# Simulating Populations IIX: Time Series and Pink Noise

On rare occasions one’s research effort may lead to a Eureka! moment. While exploring and experimenting with the latest refinement of my Zoomer model for population dynamics I stumbled over a key condition to tune the model in and out of full coherence between spatial and temporal variability from the perspective of a self-similar statistical pattern. Fractal-like variability over space and time has been repeatedly reported in the ecological literature, but rarely has such pattern been studied simultaneously as two aspects of the same population. My hope is that the Zoomer model also has a more broadened potential by casting stronger light on the strange 1/f noise phenomenon (also called pink noise, or Flicker noise), which still tends to create paradoxes between empirical patterns and theoretical models in many fields of science.

First, consider again some basic statistical aspects of a Zoomer-based simulation. Above I show the spatial dispersion of individuals within the given arena. See previous posts I-VII in this series for other examples, system conditions and technical details. The new aspect in the present post is a toggle towards temporal variation of the spatial dispersion, and fractal properties in that respect (called a self-affine pattern in a series of measurements, consistent with 1/f noise*). This phenomenon usually indicates the presence of a broad distribution of time scales in the system.

In particular, observe the rightmost log(M,V) regression. As in previous examples, the variable M in the Taylor’s power law model V = aMb represents – in my application of the law – a combination of the average number of individuals in a set of samples at different locations at a given scale and supplemented by samples of M at different scales. The latter normally contributes the most to the linearity of the log(M,V) plot, due to a better spread-out of the range of M. During the present scenario that was run for 5000 time steps after skipping an initial transient period, the self-similar and self-organized spatial pattern of slope b ≈ 2 and log(a) ≈ 0 was dominating, only occasionally interrupted by some short intervals with << 2 and log(a) >> 0. These episodes were caused by simultaneous disruption of a relatively high number of local populations in cells that exceeded their respective carrying capacity level (CC), leading to a lot of simultaneous re-positioning of these individuals to other parts of the arena in accordance to the actual simulation rules. Thus, some temporary “disruptive noise” could appear now and then, occasionally leading to a relatively scrambled population dispersion with b ≈ 1 (Poisson distribution) and log(a) >> 0 for short periods of time.

The actual time series for the present example is given above, showing local density at a given location (a cell at unit scale) over a period of 5,000 increments. The scramble events due to many simultaneous population crashes in other cells over the arena can indirectly be seen as the strongest density peaks in the series. The sudden inflow of immigrants to the given locality caused a disruption to the monitored local population, which typically crashed during the next increment due to density > CC.

In may respects the series illustrates the “frustrating” phenomenon seen in long time series in ecology: the longer the observation period, the larger the accumulated magnitude of fluctuations (Pimm and Redfearn 1988, Pimm 1991). However, in the present scenario where conditions are known, there is no uncertainty related to whether the complex fluctuations are caused by extrinsic (environmental) or intrinsic dynamics. Since CC was set to be uniform over space and time and only varied due to some level of stochastic noise (a constant + a smaller random number) the time series above is expressing intrinsically driven, self-organized kind of fluctuations over space, time and scale.

The key lesson from the present analysis is to understand local variability as a function not only of local conditions, but also a function of coarser-scale events and conditions in the surroundings of the actual population being monitored. The latter is often referred to a long-range dependence. In the Zoomer model this phenomenon is rooted in the emergence of scale-free space use of the population, due to scale-free space use at the individual level (the Multi-scaled random walk model). In the present post I illustrate long-range dependence also over temporal variability.

To remove the major magnitude of serial autocorrelation the original series above was sampled 1:80. The correlogram to the right shows this frequency to be a feasible compromise between full non-autocorrelation and a reasonably long remaining series length.

After this transformation towards observing the series at a different scale, the sampled series was subject to log(Mt,V) analysis, where Mt represents – in a sliding window manner – the mean abundance over a few time steps of the temporally coarser-scaled series, and V represents these  intervals’ respective variance.

