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Monitoring biodiversity on highly reactive rock-paper-scissors models

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Pith's one-line read In spatial rock-paper-scissors simulations, making all species equally more aggressive protects biodiversity at high mobility, while making only some species more aggressive raises extinction risk.

arxiv 2504.12054 v1 pith:C7WIULWY submitted 2025-04-16 q-bio.PE physics.bio-ph

D. Bazeia , M.J.B. Ferreira , B.F. de Oliveira , W.A. dos Santos This is my paper · ORCID
classification q-bio.PEphysics.bio-ph
keywords highlymodelsspeciesreactivenon-hierarchicalbiodiversityhierarchicalcompared
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

In the classic rock-paper-scissors setup, three species A, B and C live on a grid and beat each other in a cycle: A beats B, B beats C, C beats A. Individuals move, reproduce into empty cells, and kill prey in neighboring cells. Earlier work showed that the three species can coexist in rotating spiral patterns when mobility is low, and that too much mobility destroys biodiversity. The authors modify the rules so that one or more species react more strongly: a highly competitive species kills all prey in every neighboring cell at once, and a highly reproductive species fills all neighboring empty cells at once. They call these HC and HR models and run many random simulations on square lattices.

The simulations show that when only one or two species have this enlarged reaction, the symmetry of the game is broken: one species dominates, oscillations grow, and extinction becomes more likely at moderate mobility. These are called hierarchical models. When all three species are upgraded equally, the system stays symmetric. The striking result is that the symmetric HC-ABC model keeps all species alive at higher mobility than the standard model, while the symmetric HR-ABC model behaves about the same as the standard one. The authors interpret this as symmetric extra competition helping biodiversity, while asymmetric extra reaction harms it.

The paper also measures oscillation frequencies and spatial correlation lengths. All species in a given model oscillate at the same frequency, and that frequency increases with the number of highly competitive species. The claims are entirely simulation-based, with no analytical derivation, and the plots come with no error bars.

Extended reading notes

Core claim

The central result, stated in the Ending Comments, is that when the highly reactive model is controlled by reproduction, one notices the absence of modification in the probability of extinction, but the behavior changes when the reactive rule is competition. In this case, the most robust model is the HC-ABC. This means that the symmetric increasing of competition tends to fortify biodiversity. The paper also claims that asymmetric hierarchical models have higher extinction probability than non-hierarchical ones, so asymmetry weakens biodiversity.

Load-bearing premise

The extinction probabilities in Figs. 12 and 13 are computed after the first 1000 generations are left out of the analysis to skip the transient part of the time evolution. If early transient extinctions, which are expected to be most frequent at high mobility, are discarded rather than counted as extinctions, the reported P_ext could be systematically undercounted and could change the ordering between HC-ABC and Std. The paper gives no sensitivity check for this cutoff or for lattice size, since extinction curves use L=200 while the structural analysis uses L=500.

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Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

All assumptions of the simulation are listed. There are no fitted numbers in the central extinction claim; the autocorrelation cutoff is a diagnostic convention. The equal-rate choice pr = pc is a modeling assumption, not an empirical input.

free parameters (1)
  • Characteristic length cutoff = C(l) = 0.2
    Hand-chosen in Sec. IV.D to define l from the autocorrelation function. It affects reported lengths and qualitative comparisons, but not the direct extinction simulation curves.
assumptions (5)
  • domain assumption Stochastic May-Leonard dynamics on a square lattice with periodic boundary conditions.
    The entire model in Sec. II assumes this update scheme; results are specific to it.
  • domain assumption Equal reproduction and competition rates, pr = pc = alpha, with mobility parameterized by Eq. (1).
    Sec. II states this without loss of generality; the extinction curves depend on this rate balance.
  • ad hoc to paper Highly reactive species act on all adjacent sites simultaneously, rather than on one selected site.
    This is the new rule defining HC and HR models in Secs. II.A-II.B, introduced by the authors.
  • domain assumption Initial configurations are random and all simulations use L=200 or L=500 with periodic boundaries.
    The measured densities, patterns, and extinction probabilities are conditional on this setup; no finite-size scaling is reported.
  • ad hoc to paper The characteristic length is defined by C(l)=0.2 from the autocorrelation function.
    Sec. IV.D chooses this threshold by hand; it is a diagnostic convention, not an empirical law.

