Simulating Populations III: a New Statistical Indicator of Complex Population Kinetics

In the previous parts I-II of this series I described two variants of spatially extended population dynamics, represented by a standard Coupled map lattice (CML) model and the Zoomer model. In this post I show how a specific statistical-mechanical property of scale-free space use may reveal the difference between these two space use conditions despite an apparent similar level of spatial autocorrelation below the population’s carrying capacity.

First, a brief summary of the model conditions (for details, see Part I-II):

  • The environment is set to be homogeneous (soon to be relaxed), in order to have focus on intrinsic population kinetics.
  • The time resolution is set to be fine-grained, implying that the main driving force for change during respective time increments is individual re-shuffling rather than birth and death rates (net growth rate set to ca 1%).
  • For the CML examples (implying a scale-specific process in statistical-mechanical terms), the mean field assumption applies. Reshuffling happens by a small diffusion rate 0-1%, simplified as standard random walk at the individual level, taking place between neighbour cells (1%) at unit scale or totally hidden at finer (unobserved) scales below unit scale (0%).
  • For the Zoomer examples (implying a scale-free process), reshuffling happens by conspecific attraction over the scale range from unit scale to the extent of the simulation arena. When the environment is set to be homogeneous, conspecific attraction rate of 1% pr. scale level and 5% in total is the only remaining driving force (beyond dispersal due to local overshoot of carrying capacity) for population redistribution beyond unit scale. However, diffusion (rate 0-1%) may be present at unit scale to account for presence or absence of fine-grained stochastic noise (short term tactical response from individuals to local events, independent of the strategic zooming behaviour).

Why not including conspecific attraction under the standard CML scenario? Because one of the assumptions for mean field population kinetics, represented by a CML model, is absence of spatial memory effects on redistribution beyond the unit scale. Hence, heighbourhood re-shuffling of individuals can be represented by a standard diffusion compliant process only (assuming that advection – another scale-specific process – is also limited to finer scales than the unit grid). Like it or not! I’ll return to this topic in a later post.

While non-autocorrelated population dispersion (exponential distribution of local population density, V ∝ M)  is the typical pattern of a standard population scenario if a substantial part of the population is fluctuating close to the carrying capacity (see Part I, example 1), a similar result is difficult to achieve under scale-free space use and conspecific attraction.

Spatial autocorrelation (V ∝ M2) is swiftly restored – typically within a few time increments – following instances where a larger than normal part of the local populations are crashing simultaneously. The example above shows a snapshot close to the carrying capacity*) under condition of local overshoot crash rate of 50% (diffusion rate = 0%). A similar result appears even under a more dramatic local crash rate of 90% (not shown), which may be compared to the CML Example 1 in Part I.

Obviously, since standard and scale-free statistical mechanics for spatially extended populations are so fundamentally different i qualitative terms (for example, autocorrelation appears from different processes; growth/diffusion and intrinsically driven conspecific attraction, respectively), it should be a key goal to pinpoint statistical methods to distinguish between these two conditions for population dynamics. Here I show for the first time how spatial autocorrelation of the population’s density surface, here to be studied from the log(M,V) pattern, may provide a simple method in this respect.

Recall from Part I-II that M regards the average number of individuals pr. grid cell at a given scale, and M is changing proportionally with grid resolution (times larger linear scale gives k2 larger grid cells, which give k2 times larger M). In absence of spatial auto-correlation, the variance of M at respective scales changes proportionally with M.

To pinpoint one of the hallmarks of scale-free population dynamics, consider these log(M,V) snapshots, where I have interpolated the regression lines towards the y-intercept [giving the parameter log(a) in log(V) = log(a) + b*logM].

First, for the sake of comparison, two examples from scale-specific dynamics:


Then consider the scale-free kind of population kinetics:


In the two first of these zoomer examples, diffusion rate = 0%, while in the third example diffusion is set to 1%. All other conditions are similar to the scale-specific scenaria above.

The key difference regards the y-intercept, log(a), which – under a premise of little influence from local carrying capacity at the time of the actual snapshot – is strongly negative for a scale-specific scenario and close to zero for the zoomer conditions. The magnitude of the negative intercept from scale-specific dynamics depends on the average M; more abundant populations (from the perspective of the chosen scale range) will show a “parallel shift” towards the right-hand side in the log(M,V) plot, and thus show an even more negative log(a).

