### Simulating Populations VII: the Correlogram View of Complex Dispersion

**In my continued quest for a more realistic statistical-mechanical theory for spatially extended population dynamics I have previously pointed out a specific property of the inter-scale spatial coefficient of variation as one of the hallmarks of scale-free dispersion (see Part III). In the present post I study another statistical property, the spatial autocorrelation, which may provide additional cue about the population’s compliance with standard or complex space use.**

First, consider the standard theory, based on mean field compliant population redistribution (mixing). The following three images show a typical example, where the population is subject to 5% diffusion rate at unit (pixel) scale, net population growth of 1%, no Allée effect, and over-all population density below carrying capacity. As previously described in this series and illustrated to the left, diffusion tends to smoothen the density surface.

The log(M,V) plot over a scale range (below) typically shows y-intercept [log(a)] substantially below zero a hallmark of fine-scale smoothness – under condition where the slope β ≈ 2. I refer to previous posts in this series for technical details.

Next, consider the following scenario under the same condition, except for 5% standard diffusion at unit scale (k=1) being replaced by 5% scale-free “zooming” with 1% pr. scale level.

The snapshot of population dispersion below represents the population a few time increments after a general population crash (bottleneck episode), when the population is in the process of re-organizing itself. Despite the short time span since the episode (5-6 time increments) the log(M,V) plot has already adjusted itself from log(a) >>0 and b ≈ 1 (random mixing due to external forcing) to log(a)≈ 0 and b ≈ 2.

Observe that in this additional example of complex population dynamics the log(M,V) plot again satisfies log(a) ≈ 0 when b ≈ 2. Here the local effect at next time increment from overshooting local carrying capacity both locally and elsewhere is influenced by a rate of 50% remaining population at the actual location (as in previous examples) to zero individuals remaining. In other words, this hallmark of Zoomer-like dynamics is quite resilient to this modification of ecological conditions.

The interesting aspect in the present context is its spatial transect correlogram at unit scale k=1 (below). It shows low level of autocorrelation at all spatial lags except for scale 0, which trivially illustrates that the local population correlates with itself 100% at lag zero. Despite the non-significant autocorrelation the parameter condition for log(M,V), log(a) = 0 and b = 2.

Thus, scale-free population abundance regards both spatially autocorrelated transects (as shown in previous parts of this series) and – as shown here – non-autocorrelated transects a short period following a perturbation [during the first 3-5 increments after the event*, log(a) >> 0 and b ≈ 1].

In this respect I refer to the so-called Z-paradox (search Archive), which is resolved under the Zoomer model but not under the standard framework of population modelling. Thus, my proposed model may also provide a novel approach towards the famous and controversial Taylor’s Power law (again, search Archive).

NOTE

*) Under the condition that the bottleneck condition lasts for one time increment. If the change of condition is permanent, it may typically take 20-50 time steps to restore log(a) ≈ 0 and b ≈ 2.