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, 7, e8011

Enforced Symmetry: The Necessity of Symmetric Waxing and Waning


Enforced Symmetry: The Necessity of Symmetric Waxing and Waning

Niklas Hohmann et al. PeerJ.


A fundamental question in ecology is how the success of a taxon changes through time and what drives this change. This question is commonly approached using trajectories averaged over a group of taxa. Using results from probability theory, we show analytically and using examples that averaged trajectories will be more symmetric as the number of averaged trajectories increases, even if none of the original trajectories they were derived from is symmetric. This effect is not only based on averaging, but also on the introduction of noise and the incorporation of a priori known origination and extinction times. This implies that averaged trajectories are not suitable for deriving information about the processes driving the success of taxa. In particular, symmetric waxing and waning, which is commonly observed and interpreted to be linked to a number of different paleobiological processes, does not allow drawing any conclusions about the nature of the underlying process.

Keywords: Averaging; Conditioning; Diversity; Ecology; Extinction risk; Limit theorems; Occupancy; Paleontology; Range size; Symmetry.

Conflict of interest statement

The authors declare there are no competing interests.


Figure 1
Figure 1. An example for the data processing procedure described in the section “The Way Data are Processed”.
(A) Original trajectories; (B) x-shift; (C) x-scale; (D) y-scale; (E) Averaging. Note how the axis labels change due to the data processing.
Figure 2
Figure 2. The decomposition of a function (A) into is symmetric and asymmetric part (B).
The blue area corresponds to the quantified asymmetry (QuAsy) of the function introduced in the section “Measuring Asymmetry”.
Figure 3
Figure 3. Symmetric (red) and asymmetric (blue) part of rescaled occurrences with relative age of some Cenozoic foraminifera species.
Nine randomly selected species (A-I) are shown. Occurrences were downloaded from the Neptune Sandbox Berlin (NSB) (Lazarus, 1994; Spencer-Cervato, 1999), rescaled to have first and last occurrence at 0 and 1 resp. (x-shift and x-scale), and then binned into ten bins. The area of the bins was then rescaled to sum up to 1 (y-scale).
Figure 4
Figure 4. The distribution of asymmetry in occurrences of Cenozoic foraminifera (A) and radiolaria (B).
Histogram shows the distribution of asymmetry of individiual species, average of the asymmetry of individual species (blue dashed line) and asymmetry of the trajectory generated by averaging the trajectories of all taxa (red dashed line). Asymmetry was quantified using QuAsy (see Appendix S1), where 0 corresponds to symmetry and 1 correspond to asymmetry. Data was downloaded from the Neptune Sandbox Berlin (NSB) (Lazarus, 1994; Spencer-Cervato, 1999), data processing is as described in the caption of Fig. 3 or in Appendix S3. The distribution of asymmetry for individual species displays a tendency towards asymmetry, with a mean QuAsy value of 0.59 for foraminifera and 0.56 for radiolarians. The asymmetry of the averaged trajectories of all taxa returns a much lower value of 0.036 for foraminifera and of 0.072 for radiolarians, a decrease of more than 85% in both groups. Only 4% of individual foraminifera taxa are more symmetric than the “average foraminifera” generated by averaging all taxa. The R code used can be accessed in the Supplementary Information.
Figure 5
Figure 5. The behaviour of a random walk conditioned to be positive (blue), conditioned to be positive and go extinct after n = 20 time steps (red), and the average over multiple trajectories conditioned to be positive and go extinct (black).
In all three cases, the probability of the random walk to increase is p = 0.6. The first case shows a clear increase as expected with a transition probability larger than 0.5. This is no longer visible in the second case, and the averaged and conditioned case is symmetric and lacks any random behaviour. The distribution of asymmetry of these three cases is shown in Fig. 3.
Figure 6
Figure 6. The distribution of asymmetry for an positive random walk (blue histogram), a positive random walk conditioned to go extinct (red histogram) and averaged trajectories of a positive random walk conditioned to go extinct (black dot).
In all three cases, the probability of the random walk to increase is p = 0.6, time of extinction is after n = 20 time steps and the asymmetry was quantified using QuAsy (see Appendix S1), where 0 represents symmetry and high values correspond to high asymmetry. The distribution of asymmetry for the positive random walk going extinct is more concentrated and closer to zero than the distribution of the positive random walk, showing how conditioning increases symmetry. By additionally averaging the trajectories, this effect is emphasized even more. This is displayed by the averaged case (black dot) with a QuAsy value of below 0.1.

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