To understand this difference between scale-specific and scale-free population kinetics, observe that the power exponent b ≈ 2 implies that √V; i.e., the standard deviation (SD), is proportional with M. Further, if log(a) = 0 (≈ 1) in addition to b ≈ 2 the coefficient of variation (CV) is approximately constant over the range of M (recall that number of individuals, M,  is proportional with grid cell size, which is proportional with square root of scale, k);

CV = SD/M ≈ 1     | M ∝ √k

Only the scale-free zoomer model condition produces results under condition of spatial autocorrelation in compliance with a constant coefficient of variation over the given scale range. The population is spatially self-similar; i.e., it satisfies a statistical fractal. In contrast, the classic CML model produces a “smooth” density surface (0<CV<<1) at fine resolutions**).

Since the y-intercept under the Zoomer model scenaria approximates log(a) = 0, the CV≈1 condition is not sensitive to population abundance. In my book I describe other methods to study a population’s scaling properties under a premise of Multi-scaled random walk at the individual level, for example in the context of Taylor’s power law. I also provide empirical support for scale-free population dispersion.

In future blog posts I will bring the analysis further, by studying the influence of spatial heterogeneity and respective models’ responses to perturbation events.

I kindly ask you to give credit to my blog and my book when you refer to the statistical-mechanical theory for scale-free population dynamics! To my knowledge, for the moment these are the only sources for this departure from standard population dynamical modelling.


*) As mentioned in Part I (see its Note), the carrying capacity is given a special interpretation as the threshold whereby the local individuals (within the finest defined grid scale) are moving out, redistributing themselves or dying.

**) Recall from previous posts (and my book) that the standard framework to model spatially extended population dynamics – Coupled map lattice models and partial differential equations – in fact depend on such a smooth density surface at fine spatial resolutions; i.e., in the (hypothetical) theoretical limit of infinite local population density. Otherwise the population is not differentiable! If this premise fails, the standard approaches are doomed to fail with respect to the model’s realism and predictive power.

Simulating Populations II: Adding Spatial Memory and Scaling

In Part I of this series I presented a couple of spatially extended scenaria of intrinsically driven population dynamics under the standard statistical-mechanical premises (intrinsically scale-specific), using a parsimonious Coupled map lattice (CML) model for the simulations. In this Part II the framework will be extended with a scaling axis, orthogonal on space and time, to account for populations of individuals with space use satisfying the Multi-scaled random walk (MRW) properties. Using this scale-extended kind of CML design – the Zoomer model – I show how scale-free space use tend to generate spatial autocorrelation at the population level from the process of conspecific attraction.

“Intrinsically driven dynamics” implies that the model is simulated under the condition of a homogeneous environment. As soon as the behaviour (dynamics) under this condition is understood, environmental heterogeneity can be introduced for easier interpretation and improved realism. MRW-based behaviour at the individual level implies spatial memory and scaling under a premise of the parallel processing conjecture. As in Part I, the starting point for the present post is a citation from a previous post:

The Zoomer model  includes all the four standard BIDE rates (Birth, Immigration, Death and Emigration), and it is also spatially explicit. However, contrary to standard coupled map lattice models, spatial scale (the “lattice”) is implemented in a multi-scaled manner. This “scale range” approach allows for formulation of various aspects of complex population kinetics.

For example, the Zoomer model allows for explicit modelling of intraspecific cohesion (conspecific attraction), which is a complex process that depends on both temporal and spatial memory utilization at the individual level. In a simplified scenario, consider that a tendency for conspecific attraction is the main driver of the population kinetics. Further, consider that we study the process at sufficiently fine temporal and spatial scales to allow us to disregard the “slower” rate BIDE terms (it will be trivial to account for these processes as well).

I refer to the link above, and my book, for mathematical details of the model formulation. Below I take previous presentations a step further by extending the Zoomer model with fine-scale diffusion and a weak level of birth and death rates. Diffusion represents local randomization of individuals, based on memory-less, Markovian compliant moves. In the present simulations I assume that this kind of standard-model redistribution of individuals for the most part is contained at finer spatial scales than the finest grid cells in the simulation arena. Below I explore two scenaria; 0% and 1% nearest neighbour diffusion rate, respectively, at the unit scale (smallest grid cells) during a time increment. I also keep the weak growth rate of 1% (survival rate 99% and reproduction rate 2%), as in the Part I simulations under the standard CML model.