Interestingly, the log(Mt,V) pattern was reasonably similar to the log(M,V) pattern from the spatial transect above; i.e., the marked line with b ≈ 2 and log(a) ≈ 0, under the condition that the series was analyzed at a sufficiently coarse temporal scale from sampling to avoid most of the serial autocorrelation.

The most intriguing result is perhaps the Figure below, showing the log-log transformed power spectrogram of the original time series**. Over a temporal scale range up to frequency of 1500 (the x-axis) the distribution of power (the y-axis); i.e., amplitudes squared, satisfies 1/f noise!

See my book for an introduction to this statistical “beast” ***, where I devote several chapters to it. At higher frequencies further down towards the “rock bottom” of 4096 (the dimmed area), the power shows inflation. This is probably due to an integer effect at the finest resolutions of the time series, since individuals always come as whole numbers rather than fractions. Thus, the finest-resolved peak in power was probably influenced by this effect.

To my knowledge, no other model framework has simultaneously been able to reproduce the empirically often reported fractal-like spatial population abundance (the spatial expression of Taylor’s power law) with a fractal-like temporal variation (the temporal Taylor’s power law). Only either-or results have been previously reproduced by models among the thousands of papers on this topic over the last 60 years or so.

For an introduction to 1/f noise in an ecological context, you may find this link to a review by Halley and Inchausti (2004) helpfull. I cite:

1/f-noises share with ecological time series a number of important properties: features on many scales (fractality), variance growth and long-term memory. (…) A key feature of 1/f-noises is their long memory. (…)  Given the undoubted ubiquity and importance of 1/f-noise, it is surprising that so much work still revolves about either observing the 1/f-noise in (yet more) novel situations, or chasing what seems to have become something of a “holy grail” in physics: a universal mechanism for 1/f-noise. Meanwhile, important statistical questions remain poorly understood.

What about ecological methods based on the theory above? By exploring the statistical pattern of animal dispersion over space and its variability over time – and mimicking this pattern in the Zoomer model – a wide range of ecological aspects may be studied within a hopefully more realistic framework than the presently dominating approach using coupled map lattice models or differential equations. Statistical signatures of quite novel kind can be extracted from real data and interpreted in the light of complex space use theory.

In upcoming posts I will on one hand dig into the set of Zoomer model “knobs” that steer the statistical pattern in and out of – for example – the 1/f noise condition, and on the other hand I will exemplify the potential for ecological applications of the model.

REFERENCES

Halley, J. M. and P. Inchausti. 2004. The increasing importance of 1/f noises as models of ecological variability. Fluctuation and Noise Letters 4:R1–R26.

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

Pimm, S. L., and A. Redfearn. 1988. The variability of animal populations. Nature 334:613-614.

NOTES

*) “1/f-noise” refers to 1/fν-noise for which 0 ≤ ν ≤ 2; “near-pink 1/f-noise” refers to cases where 0.5 ≤ ν ≤ 1.5 and “pink noise” refers to the specific case were ν = 1. All 1/fν-noises are defined by the shape of their power spectrum S(ω):

S ∝ 1/ων

Here ω = 2πf is the angular frequency. “White noise” is characterized by ν ≈ 0, and its integral, “red” or “brown” noise, has ν ≈ 2. A classical random walk along a time axis is an example of the latter, and its derivative produces white noise.

**) Since a FFT analysis requires a series length satisfying a power or 2, only the first 4096 steps were included.

***) 1/f noise is challenging, both mathematically and conceptually:

The non-integrability in the cases ν ≥ 1 is associated with infinite power in low frequency events; this is called the infrared catastrophe. Conversely for ν ≤ 1, which contains infinite power at high frequencies, it is called the ultraviolet catastrophe. Pink noise (ν = 1) is non-integrable at both ends of the spectrum. The upper and lower frequencies of observation are limited by the length of the time series and the resolution of the measurement, respectively.
Halley and Inchausti (2004), p R7.

# Simulating Populations VII: the Correlogram View of Complex Dispersion

In my continued quest for a more realistic statistical-mechanical theory for spatially extended population dynamics I have previously pointed out a specific property of the inter-scale spatial coefficient of variation as one of the hallmarks of scale-free dispersion (see Part III). In the present post I study another statistical property, the spatial autocorrelation, which may provide additional cue about the population’s compliance with standard or complex space use.

