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arxiv: 1907.06965 · v1 · pith:NPPSGZ5Ynew · submitted 2019-07-16 · 🧮 math.PR · math.ST· stat.TH

Stochastic Evolution of spatial populations: From configurations to genealogies and back

Andreas Greven This is my paper

Pith reviewed 2026-05-24 20:51 UTC · model grok-4.3

classification 🧮 math.PR math.STstat.TH
keywords spatial population modelsgenealogiesFleming-ViotCannings modelshierarchical mean-field limituniversality classesFisher-Wright diffusionseedbanks
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The pith

Genealogies of spatial populations can reconstruct their type and location structures, and configurations can reconstruct genealogies.

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

The paper reviews results on spatial population models showing that genealogical information reveals type and location structures while the reverse also holds. Large-scale behavior is analyzed through hierarchical mean-field limits, spatial continuum limits, and finite system schemes to identify universality classes. The approach extends to new classes including Fleming-Viot models in continuum space, Cannings models with block resampling, Fisher-Wright models with coloured seedbanks, and Fleming-Viot models with selection and rare mutation. A reader would care because the duality supplies tools to study evolution and diversity in biological populations at large scales.

Core claim

The results obtained show that genealogical information can be used to analyze the type and location structure and vice versa in spatial population models. The large scale behaviour is captured by the hierarchical mean-field limit or the spatial continuum limit and the finite system scheme. This is applied to extended model classes: Fleming-Viot genealogies in continuum geographic space, Cannings models with block resampling, Fisher-Wright diffusion with coloured seedbanks, and evolving genealogies of Fleming-Viot models with selection and rare mutation.

What carries the argument

Duality between population configurations and genealogies, extracted via hierarchical mean-field limit, spatial continuum limit, or finite system scheme.

If this is right

  • Universality classes of large-scale behaviour can be identified for the extended models.
  • Effects such as reduced diversity from block resampling or enhanced diversity from seedbanks can be explained via genealogies.
  • The methods apply to models with selection and rare mutation in continuum space.
  • Biological situations can be analyzed by switching between configuration and genealogy viewpoints.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The duality may allow inference of historical migration patterns from current genetic samples in continuous habitats.
  • Similar bidirectional analysis could apply to other interacting particle systems with dual descriptions.
  • Numerical simulations of the extended models could test whether the limits produce the claimed universality in finite but large systems.

Load-bearing premise

The hierarchical mean-field limit, spatial continuum limit, and finite system scheme accurately capture the large-scale universality classes of the extended population models in biological situations.

What would settle it

A spatial population where the predicted genealogical distances or coalescence times fail to match the observed spatial clustering of types under the described limits.

Figures

Figures reproduced from arXiv: 1907.06965 by Andreas Greven.

Figure 1
Figure 1. Figure 1: 2 Description of genealogies The first next and new point is now to construct for the mechanisms from the paragraph "mod￾els", the corresponding genealogical processes as Markov process on some Polish state space. A survey of our approach to genealogies is in [DG18]. 2.1 The state space of genealogies The first point is to specify a state space which allows to describe the state of the genealogy of the cur… view at source ↗
Figure 2
Figure 2. Figure 2: • What does the theory of populations in infinite geographic space have to do with that of systems on large finite spaces, we may see in reality? The first point is tackled using two different approaches. For one the multi-scale renormal￾ization combined with the hierarchical mean field limit and second the continuum space limit. The second point is approached via the finite system scheme. 3.1 The hierarch… view at source ↗
read the original abstract

The paper reviews the results obtained for spatial population models and the evolution of the genealogies of these populations during the last decade by the author and his coworkers. The focus is on their large scale behaviour and on the analysis of universality classes of large scale behaviour via the methods of the hierarchical mean-field limit or via the spatial continuum limit and from another angel, the finite system scheme . We use genealogical information to analyze the type and location structure and vice versa. To apply this approach and to explain effects in biological situations we extend the classical model classes in new directions. Namely we look as population models here at: Fleming-Viot genealogies in continuum geographic space, Cannings models with block resampling (reducing diversity), Fisher-Wright diffusion with coloured seedbanks (enhancing diversity) and evolving genealogies of Fleming-Viot models with selection and rare mutation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript reviews a decade of results by the author and collaborators on spatial population models, focusing on the large-scale behavior of populations and their genealogies. It highlights the duality between genealogical information and type/location structure, analyzed via hierarchical mean-field limits, spatial continuum limits, and the finite system scheme. Extensions are presented to new model classes including Fleming-Viot genealogies in continuum space, Cannings models with block resampling, Fisher-Wright diffusions with coloured seedbanks, and Fleming-Viot models incorporating selection and rare mutation.