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Pith. "Pith review of Monitoring biodiversity on highly reactive rock-paper-scissors models." pith.science (2026). https://pith.science/paper/C7WIULWY

@misc{pith2026250412054,
  author       = {Pith},
  title        = {Pith review of: Monitoring biodiversity on highly reactive rock-paper-scissors models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/C7WIULWY}},
  note         = {Machine review of arXiv:2504.12054}
}
read the original abstract

This work investigates how biodiversity is affected in a cyclic spatial May-Leonard model with hierarchical and non-hierarchical rules. Here we propose a generalization of the traditional rock-paper-scissors model by considering highly reactive species, i. e., species that react in a stronger manner compared to the others in respect to either competition or reproduction. These two classes of models, called here Highly Competitive and Highly Reproductive models, may lead to hierarchical and non-hierarchical dynamics, depending on the number of highly reactive species. The fundamental feature of these models is the fact that hierarchical models may as well support biodiversity, however, with a higher probability of extinction than the non-hierarchical ones, which are in fact more robust. This analysis is done by evaluating the probability of extinction as a function of mobility. In particular, we have analyzed how the dominance scheme changes depending on the highly reactive species for non-hierarchical models, where the findings lead to the conclusion that highly reactive species are usually at a disadvantage compared to the others. Moreover, we have investigated the power spectrum and the characteristic length of each species, including more information on the behavior of the several systems considered in the present work.

Figures

Figures reproduced from arXiv: 2504.12054 by the authors.

Figure 1
Figure 1. In this scheme, A dominates B, B dominates C, and C dominates A. This is the Standard RPS model, which we will refer to as the Std model throughout this work. It should be noted that the Std model exhibits symmetric interactions among species, which means that there are no privileged species in this case. The RPS model has been extensively studied in the past few decades, since it captures a bunch of inter￾esting fe… view at source ↗
Figure 1
Figure 1. Non-hierarchical (or symmetric) competition rule [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Illustration of an example where species [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figures from the paper (10 more)
Figure 3
Figure 3. Figure 3: Illustration of an example where species [PITH_FULL_IMAGE:figures/full_fig_p003_3.png]
Figure 4
Figure 4. Figure 4: Snapshots of both Std and HC models after 6000 generations, highlighting the emergence of spatial patterns. For these simulations we have set L = 500 and M = 4×10−6 , meaning that pm = 0.5 and pc = pr = 0.25. The colors of species A, B and C follows the same color pale…
Figure 5
Figure 5. Figure 5: Time evolution of species densities displayed for [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]
Figure 6
Figure 6. Figure 6: Snapshots of both Std and HR models after 6000 generations, highlighting the emergence of spatial patterns. For these simulations we have set L = 500 and M = 4×10−6 , meaning that pm = 0.5 and pc = pr = 0.25. The colors of species A, B and C follows the same color pale…
Figure 7
Figure 7. Figure 7: Time evolution of species densities displayed [PITH_FULL_IMAGE:figures/full_fig_p006_7.png]
Figure 8
Figure 8. Figure 8: Power spectrum of each species in both Std and [PITH_FULL_IMAGE:figures/full_fig_p007_8.png]
Figure 9
Figure 9. Figure 9: Power spectrum of each species in both Std and [PITH_FULL_IMAGE:figures/full_fig_p008_9.png]
Figure 10
Figure 10. Figure 10: Autocorrelation function of both Std and HC [PITH_FULL_IMAGE:figures/full_fig_p009_10.png]
Figure 12
Figure 12. Figure 12: Probability of extinction as a function of mobility [PITH_FULL_IMAGE:figures/full_fig_p009_12.png]
Figure 13
Figure 13. Figure 13: Probability of extinction as a function of mobility [PITH_FULL_IMAGE:figures/full_fig_p010_13.png]

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