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Abstract and Figures

We study the large-time behavior of an ensemble of entities obeying replicator-like stochastic dynamics with mean-field interactions as a model for a primordial ecology. We prove the propagation-of-chaos property and establish conditions for the strong persistence of the N-replicator system and the existence of invariant distributions for a class of associated McKean–Vlasov dynamics. In particular, our results show that, unlike typical models of neutral ecology, fitness equivalence does not need to be assumed but emerges as a condition for the persistence of the system. Further, neutrality is associated with a unique Dirichlet invariant probability measure. We illustrate our findings with some simple case studies, provide numerical results, and discuss our conclusions in the light of Neutral Theory in ecology.
Graphical hypothesis testing for contrasting H0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_0$$\end{document}: “the law of the first coordinate trajectories of (23) follows a Beta(a, b) at time t”. The statistical test is based on the work of Ebner and Liebenberg (2021). The upper panels show the results for PS1 and the lower panels are the corresponding results for PS2, with their corresponding theoretical s given in the main text. The left column corresponds to time t=40\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t=40$$\end{document} and the right one to time t=50\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t=50$$\end{document}. The test is based on the statistic Tn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_n$$\end{document}, constructed according to a L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document} distance given in Equation (26), whose observed values are shown by the red dashed lines. We can notice that, in all the cases, they are much lower than the quantiles that delimit the rejecting region at each indicated (1-α)×100\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1-\alpha )\times 100$$\end{document} percentile, with α=0.1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha =0.1$$\end{document} (yellow dashed lines), α=0.05\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha =0.05$$\end{document} (purple dashed lines) and α=0.01\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha =0.01$$\end{document} (green dashed lines), regarding the theoretical probability density of the statistic Tn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_n$$\end{document} under H0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_0$$\end{document}, approximated by simulations
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Journal of Mathematical Biology (2025) 90:15
https://doi.org/10.1007/s00285-024-02174-w
Mathematical Biology
Persistence and neutrality in interacting replicator
dynamics
Leonardo Videla1·Mauricio Tejo2·Cristóbal Quiñinao3·
Pablo A. Marquet3,4,5,6 ·Rolando Rebolledo7
Received: 30 May 2023 / Revised: 20 August 2024 / Accepted: 10 November 2024 /
Published online: 3 January 2025
© The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2024
Abstract
We study the large-time behavior of an ensemble of entities obeying replicator-like
stochastic dynamics with mean-field interactions as a model for a primordial ecology.
We prove the propagation-of-chaos property and establish conditions for the strong
persistence of the N-replicator system and the existence of invariant distributions
for a class of associated McKean–Vlasov dynamics. In particular, our results show
that, unlike typical models of neutral ecology, fitness equivalence does not need to
be assumed but emerges as a condition for the persistence of the system. Further,
neutrality is associated with a unique Dirichlet invariant probability measure. We
illustrate our findings with some simple case studies, provide numerical results, and
discuss our conclusions in the light of Neutral Theory in ecology.
Keywords Stochastic replicator dynamics ·Propagation of Chaos ·Stochastic
persistence ·McKean–Vlasov equation ·Invariant distributions ·Emergence of
ecologies
Mathematics Subject Classification 60H10 ·92D25
1 Introduction
The emergence of life corresponds to the emergence of an ecological system of
interacting self-replicating entities. Several characteristics of early life have endured
through time and are shared by all forms of life today, from the near universality
This work was funded by project FONDECYT 1200925 The emergence of ecologies through metabolic
cooperation and recursive organization, Centro de Modelamiento Matemático (CMM), Grant FB210005,
BASAL funds for centers of excellence from ANID-Chile, Exploration-ANID 13220168, Biological and
quantum Open System Dynamics: evolution, innovation and mathematical foundations, FONDECYT
Iniciación Project No. 11240158-2024 Adaptive behavior in stochastic population dynamics and
non-linear Markov processes in ecoevolutionary modeling, and FONDECYT Iniciación 11200436,
Excitation and inhibition balance as a dynamical process.
Extended author information available on the last page of the article
123
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