# 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, diffusion tends to smoothen the density surface. The log(M,V) plot over a scale range 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.

The correlogram (above) reflects the undulating density surface.

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 present  snapshot of population dispersion 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. 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, 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.

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.