A sardinian warbler Sylvia melanocephala (Southern Spain) has tactical focus on chasing insects, watching out for raptors and relating to other inputs within its current field of perception. Simultaneously, the present model assumes that the warbler is also “background processing” past experiences in a manner which makes it able to relate to its environment over a wide range of spatial and temporal scales (spatial map utilization). One such environmental cue could be conspecific whereabouts. Photo: AOG.

Zero diffusion implies that any random walk-like movement (in a statistical-mechanical sense) of individuals takes place inside the unit grid cells only, on a 32×32 cell arena. One percent diffusion to nearest-neighbour cells makes this process substantially weaker than the zooming process from complex dynamics, where 5% of individuals at each time increment are “reshuffled” with equal weight (distributed equally) at respective scales 21, 22, 23, 24 and 25 relative to the smallest grid cells; i.e., 1% pr. scale level beyond the unit scale 20=1. This difference in model design makes sense, given the recent empirical confirmation of complex animal space use at the individual level, over a wide range of taxa and ecological conditions.

While the “zoomers” during a given time increment are distributed equally over the actual scale range, they are “zooming in” from scale level k+1 to to their neighbourhood at respective scales k-1 where the local population density (from that scale’s perspective) is highest at that moment. Hence, it is assumed that the individuals have the cognitive capacity to interpret respective scales’ general level of conspecific density with the help of historic experience from the environment (as supported by empirical tests and anecdotal support of the MRW model). In other words, the memory map is providing the raw data for these individual considerations what neighbourhood to zoom in to.

An important aspect of the zooming process is that individuals moving (“zooming out” from a locally high density region will have a tendency (statistically) to “zoom in” towards the same region, while individuals starting from a relatively low density region will tend to zoom into the neighbourhood with higher density.

In summary, standard diffusion-compliant reshuffling of individuals (tactical, memory-independent behaviour) is constrained to very fine spatial resolutions, while multi-scaled reshuffling is distributed over a scale range covered by the model’s grain to extent range.

The intriguing property of this construction is model coherence between scale-free distribution of displacement lengths at the individual level (a power law, with tail exponent β ≈ -2; i.e., Lévy walk-like if we ignore the spatial memory aspect) and scale-free inter-scale redistribution of the density surface at the population level. In my book and in several blog posts I have provided many examples supporting the present model assumption that the tail of individual movement lengths can span a decently large population range (for example, recall this post for the lesser kestrel example, and this post for the Florida snail kite).

Since the zooming rate between scale k and scale k+1 is 1% for all zoomer scales 1, 2, …, k, …, 5 included in the present model, this implies that individuals during an increment on average has a 1% chance of moving a distance of magnitude (k+1) relative to distance (k); which contains 22 = 4 times as large cells. However, this lower probability for long-distance moves is compensated for by (21)2 = 4 times as many individuals embedded in a k+1 grid cell than a cell at scale k (Gautestad and Mysterud, 2005). Hence, a constant cross-scale zoomer rate at the population level reflects power exponent β ≈ -2 in the step length distribution at the individual level, under the assumption that individuals on average are putting “equal weight” into space use at respective scales.

Further, in the present illustrative simulations with weight on intrinsic population dynamics the main driver for spatial redistribution from zoooming is a tendency for conspecific attraction. Avoiding getting lost in space relative to the local population is obviously an important driver for animal space use. Zoomers to a specific level k+1 during Δt are assumed to prefer to move to the highest local population density in the individual’s neighbourhood at scale k. Again, I have to refer to my papers, my book and other blog posts for details.

Finally, let’s turn to the Zoomer results. Activating zooming to all levels beyond the unit scale, and setting diffusion rate to 0%, leads to the following snapshots after a couple hundred iterations:


Setting diffusion rate to 1% and confined to scale k=1, leads to a somewhat smoother density surface:


As was the case for a non-zooming condition (Part I), also the Zoomer model shows a steep log(M,V) plot, which reflects a high degree of spatial autocorrelation in the density surface of the population. Small-scale diffusion tend to smoothen the density surface and reduce the y-intercept [log(a) becomes somewhat smaller]. However, as long as the population is by the most part undisturbed by perturbations from the local carrying capacity ceiling the power exponent remains close to b=2, as for the non-complex CML scenaria in Part I.

