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.
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.
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.
*) 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.
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, with 1% 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 – 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.
*) 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.
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 from 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 β ≈ 2 under the standard CML condition, and log(a) ≈ 0 when slope β ≈ 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.
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):
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 (k 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) + β*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 β ≈ 2 implies that √V; i.e., the standard deviation (SD), is proportional with M. Further, if log(a) = 0 (a ≈ 1) in addition to β ≈ 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.
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.
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 β=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.
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.”
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.