Population Dynamics - the Bold Approach

Readers of my book and my blog are well aware of a critical attitude towards traditional population modelling, in particular because individual memory effects on long distance movement with a potential for returns are not accounted for. Thus, I have developed an alternative approach, the "Zoomer model", which has been presented in various posts. In this post you find short videos of simulations where the classical and the alternative approach are contrasted under very basic conditions.

In the videos I  illustrate the profound difference between memory-less, Markov-compliant, scale-specific dynamics (the Paradigm) and scale-free, memory-influenced dynamics (the Zoomer model). These two classes of dynamics are exemplified to show the spatial redistribution effect on conspecific attraction, which is presented in two variants; 

• population re-distribution where the emerging pattern primarily is driven by such "intraspecific cohesion", and 
• the population kinetics is at some point during each series switching from intrinsic cohesion alone to becoming additionally influenced by some extra rate of local, non-directed and thus more erratic movement; i.e., diffusion-like. This illustrates that animal space use and re-distribution is influenced by many environmental factors; not only the distribution of conspecifics. The balance between these forces, here simplified by intra-specific cohesion and diffusion*), will determine the overall population kinetics and spatial dispersion. 

The Paradigm-based simulations are based on coupled map lattice modelling within a 64x64 grid of local population densities. The environment is set to be homogeneous at this unit scale of grid cells.**)  During each time step a small rate of population re-distribution takes place, whereby some individuals are moving to the local neighbourhood cell where the abundance is higher than at their respective current location***). The main difference between the two Paradigm examples (Video 1 and 2) and the two Zoomer counterparts (Video 3 and 4) is given by the term "local neighbourhood". A "zoomer" individual is assumed to invoke a memory capacity to allow for occasional and directed movement to a neighbourhood at a coarser intra-arena scale than its current field of perception, which is assumed to be limited to a finer resolution than the unit grid cell size, u. Further, such "strategic" moves take place at a rate that is distributed in a scale-free manner and with equal strength over the range u, 4u, 16u, 32u and 64u (Gautestad and Mysterud, 2005). In other words, the latter (larger) cell receives on average the same amount of zoomers, N_z, pr. time step as each of the embedded cells at finer resolutions (N_z*u =constant). For details, please search for posts under the key phrase "Zoomer model".

About half-way through each of the four videos you will see the added influence from strong dispersal, counteracting the conspecific attraction that rules the first part of respective series: the pattern is at this point switching from intrinsic cohesion to a strong rate of random dispersal, resulting in a gradually increasing random redistribution that is counteracting the clumping effect from intraspecific cohesion during the first phase in each series. In behavioual-ecological terms, the population is under strong influence of some kind of external perturbation, making the individual space use more erratic and tactical at the intra-u scale****).

Scenario 1: scale-specific dynamics. Individuals are aggregating in the neighbourhood cells at unit scale with highest abundance, on expense of lower-density cells.

Scenario 2: scale-specific dynamics with strengthened local diffusion, which under the given conditions effectively eliminates "clumping" from conspecific attraction.

Scenario 3: scale-free dynamics over a 4-level zooming" range.

Scenario 4: scale-free dynamics with local diffusion effect added.

Why focusing on conspecific attraction? It is a fact that most populations live in an open environment. Without the helping hand from intraspecific cohesion (i.e., in the absence of a very strong influence from some kind of external forcing, referred to as advection, in combination with a strong net birth rate), a population would generally be doomed to drift towards extinction due to a combination of diffusion along the open border zone towards regions where the abundance is lower or zero. Here the individuals are additionally prone to the hazardous influence from various Allée effects, which may spiral the population even faster towards extinction. 

As shown in previous Zoomer model examples, memory-influenced dynamics allows for individuals to return to conspecifics even after events involving long-distance occasional sallies. Hence, a population under influence of a strategic and memory-dependent glue effect may thrive even an environment where it floats freely in a region where it is only locally abundant.

NOTES

*) The birth and death rates are set to be sufficiently small to be of minor importance for the redistribution of individuals at the presently defined temporal resolution of these simulations (as they should be, realistically). Thus, cohesive versus diffusive factors are dominating in the population redistribution trends. 

**) It is a trivial expansion to make the environment heterogeneous in various ways, however such extensions it will not influence the basic system properties that are highlighted here.

***)  At a defined maximum local density, some portion of the individuals are redistributing themselves randomly within the arena, and some are leaving the arena all-together (or dying).

****) While adding the effect from diffusion-like movement in the last part of each series is compliant with the general conditions under the Paradigm, classic diffusion is NOT compatible with the Zoomer model's premises! However, the combination of scale-free redistribution from "zooming" and standard scale-specific dispersal is included in the present examples to show how the latter tend to smear out the characteristic scale-free (fractal-like) statistics that emerge from the default Zoomer model. Thus, the influence from standard diffusion on a real population's spatial dispersion can be statistically tested for (see previous Zoomer model posts or my book). In other words, a nice example of how to distinguish Paradigm-like and Zoomer-like population dispersion in situ.

REFERENCE

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