue=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 21.5 = 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 log2(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).
*) 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.
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:
*) 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).
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/fα (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/f0), while the extremely clumped dispersal shows more like red noise (1/f1.8).**
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 kb larger grid area, with b=1). However, while zoomers re-distribute themselves contagiously in a spatially explicit manner from a process that depends on the individuals’ spatial memory of past experiences, the swarms of emigrants in the current Zoomer design settle randomly. “Stressed” individuals (emigrants from local crunch events) are assumed to redistribute themselves in a tactical manner, temporarily following the collective behaviour of a swarm during the actual time increment.
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.
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.
*) 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.