Significance. This review synthesizes methods for establishing universality classes in stochastic spatial population models through duality and scaling limits. The extensions to models addressing reduced diversity (block resampling), enhanced diversity (seedbanks), and selection/mutation provide a framework applicable to biological scenarios, consolidating prior work into a coherent picture of large-scale behavior.

minor comments (3)
  1. [Abstract] Abstract: 'from another angel' should read 'from another angle'.
  2. [Introduction] The manuscript would benefit from a brief explicit statement in the introduction or conclusion on the scope of the review (e.g., which key papers are omitted and why) to help readers assess completeness.
  3. Ensure consistent notation for the various limits (hierarchical mean-field, spatial continuum, finite system scheme) across sections, with a short glossary or reference table if space permits.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary and significance assessment of our review manuscript, as well as the recommendation for minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

Review of prior results; no new derivations or load-bearing claims

full rationale

The document is explicitly a review summarizing a decade of results obtained by the author and coworkers on spatial population models, genealogies, and limits (hierarchical mean-field, spatial continuum, finite system scheme). It extends classical models in listed directions but presents these as reviewed outcomes rather than new derivations. No equations, predictions, or uniqueness theorems are introduced within the paper that reduce to self-definition, fitted inputs renamed as predictions, or self-citation chains. The duality between genealogies and type/location structure is stated as a reviewed fact, not re-derived here. This matches the default expectation for a non-circular review paper.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

This is a review paper; the abstract introduces no new free parameters, axioms, or invented entities.

pith-pipeline@v0.9.0 · 5669 in / 1029 out tokens · 21536 ms · 2026-05-24T20:51:54.868202+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages · 6 internal anchors

  1. [1]

    Dawson and Andreas Greven

    [DG14] Donald A. Dawson and Andreas Greven. Spatial Fleming-Viot models with se- lection and mutation , volume 2092 of Springer Lecture Notes in Math. Springer V erlag,

    work page 2092

  2. [2]

    Stochasti c evolution of genealo- gies of spatial populations: state description, character ization of dynamics and properties

    [DG18] Andrej Depperschmidt and Andreas Greven. Stochasti c evolution of genealo- gies of spatial populations: state description, character ization of dynamics and properties. Genealogies of Interacting Particle Systems (M. Birkner , R . Sun and J. Swart, Eds.), Lect. Notes Ser . Inst. Math. Sci. Natl. Univ . Singap. (forthcom- ing, 2019), World Sci. Publ...

    work page 2019

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    Stochastic evolution of genealogies of spatial populations: state description, characterization of dynamics and properties

    arXiv submitted July 2018, http://arxiv.org/abs/1807.03637. [DG19] Andrej Depperschmidt and Andreas Greven. Tree-valu ed Feller di ffusion. http://arxiv.org/abs/1904.02044, submitted April

    work page internal anchor Pith review Pith/arXiv arXiv 2018

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    Dawson, Andreas Greven, and Iljana Zähle

    [DGZ] Donald A. Dawson, Andreas Greven, and Iljana Zähle. Co ntinuum limits of mul- titype population models and renormalization. In preparat ion (2019). [EK86] S.N. Ethier and T. Kurtz. Markov Processes. Characterization and Convergence. John Wiley, New Y ork,

    work page 2019

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    The hierarchical Cannings process in random environment

    [GdHK17] Andreas Greven, Frank den Hollander, and Anton Kli movsky. The hierarchical Cannings process in random environment. ALEA, Lat. Am. J. Probab. Math. Stat. 15, 295–351 (2018) DOI: 10.30757 /ALEA.v15-14, Preprint

    work page 2018

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    The hierarchical Cannings process in random environment

    Available at https://arxiv.org/abs/1703.03061. [GdHKK14] Andreas Greven, Frank den Hollander, Sandra Klie m, and Anton Klimovsky. The continuum limit of the hierarchical Cannings process: from configurations to ge- nealogies. ALEA, Lat. Am. J. Probab. Math. Stat. 11 (1), 43–140 ,

    work page internal anchor Pith review Pith/arXiv arXiv

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    Renormalisation of hierarchically interacting Cannings processes

    arXiv: 1209.1856v2. [GdHO19a] Andreas Greven, Frank den Hollander, and Margrie t Oomen. Spatial populations with seed-bank: cluster formation and genealogy. Preprint,

    work page internal anchor Pith review Pith/arXiv arXiv

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    Branching trees I: Concatenation and infinite divisibility

    REFERENCES 29 [GGR19] Andreas Greven, Patric Glöde, and Thomas Rippl. Bra nching trees I: Concate- nation and infinite divisibility. http://arxiv.org/abs/1612.01265; EJP V ol.24, paper 52,1-55,

    work page internal anchor Pith review Pith/arXiv arXiv

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    The fixation time of a strongly beneficial allele in a structured population

    arXiv http: //arxiv.org/abs/1402.1769. [GPW09] Andreas Greven, Peter Pfa ffelhuber, and Anita Winter. Convergence in distribution of random metric measure spaces ( Λ-coalescent measure trees). Probab. Theory Related Fields, 145(1-2):285–322,

    work page internal anchor Pith review Pith/arXiv arXiv

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    Branc hing processes — a gen- eral concept

    [GRG] Andreas Greven, Thomas Rippl, and Patric Glöde. Branc hing processes — a gen- eral concept. submitted July 2018, http: //arxiv.org/abs/1807.01921. [Gri17] Max Grieshammer. Measure Representations of Genealogical Processes and Applications to Fleming-Viot Models . PhD thesis,

    work page arXiv 2018

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    Family size decomposition of genealogical trees

    http: //nbn- resolving.de/urn:nbn:de:bvb:29-opus4-85653. [Gri19] Max Griesshammer. Family size decomposition of gen ealogical trees. https://arxiv.org/abs/1903.02782,

    work page internal anchor Pith review Pith/arXiv arXiv 1903

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    The historical process of the spatial Moran model with selec tion and mutation

    [Sei15] Peter Seidel. The historical process of the spatial Moran model with selec tion and mutation. PhD thesis, Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), http://nbn-resolving.de/urn:nbn:de:bvb:29-opus4-59538, 2015

    work page 2015