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Lotka-Volterra versus May-Leonard formulations of the spatial stochastic rock-paper-scissors model: The missing link

P. P. Avelino, B. F. de Oliveira, and R. S. Trintin
Phys. Rev. E 105, 024309 – Published 22 February 2022
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  • INTRODUCTION
  • LV AND ML FORMULATIONS OF THE RPS MODEL
  • RESULTS
  • CONCLUSIONS
  • ACKNOWLEDGMENTS
    • References

    Abstract

    The rock-paper-scissors (RPS) model successfully reproduces some of the main features of simple cyclic predator-prey systems with interspecific competition observed in nature. Still, lattice-based simulations of the spatial stochastic RPS model are known to give rise to significantly different results, depending on whether the three-state Lotka-Volterra or the four-state May-Leonard formulation is employed. This is true independently of the values of the model parameters and of the use of either a von Neumann or a Moore neighborhood. In this paper, we introduce a simple modification to the standard spatial stochastic RPS model in which the range of the search of the nearest neighbor may be extended up to a maximum Euclidean radius R. We show that, with this adjustment, the Lotka-Volterra and May-Leonard formulations can be designed to produce similar results, both in terms of dynamical properties and spatial features, by means of an appropriate parameter choice. In particular, we show that this modified spatial stochastic RPS model naturally leads to the emergence of spiral patterns in both its three- and four-state formulations.

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    • Received 6 October 2021
    • Accepted 4 February 2022

    DOI:https://doi.org/10.1103/PhysRevE.105.024309

    ©2022 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Biological self-organizationPopulation dynamics
    1. Physical Systems
    Stochastic dynamical systems
    1. Techniques
    Monte Carlo methods
    Interdisciplinary PhysicsPhysics of Living SystemsNonlinear Dynamics

    Authors & Affiliations

    P. P. Avelino1,2, B. F. de Oliveira3, and R. S. Trintin1,3

    • 1Instituto de Astrofísica e Ciências do Espaço, Universidade do Porto, CAUP, Rua das Estrelas, PT4150-762 Porto, Portugal
    • 2Departamento de Física e Astronomia, Faculdade de Ciências, Universidade do Porto, Rua do Campo Alegre 687, PT4169-007 Porto, Portugal
    • 3Departamento de Física, Universidade Estadual de Maringá, Av. Colombo 5790, 87020-900 Maringá, PR, Brazil

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    Issue

    Vol. 105, Iss. 2 — February 2022

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    Images

    • Figure 1
      Figure 1

      Snapshots of the spatial distribution of the different species on a 5002 lattice at t=2000 for LV and ML formulations of the standard and modified versions of the spatial stochastic RPS model. Notice the absence of spiral patterns in the LV formulation of the standard spatial stochastic RPS model and that the significant discrepancies observed for the LV and ML formulations are no longer present in the modified version.

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    • Figure 2
      Figure 2

      The spatial autocorrelation functions C(r) of the models considered in Fig. 1—the arrow represents the lengthscale ℓ defined by C(ℓ)=0.5. Notice the similarity between the two curves obtained using the LV and ML formulations of the modified version of the spatial stochastic RPS model (bottom panel), in sharp contrast to the significant differences observed when the standard version is employed (top panel).

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    • Figure 3
      Figure 3

      The time evolution of the densities ρi of the different species over time for the realizations of the spatial stochastic RPS model considered in Fig. 1. Again notice that the observed qualitative and quantitative differences between the results obtained assuming LV and ML formulations of the standard spatial stochastic RPS model are essentially absent in the modified version.

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    • Figure 4
      Figure 4

      Power spectra of ρ1(t) obtained using LV and ML formulations of the standard (top panel) and modified (bottom panel) versions of the spatial stochastic RPS model—a similar behavior is found for the species 2 and 3 (see Fig. 3). Notice the similarity between the curves and the absence of triplen harmonics.

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