**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.

NOTE

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

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