However, there is a specific difference between auto-correlation from a scale-specific (CML model) scenario and a scale-free (Zoomer model) scenario! In the next post, Part III, this intrinsic difference in population kinetics will be explored.

The main take-home message from this Part II is the inclusion of conspecific attraction at “strategic” scales; i.e., at spatial resolutions coarser than the individuals’ field of perception. Such a behaviour is well documented, but it has been problematic to implement in traditional model designs due to (a) the classic statistical mechanical framework is void of spatial memory effects, which strategic conspecific attraction depends on; and (b) conspecific attraction should be implemented in a scale-free manner to make the model coherent with complex space use at the individual level.


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

Simulating Populations I: the Bridge Towards Standard CML

My book’s title reads: “Animal Space use: Memory Effects, Scaling Complexity, and Biophysical Model Coherence“. The latter part refers in particular to model compliance between individual- and population-level dynamics in spatially extended systems. Within the standard statistical-mechanical framework there is a well-developed theory for such coherence, based on memory-free and non-scaling (Markov-compliant) dynamics. However, as my book and blog is highlighting, the standard approaches towards modelling animal space use are often struggling when validated against high-quality spatio-temporal data. In a series of posts I plan to illustrate challenges and potential solutions at the population level by exploring the Zoomer model – a parsimonious variant of the Multi-scaled random walk model.

First, I want to recap a citation from a previous post:

“Parsimonious models are simple models with great explanatory predictive power. They explain data with a minimum number of parameters, or predictor variables. The idea behind parsimonious models stems from Occam’s razor, or “the law of briefness” (sometimes called lex parsimoniae in Latin). The law states that you should use no more “things” than necessary; In the case of parsimonious models, those “things” are parameters. Parsimonious models have optimal parsimony, or just the right amount of predictors needed to explain the model well.”

In the previous post I described the hidden layer concept at the individual level. In the present series of posts I turn towards the hidden layer at the population level; i.e., studying the statistical mechanics of populations. Migrating dunlins Calidris alpina, Western Norway. Photo: AOG.

A parsimonious model is the natural starting point when stepping away from the standard methods. By exploring a system’s behaviour at its very basic level, the core properties may be compared and tested against an alternative framework. Later on, additional layers of details may then be successively added to the basic model, for the sake of improved realism and case-specific tailor-making.

I have already described the Zoomer model’s basic properties in a range of posts (search for “Zoomer model” in the upper right search field). A more detailed walk-through is provided in my book. Below I spin a thread from the Zoomer model towards the standard approaches of spatio-temporal population dynamics by tuning the simulation condition towards a scale-specific and memory-less variant. In short: I switch off zooming (multi-scaled space use), and achieves a standard and parsimonious coupled map lattice (CML) model. This presentation then represents the entry point towards the extended system to be explored in the follow-up posts.

For the sake of exploring the populations’ intrinsic and most basic space use behaviour (prior to studying their responses to environmental heterogeneity and perturbations) I define a condition of a homogeneous environment. Generally I also go for a high frequency time axis, to underscore the fact that individual reshuffling generally has a more volatile effect on population dispersion of non-sessile animals than intrinsic death and growth rates.


First, consider an extreme case of spatially extended “boom and bust” dynamics, as illustrated above. The CML scenario describes a 32×32 cell arena where individuals show strong local reproduction and weak local survival during a time increment (survival rate: 0.6; reproduction rate: 0.5, net growth 0.1). When a local population reaches the defined carrying capacity*), on average 90% of the individuals are emigrating or dying (90% of the emigrants are redistributing themselves randomly within the arena, and the rest are either leaving the arena or dying). The leftmost image shows a spatio-temporal snapshot, the middle image shows a transect of local population density, and the rightmost image presents a multi-scaled snapshot of log(mean number of individuals) versus log(variance) of transects at respective spatial resolutions.

In the present context, the mean M is calculated in a rather non-conventional manner by summing individuals over local groups of cells (coarse-graining local density), and plotting the respective scales’ log(M,V) results in the same graph. In this manner the population’s variability characteristics are visualized over a scale range. In other words, M is proportional with the degree of spatial coarse-graining.