First, consider the standard theory, based on mean field compliant population redistribution (mixing). The following three images show a typical example, where the population is subject to 5% diffusion rate at unit (pixel) scale, net population growth of 1%, no Allée effect, and over-all population density below carrying capacity. As previously described in this series, diffusion tends to smoothen the density surface. The log(M,V) plot over a scale range typically shows y-intercept [log(a)] substantially below zero  a hallmark of fine-scale smoothness – under condition where the slope β ≈ 2. I refer to previous posts in this series for technical details.

The correlogram (above) reflects the undulating density surface.

Next, consider the following scenario under the same condition, except for 5% standard diffusion at unit scale (k=1) being replaced by 5% scale-free “zooming” with 1% pr. scale level.

The present  snapshot of population dispersion represents the population a few time increments after a general population crash (bottleneck episode), when the population is in the process of re-organizing itself. Despite the short time span since the episode (5-6 time increments) the log(M,V) plot has already adjusted itself from log(a) >>0 and b ≈ 1 (random mixing due to external forcing) to log(a)≈ 0 and b ≈ 2.

Observe that in this additional example of complex population dynamics the log(M,V) plot again satisfies log(a) ≈ 0 when b ≈ 2. Here the local effect at next time increment from overshooting local carrying capacity both locally and elsewhere is influenced by a rate of 50% remaining population at the actual location (as in previous examples) to zero individuals remaining. In other words, this hallmark of Zoomer-like dynamics is quite resilient to this modification of ecological conditions.

The interesting aspect in the present context is its spatial transect correlogram at unit scale k=1. It shows low level of autocorrelation at all spatial lags except for scale 0, which trivially illustrates that the local population correlates with itself 100% at lag zero. Despite the non-significant autocorrelation the parameter condition for log(M,V), log(a) = 0 and b = 2.

Thus, scale-free population abundance regards both spatially autocorrelated transects (as shown in previous parts of this series) and – as shown here – non-autocorrelated transects a short period following a perturbation [during the first 3-5 increments after the event*, log(a) >> 0 and b ≈ 1].

In this respect I refer to the so-called Z-paradox, which is resolved under the Zoomer model but not under the standard framework of population modelling. Thus, my proposed model may also provide a novel approach towards the famous and controversial Taylor’s Power law.

NOTE

*) Under the condition that the bottleneck condition lasts for one time increment. If the change of condition is permanent, it may typically take 20-50 time steps to restore log(a) ≈ 0 and b ≈ 2.

# Simulating Populations VI: the Unrealism of Standard Models

In the previous post (Part V) I suggested that the standard theoretical framework for simulating population dynamics/kinetics is fundamentally unrealistic, since in any open environment it dooms local population abundance to approaching zero and extinction due to memory-less dispersal and further lubricated by Allée effects. The only apparent rescue conditions for a population are (a) the population lives in a closed rather than open environment, which is an unrealistic scenario in general terms; or (b) on average the net growth rate is larger than the dispersal (diffusion) rate. In this post I argue that even the latter assumption is flawed.

The traditional Coupled map lattice (CML) modelling, partial differential modelling, and other standard spin-offs from classical statistical mechanics may apparently be “rescued” with respect to model realism by defining unit spatial scale coarse enough to ensure that the net growth rate is stronger than diffusion rate at this level. Since diffusion is a scale-specific process – see below – and growth (specific birth and death rates) are not linked to a specific scale, the diffusion rate will apparently be reduced when observed from coarser spatial resolutions. For example, a ten times larger spatial resolution (unit pixel size) for a given population study on average embeds ten times more individuals. Hence, the percent of the population that disperses to the coarser-scaled neighbourhood becomes smaller from this perspective due to smaller perimeter/area, and may even be smaller than the population’s net growth rate. Paradox solved? Unfortunately not, because the reasoning is flawed.

Together we thrive. Photo: AOG.

