DISCUSSION
SAR and SAD results . Over a six-year period, both the observed SAR and the SAD at the field site exhibit a departure over time from METE predictions. The METE-predicted SAR is not a power-law; rather, the predicted slope decreases with area. As is evident in Fig. 1, the general pattern of deviation from METE in this study is in the direction of even further deviation from power law behavior, with the slope decreasing more rapidly with scale than METE predicts. This is in contrast with results from another study (Newman et al. 2020) that compared the SAR in two nearby Bishop pine forests: one that had recently burned and was undergoing post fire succession, and a mature forest that had not burned in many decades. The mature forest SAR was accurately predicted by Eq. 1, whereas the SAR in the recently burned forest was more consistent with a power law. This informs us that we cannot expect any simple universality in the pattern of deviation from the static prediction. Different types of disturbance regime can produce different patterns of deviation.
The change over time in the SAD, and its pattern of deviation from the METE prediction, is also illuminating. We observed agreement with METE in 2014 followed by poor performance in the following years. On the one hand, a decrease in the number of species with very few individuals was observed. Whereas there were, for example, 4 species with either one or two individuals in 2014, there were an average of 1.2 such species over the years from 2015-2018. Less anticipated was the decrease in abundances of the most abundant species, resulting in a SAD that is not described by the METE-predicted log-series distribution nor by a lognormal distribution. The bow-shaped appearance of the SAD in Fig. 3 is more characteristic of an exponential distribution. This contrasts with evidence from another study suggesting that disturbance results in a lognormal distribution (Kempton & Taylor 1974). It also contrasts with evidence from the Bishop pine study (Newman et al. 2020), which shows a rank-log(abundance) curve that bows in the opposite direction (that is, a concave curve) from the SADs in Fig. 3.
Thus, both the SAD and the SAR findings reported here suggest that disturbance can exert different effects on macroecological patterns. Further investigation is needed to catalog the patterns that macroecological metrics can exhibit under various types of disturbance, and to develop dynamic theory that can predict such a variety of patterns. While White et al. (2012) looked at the SAD in 15,848 plant, mammal, arthropod, and bird communities and concluded that the log-series outperformed the lognormal, no comprehensive analysis of the relationship between the shape of the SAD, the SAR, and the disturbance level or degree of departure of the ecosystem state variables from steady state has been carried out.
Compressed SAR. In our analysis we made the choice to ignore the corridors between the 50 plots and compress them into one continuous plot that we then sectioned into quadrats more finely. Consider the difference between what we observed and what would be observed if we had a full data set that included individuals present within the corridors. When we compress the plots as we did, individuals in different plots are brought closer together than they really are in the full dataset. Generally, as a consequence of species turnover, the number of species in common between plots decreases as the distance between them increases. Thus, within areas of the same size in the compressed and full dataset, we would expect there to be more species in the compressed area since it spans a larger distance. In other words, the observed slopes computed from the compressed data will be higher than in a full data set at corresponding areas. This might appear to introduce an artifact, but in the MaxEnt procedure that METE uses, this will not matter. In particular, MaxEnt predicts the shape of the SAR from the values of state variables (number of species and number of individuals) that impose constraints on information entropy maximization. At the same spatial scale, these state variables will differ between the compressed and full data sets, with number of individuals independent of the compression and number of species increased, at fixed scale, in the compressed data. Because the predicted slopes depend upon the state variables, both they and the observed slopes are changed as a consequence of compression.
If our goal was to predict the slopes for the uncompressed data, this would pose a problem; however, our goal is to study the errors between the METE predicted slopes and observed slopes over the years. The errors should follow the same trend for the full and compressed dataset, because in the compressed dataset, the observed and predicted slopes have both simply been shifted upwards and to the left on the plot of log(N/S ) versus slope, since both slope and S at each scale have increased.
Predicted rank abundance curves. In our analysis we generated and ranked 1,000 samples from the predicted SAD and observed that our predicted rank abundance curve falls roughly in the middle of the samples (Fig. 3). This shows that the method we used to calculate the predicted rank abundance curve from the predicted SAD is reasonable and an appropriate measure to use when computing the errors for the SAD. The samples additionally provide a visualization of how far from METE’s prediction the observed data are as many of the points fall outside the range of the samples in every year but 2014.
Demographic evidence for stress. The stress factors offer insight into the SAR error, but less so into the SAD error (Table 1). In terms of the signs of the correlation coefficients, we expected recruitment, number of individuals, and species richness to negatively correlate with error because these values tend to be lower in stressed and disturbed systems. Conversely, we expected mortality and net loss to positively correlate with error because these values tend to be higher in stressed and disturbed systems. The signs on the SAR error correlations with recruitment, abundance and species richness are as expected (-0.56, -0.51, -0.50 respectively) and similarly for mortality and net loss (+0.91, + 0.97 respectively). The only even moderately explanatory factor for the SAD error, species richness (r = -0.80), also has the expected negative sign. It is interesting to note that the SAR is derived from the SAD and a spatial function describing species-level aggregation (Harte 2011). Because net loss does not have the expected sign or a strong correlation with the SAD error, the net loss is likely impacting spatial aggregation. In stressful environments, there is evidence that facilitation reduces mortality for clustered individuals (Jia et al. 2011; Brooker et al. 2008), which would lead to a disproportionate number of deaths for unclustered individuals and an increase in overall aggregation. Further work is required to understand how mortality and recruitment affect spatial patterning at this site.
Our findings point to the importance of extending the static MaxEnt-based macroecological theory to the dynamic regime. By combining the MaxEnt inference procedure with explicit stress mechanisms acting via higher death rates, lower growth rates, reduced immigration, etc., a theoretical foundation can be established upon which it might be possible to understand how disturbance alters macroecological patterns and conversely how altered patterns allow attribution of stress factors.
Conclusion. Measurement of macroecological metrics like the SAD and the SAR have the potential, as yet not widely exploited, to achieve the goal of providing critical early warning information about an ecosystem experiencing rapid change. It remains to be seen whether such information is more readily obtainable from analysis of macroecological metrics or if it is necessary to make more traditional repeated observations on very fine scale, in which the taxonomic identity, location, and traits of the individuals in a plant community, along with local variability in abiotic conditions, are recorded. Based on the clear trends observed in our analysis of the Mt. Baldy data, and the possibility that such trends are unique to different types of disturbance, we can conclude that further pursuit of this goal could expedite success in the search to better predict ecosystem responses in the Anthropocene.