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 statistical effect of 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, to the right, 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 the slope b ≈ 2 under the standard CML condition, and log(a) ≈ 0 when b ≈ 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.