First, consider the condition where the individuals are living in an open environment, which is the general condition. No species is abundant everywhere, meaning that a given population is surrounded by unoccupied space*).

The correct scale to study the diffusion rate is not an arbitrary scale (pixel size), defined as unit scale, but the actual population’s characteristic diffusion scale under the given environmental conditions. In other words, diffusion is an intrinsic property of the population, and not something that can be defined by the observer at will.

To understand this concept of characteristic diffusion rate we need to turn to statistical mechanics. The characteristic diffusion scale is given by the diffusion coefficient, which is proportional with the square of average step length by the average individual divided by the time interval for non-correlated successive directional change for the individuals’ random walk process (Brownian motion-like when we consider the statistical property of a memory-less kind of movement, which is a premise for the standard modelling framework).

The diffusion coefficient-determined rate; i.e., the square of the net length moved pr. unit time, as determined by the individual random walk properties**), is the parameter to be compared to the net growth rate of the population. Even if we consider that a diffusion in general terms is a slow process, the diffusion’s characteristic rate at its unit time scale becomes substantially larger when we re-scale the time axis to the characteristic reproduction interval for a new individual to be born (generation time divided by number of individuals in the brood, and adjusting for death rate effects). Under this biologically adjusted diffusion rate by temporal rescaling to the system’s population dynamical scale, it may easily become very much larger than the net reproduction rate. In short, such temporal rescaling ensures full “local mixing” of individuals. However, in an open environment such high rate of individual reshuffling will doom the population to extinction.

In an open environment the population is typically surrounded by suitable habitat, but for the moment it is not utilized by the species. This is in accordance to common empirical experience, not from reading Ecology text books… Thus, the population’s fringe zone “leaks” individuals to the surrounding area, due to non-zero outward dispersal in net terms. Outward-directed flow of individuals is larger than inward-directed flow, since the latter has no or negligible population source. Based on the argument above that the dispersal rate generally is substantially larger than the population’s net growth rate when diffusion is re-scaled to the characteristic time scale of population growth (see above), the population is doomed under standard model premises. Another way to look at it is to consider diffusion  rate at its correct scale given by its diffusion coefficient, and only consider the (fuzzy) zone along the population perimeter where there is a net flow of individuals. As illustrated in my previous post (Part V), unless you put up a fence the local abundance drifts towards zero!

Consider dropping a single drop of ink into a water-filled bathtub – representing our “open environment”. The black dot will over time spread out and become uniformly dispersed. Consider that the ink molecules have a limited life span, and (hypothetically!) two molecules need to meet within their respective life span to produce additional ink molecules. Since we are considering a bathtub; a constrained space, the ink “population” may still survive. However, the larger the bathtub the smaller the rate of inter-molecular encounters. The critical space size is given by the level (bathtub size) where the given diffusion rate – after temporal rescaling to match the net reproductive rate (see above) – becomes larger than the reproduction rate pr. individual of new ink molecules! This balance between diffusion rate and reproduction rate is diminished by increasing bathtub size, invoking an Allée effect in larger bathtubs!

What about a metapopulation system, where migration between sub-populations (a representative for diffusion at the scale of sub-populations) is relatively small? Such a “semi-permeable” kind of population sectioning puts constraint on the much higher diffusion rate inside sub-populations (local high-speed mixing as seen from the perspective of the net reproductive rate). In fact, in standard models for spatially extended populations it is assumed that the characteristic scale for diffusion ensures a high rate of population mixing at relatively fine spatial scales, relative to the extent of the population under study. For example, it ensures a “smooth” and thus differentiable density surface of the population’s spatial dispersion; a premise for realistic application of partial differential equations. However, even if a metapopulation system with mixing constraint on intermediate spatial scales may avoid the paradox of population extinction as outlined above, such a metapopulation system is doomed since it exists in an open environment at its fringes! Huffaker’s 1958 experiments on sub-populations of mite were run in a constrained environment, the size of the terrarium!