In the example above, the variance is strong (V>>M), and V varies approximately proportionally with M (and thus with spatial scale in this plot). This is indicative of an exponential distribution of local population density, which is to be expected due to the “boiling” boom and bust condition.

However, what if the population as a whole for most of its local parts are some distance from reaching the actual carrying capacity? Recall from standard ecological theory that local populations that are subject to relative (percent-wise) fluctuations – like variability in growth rate – are expected to show a log-normal distribution with V∝M2.


This scenario is shown above, using the same carrying capacity (however, at the present point in time the population density is relatively low and thus not influence by it), and with more subdued dynamics than in the first example: local overshoot emigration reduced from 90% to 50%. Further, local populations are dynamically coupled by a two-dimensional diffusion rate of 2% pr. time increment, which represents random immigration and emigration (two-way reshuffling) to a given cell during a given time increment.

Diffusion tends to “smoothen” the total population’s density surface by introducing strong spatial auto-correlation, which is clearly visible in both the density surface and the transect snapshots (image 1 and 2). Further, diffusion – like local growth and birth rate – influences local density in a relative manner and thus supporting a log-normal distribution with V∝M2. Since the current process is defined to be scale-specific (we are accepting a CML model to represent it and we are using diffusion for local re-shuffling, right?), diffusion to more distant cells during one time increment can be ignored if neighbour cell diffusion is small. In the present example, next-nearest neighbour diffusion rate is 0.02*0.02=0.0004, or 0.04%.

The dynamics of both scenaria are basic stuff for population dynamical modelling, and solidly explored under standard CML designs. The two process variants are also well understood within the complementary standard statistical mechanical theory. In particular, the respective log(M,V) plots provide on one hand the expected characteristics of a system satisfying ergodicity (first scenario above; V≈aM, spatially non-autocorrelated) and on the other hand a linear local coupling leading to spatial autocorrelation and V≈aM2 due to diffusion and most or all local populations below the carrying capacity (second scenario above; ergodicity only satisfied within grid cells). For an ergodic system one expects a power exponent ß≈1 [slope ≈ 1 as in the uppermost log(M,V) plot], while autocorrelation leads to 1<ß<≈2, and limits population ergodicity to fine scales.

Another characteristic property of a scale-specific process is the tendency even for conditions leading to autocorrelation with  ß≈2 (in practice, in the range 1.3<ß<1.8 due to influence from some local population crashes at any point in time) to drift towards ß≈1 (more synchronous crashes) as over-all population density approaches the carrying capacity. The parameter a also tends to increase. The degree of over-all autocorrelation vanishes during these events. An example is shown below, reflecting the state of Example 2 above shortly after many local populations have crashed simultaneously:


In general, under systems where the conditions for a CML model are satisfied the magnitude of the parameters ß and the log(V) intercept a reflect interesting statistical-mechanical states of the actual population. For example, a time series of (M,V,t) may reveal presence or absence of density dependent regulation by studying the behaviour of the a and β parameters over a scale range as time goes by (observe the special definition of M ∝ scale, as described above).

What if the CML conditions are not satisfied, for example, if spatio-temporal memory is influencing individual space use over a range of scales, as illustrated by the Multi-scaled random walk model? If individual scale-free space use exceeds the finest scale in the CML system, the statistical-mechanical framework needs to be extended to allow for scale-free population dynamics. “Zooming” effects will be switched on! Examples are coming in the upcoming parts of this series.


Traditionally, the carrying capacity of a species is defined as the maximum population size of the species that the environment can sustain indefinitely, given the food, habitat, water, and other necessities available. In the present context I modify this concept, which is better suited to describe the population as a whole, to allow for specific behavioural shift by the local population when a given threshold is reached. In short: a larger or a smaller part of the individuals at the given local site is moving out to search for better opportunities elsewhere.




The Hidden Layer

Focusing on the statistical pattern of space use without acknowledging the biophysical model for the process will create much confusion and unnecessary controversy. Ecologists are now forced to get a better grip on concepts from statistical mechanics than earlier generations. For example, to understand the transformation from data on actual behaviour to pattern analysis of space use, the concept of the hidden layer represents the first gate to pass.

