Simulating Animal 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.


*) 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 Animal 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, 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.

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 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.