To conclude, in my view ecological theory for population dynamics needs a more realistic kind of modelling platform. At the individual level spatial memory and scale-free space use is now being empirically verified with a high pace. This insight needs to be reflected in theory of population level space use. As stated in my note below I propose that intraspecific cohesion, for example as implemented by the individual-level conspecific attraction property of the Zoomer model, may resolve the “doomed to extinction” paradox of standard population models for populations in an open environment. In this framework, the standard diffusion is replaced by the scale-free redistribution process termed zooming. However, this process is not compatible with the standard statistical-mechanical theory for population modelling.

NOTE

*) In some cases this fringe zone may be easily understood from the perspective of unfriendly neighbourhood in habitat terms, but often the surroundings appear quite similar to the conditions inside the distribution range. For example, the over-all population may be spatially fragmented with respect to abundance; particularly along the core areas’ perimeters, with no apparent reason why small and large chunks of the intermediate areas should not be habitable. Pick any well-studied species, and ask an expert on its distributional range to explain population absence in some regions with apparently similar ecological conditions to the population’s present range. In an in my view unsatisfactory manner the way out of this dilemma (the “shoot from the hip” answer) is normally to point to some environmental factor still not revealed, or some kind of statistical chance effect. In my Zoomer model such apparently inexplicable “clumping” of a population is explained as an emergent property of conspecific attraction and scale free redistribution of some individuals. A given number of individuals cannot be everywhere all the time. However, in the present post I’m concerned about the basic premises of the standard framework for population dynamical modelling, not how the empirical paradox may be resolved.

**) As repeatedly underscored in my book and in this blog, individuals generally do not move in a stochastic manner. However, the animal’s path may be represented by a statistical function in over-all terms. In the standard framework this function is assumed to satisfy the parameterized random walk model of the Brownian motion type.

# Simulating Populations V: Bottlenecks and Recovery

Time to simulate a stress-test of the two population-kinetic frameworks, the Coupled map lattice model and the Zoomer model! Consider a scenario where some kind of environmental event has crushed the population to about 1% of its normal carrying capacity. In addition, the remaining population has also become spatially fragmented during this catastrophe. Then consider that the condition improves to the pre-event level. What is the population’s potential to recover under the two scenaria you have become familiar with in Parts I-IV, scale-specific and scale-free kinetics?

The map to the right shows the small population’s spatial dispersion at the start of the potential recovery phase. Isopleths indicate local population density, which shows an average of 165 individuals pr. occupied cell at unit scale while the carrying capacity (CC) has been restored to a potential for 5,000 individuals at this scale. In other words, most local populations have gone extinct as a consequence of the recent crunch event.

Then the recovery phase begins to run. Starting with the standard condition of scale-specific population dynamics/kinetics (Coupled map lattice model) and setting diffusion rate at unit scale to 5% and net population growth of 2%, the following image shows the population dispersion after 20 iterations.

Since the diffusion rate is larger than the local growth rate (the general condition of spatially unconstrained animal populations) and the population is now surrounded by unoccupied area, the population is drifting towards extinction!

This faith is also facilitated by an additional model condition, Allée effects. At this low level of population abundance it is important to consider and implement three aspects: First, accelerated extinction at very low abundance levels have to be introduced. Here I set the critical level to 50 individuals pr. unit cell*). Below this level, the population is reduced by 10% pr. time increment. Second, due to the low abundance levels, one has to consider that individuals exist as discrete entities, not fractions of numbers (at high abundance the difference between discrete and continuous numbers are insignificant). Third, at very low population densities random events take its toll. I implement this as some noise level on the survival rate in the Allée zone; i.e., in cells with less than 50 individuals.

In contrast, does scale-free and memory-influenced zooming influence the population’s otherwise dire faith after the catastrophic event? Obviously it does. The Zoomer snapshots below at t=20, t= 200, t= 500 and t= 1,000 shows a population in healthy recovery, despite  being surrounded by a wide zone of unoccupied space. The 5% diffusion rate under the CML condition above is replaced by a 5% zooming rate (“exploratory steps”, as seen from the individual perspective), with 1% conspecific attraction kind of redistribution pr. scale level (see previous parts of this series). In other respects the conditions are similar to the CML model, including net growth being smaller than individual reshuffling at unit scale.