Research on animal movement and space use has always had a central place in ecology. However, as more field data, better computers and more sophisticated statistical methods have become available, some old dogma have come under attack. Specific theoretical aspects of this quest for improved model realism have emerged from the rapidly growing cooperation between biologists and physicists in the emerging field of macro-level biophysics. The so-called Lévy flight foraging hypothesis is one example. And, of course, I can’t resist mentioning the MRW theory.

A booted eagle Hieraaetus pennatus is triggering a flock of spotless starlings Sturnus unicolor to show swarming behaviour. Malaga river delta, December 2017. Photo: AOG.

In 1985 Charles Krebs described ecology as the scientific study of the interactions that determine the distribution and abundance of organisms. In an ethological context animal space use is studied on two levels – tactical and strategic. The tactical level regards understanding individual biology and behavioural ecology on a moment-to-moment temporal scale. Strategic space use adds an extra layer of complexity to the tactical behaviour. In a simplistic manner we may refer to this layer as the animal’s state at a given moment in time; for example whether it is hungry or not (e.g., in hunting mode). Strategy also involves processing of memory-based goals. Strategies executed at coarser time scales than tactics. Some of the interaction between tactics and strategy may then – under specific conditions (see below) – be transformed to dynamic models at the tactical level; so-called mechanistic models, which consists of a set of executable rules covering respective cognitive and environmental conditions. Validating the model dynamics and resulting statistical patterns against real animal data then rates the degree of model realism. For example; realistic, tactical models have been developed to cast light on the “clumping behaviour” (dense swarming) of flock of birds that are threatened by a raptor.

The myriad of rules that influence animal movement makes detailed modelling an impossible task, and would anyway only lead to a descriptive picture with no value to ecological hypothesis testing. In fact, the signature of successful modelling is simplification. Thus, only specific aspects of the reference individual’s behaviour can be included and scrutinized.

The present post addresses one particular aspect of system simplification; coarse-graining the temporal scale. This approach implies a qualitative change of how the space use system is observed and analyzed. Actually, temporal coarse-graining is forced upon us when studying animal space use from sampling an individual’s successive displacements as a series of locations (fixes) during a given period of time. During each inter-fix interval the observed displacement regards the resultant vector from a myriad of intermediate and unobserved events. What has happened to the moment-to-moment kind of behavioural ecology? It has become buried below the hidden layer.

At the surface of this hidden layer you lose sight of behavioural details (like raptor response and swarming rules) but you gain access to an alternative perspective of movement and space use. Alternative statistical descriptors are emerging at this temporally coarser scale, following the laws of statistical mechanics. What is analyzed above the hidden layer is the over-all pattern from many displacements events that are aggregated into a spatial scatter of fixes.

For example, you may coarse-grain both the temporal and spatial system dimensions, and study the aggregated distribution of fixes at the spatial scale of virtual grid cells (pixels) and temporal scale of the fix sampling period. The spatio-temporal variations in intensity of space use within the actual space-time extents then allows for modelling and hypothesis testing, but now using statistical-mechanical descriptors of space use intensity. These descriptors are either not valid below the hidden layer (e.g., the information content of local density of fixes) or they have an alternative interpretation (e.g., movement as a “step” versus movement as a resultant vector for a given interval and location). Both levels of analysis require large sets of input to allow for statistical treatment.

Why is the hidden layer concept and the statistical-mechanical approach more important to relate to today than in earlier decennia? The short answer is the realization – seeded by better and more extensive data – that animal space use involves more than a couple of universality classes of movement (see this post). In fact, in my book, papers and blog posts I have detailed eight classes, most of which are unfamiliar to you.

To understand space use that is influenced by spatial memory and scale-free movement, statistical mechanical modelling is a prerequisite for realistic representation of such complex systems, unless you limit your perspective to a short-term behavioural bout within a very localized arena. In other words, “a single piece of a jigsaw-puzzle of space use dynamics”. For example, if you zoom closely into a small segment of a circle you observe an approximately straight line. Take a step outwards, and you are facing the qualitatively different geometry – the mathematics of a curve and finally a full circle. Stubbornly staying within the linear framework when analyzing more extensive objects than what you observe at fine scales will force you into a corner filled with paradoxes.

Fine-grained and coarse-grained analyses of animal space use are complementary approaches to the same system.