The log(M,V) result at t = 1,000 (as in the earlier parts of recovery; not shown) shows full compliance with intercept ≈ 0 and slope ≈ 2, as predicted by the Zoomer model.

In this manner the Zoomer model illustrates – and potentially resolves – some crucial but under-communicated issues with respect to the standard modelling framework.

Inclusion of spatial memory and strategic space use – in particular the capacity for individuals to include conspecifics as part of their resource map at strategic scales (conspecific attraction) – counteracts the otherwise detrimental effect of living in an open world.  At the fringe of any animal population, under the standard modelling paradigm local abundance is constantly threatened by individuals getting lost in space, by drifting away from sufficiently strong contact with conspecifics (ref: diffusion and Allée effects). The Zoomer design, by implementing spatio-temporal memory, formulates a solution to this core problem for population dynamical modelling. However, the solution requires an extended kind of statistical mechanics. Read my book – for the time being the main source (and for some parts the only source) for a theoretical overview of this approach!

In the next post I will address an expected primary objection to my quite far-fetching conclusions above, that traditional population dynamical modelling is based on shaky assumptions with respect to realism. In future posts I will also present empirical support for the Zoomer model.

NOTE

*) There is nothing magic about N=50, but to avoid a more complicated formula for the Allée effect – with little or no advantage with respect to model realism in over-all terms – I have just chosen a “small abundance number” relative to the carrying capacity.

# Simulating Populations IV: Environmental Heterogeneity

In the foregoing Parts I-III model complexity was increased in a stepwise manner for the sake of exploring intrinsic population behaviour one factor by the time. In this post I take one additional step by studying the overall effect of environmental heterogeneity.

For the time being I consider spatial heterogeneity only, leaving temporal fluctuations to a later post. Anyway, a new level of realism is hereby added relative to the scenaria in the previous posts: external influence is now adding to intrinsic processes with respect to variations in local population abundance.

Consider the Zoomer snapshot to the right (zero diffusion, 5% zooming over a scale range, as in previous examples), after the population has progressed 100 time steps in an environment where the local carrying capacity varied over space (CC=2942 individuals pr. cell at unit scale within the arena, on average). Due to 50% overshoot survival (see
Part I for a definition of CC), the population fluctuates between CC and CC/2.

The net growth rate is small at the defined time resolution (1%), and a given bust event at local density is passing local CC brings the local population down to CC/2 in a single time step. While it takes only one iteration to bring the local population down to CC/2, it takes many increments to bring it up towards CC again.

The standard, non-scaling Coupled map lattice condition given the same heterogeneous map for local CC variability is shown by the two images below.

The important pattern in the two sets of Figures above and below is the apparent similarity of the log(M,V) plot in the present condition of habitat heterogeneity and the condition of habitat homogeneity that was presented in Parts I-III. This similarity makes sense, since local variability as a consequence of local habitat heterogeneity needs to be analyzed at  a finer scale than the entire arena size and then compared between sections. Under all scenaria so far, the log(M,V) plots regard population abundance within the arena as a whole.

Thus, in both scenaria above the local variability is hidden; i.e., “averaged out”.

Anyway, there is a crucial difference to observe between scale-specific (CML compliant) and scale-free statistics. The intercept log(a) << 0 while the slope b ≈ 2 under the standard CML condition, and log(a) ≈ 0 when b ≈ 2 under the Zoomer condition. Thus, we can conclude that the novel indicator of complex population dynamics – self-similar population dispersion due to CV≈1 (see Part III) apparently stands the heterogeneity test!

Such resilience to environmental conditions when is comes to distinguishing standard from complex space use is of course crucial for the realism of this system property when we later on are confronting the theory with real data.

After these step-wise system introductions for the sake of revealing the respective systems’ intrinsic population kinetics, time has come to throw additional realism into the model conditions. As a starter with respect to cruising towards ecological aspects I study the populations’ response to bottleneck events (population crunches) and their ability to recover under standard and alternative statistical-mechanical premises! Look forward to Part V.