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Stochastic individual-bases models

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Pith's one-line read Scaling limits of stochastic individual-based models recover the main predictions of adaptive dynamics.

desk verdict Solid lecture notes that cleanly organise 25 years of rigorous stochastic adaptive-dynamics limits; pedagogical synthesis, not new theorems. read the letter →

arxiv 2607.11520 v1 pith:PABX4KQT submitted 2026-07-13 q-bio.PE math.PR

Anton Bovier , Anna Kraut This is my paper · ORCID
classification q-bio.PEmath.PR MSC 60J8060F1792D1592D25
keywords adaptivedynamicsindividual-basedmodelsscalinglimitspolymorphicevolutionsequencecanonicalequationevolutionarybranchingfitnessvalleysLotka–Volterra
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

These lecture notes show that a single class of Markovian individual-based population models, scaled in population size, mutation rate, mutation step size and time, produces a hierarchy of limit processes that match the qualitative picture of adaptive dynamics. Starting from births, deaths, competition and mutations on a trait space, large-population limits yield competitive Lotka–Volterra ecology; rare mutations produce pure-jump trait-substitution or polymorphic evolution sequences; small mutation steps produce the canonical equation of adaptive dynamics and evolutionary branching; moderately rare mutations and longer times produce metastable escapes across fitness valleys. The notes also treat diploid genetics, phenotypic plasticity and changing environments. A reader who cares about evolution as a multi-scale stochastic process obtains a mathematically controlled map from microscopic rules to macroscopic evolutionary dynamics.

What carries the argument

The individual-based Markov process on finite point measures, generated by births without/with mutation, natural death and competition, then rescaled by carrying capacity K (competition of order 1/K), mutation probability µ and step size σ; its law-of-large-numbers limit is a competitive Lotka–Volterra system whose invasion fitnesses and branching-process approximations control successive invasions and the resulting pure-jump or continuous limit processes.

What would settle it

Construct a finite-trait competitive Lotka–Volterra system (with parameters allowed by the model) that, after addition of one positive-invasion-fitness mutant, fails to converge to a unique locally stable equilibrium (e.g., exhibits a stable cycle or multiple attractors) and check whether the stochastic process still produces a well-defined pure-jump polymorphic evolution sequence on the evolutionary time scale.

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Extended reading notes

Core claim

Under stated joint limits of carrying capacity K, mutation probability µ_K and mutation-step size σ_K, the rescaled individual-based process converges (in finite-dimensional distributions or Skorokhod topology) to the polymorphic evolution sequence, the canonical equation of adaptive dynamics, or a metastable jump process across fitness valleys, thereby giving rigorous stochastic foundations for the principal qualitative claims of adaptive dynamics.

Load-bearing premise

After a successful mutant invasion the deterministic competitive Lotka–Volterra system is assumed always to converge to a unique locally strictly stable equilibrium; without that global attractivity the polymorphic evolution sequence is not well-defined as a pure jump process.

Editorial extensions

If this is right

  • Trait-substitution sequences and polymorphic evolution sequences emerge as rigorous K→∞, µ_K≪1/(K ln K) limits of the individual-based process.
  • The canonical equation of adaptive dynamics is recovered as a simultaneous small-step limit under K^{-1/2+α}≪σ_K≪1 and exp(-K^α)≪µ_K≪σ_K^{1+α}/(K ln K).
  • Escape from evolutionary stable conditions across fitness valleys of width L>α occurs on the explicit exponential time scale 1/(K µ_K^L) with explicitly computable rates and success probabilities.
  • Diploid Mendelian models, phenotypic plasticity via fast switches, and time-varying environments admit the same scaling analysis and produce prolonged survival of unfit alleles or accelerated valley crossing.
  • The hierarchy of limits supplies a concrete dictionary between microscopic rates and macroscopic evolutionary regimes (monomorphic walks, branching, metastability).

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same generator-plus-scaling template can be reused to derive hybrid limits in which ecological fluctuations remain of order one while evolutionary jumps still occur, producing stochastic corrections to the canonical equation.
  • Quantitative control of the exit time from an evolutionary stable condition supplies a natural clock for comparing the relative likelihood of valley crossing versus environmental change.
  • Because the notes already treat diploid inheritance, the same programme can be applied to models with recombination or sexual selection to test when genetic diversity persists beyond the monomorphic regime of haploid adaptive dynamics.
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Editorial analysis

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Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 4 minor

Summary. These lecture notes give a rigorous, self-contained synthesis of the stochastic individual-based theory of adaptive dynamics developed over the last 25 years. Starting from a continuous-time Markov process on finite point measures whose generator encodes birth, death, competition and mutation, the notes derive, under successive scalings of carrying capacity K, mutation probability µ_K and mutation step size σ_K, the principal macroscopic objects of adaptive dynamics: the law of large numbers (competitive Lotka–Volterra), the polymorphic evolution sequence (PES), the canonical equation of adaptive dynamics (CEAD), evolutionary branching, and metastable jumps across fitness valleys. The exposition is organised in three parts—background and tools, the various scaling regimes, and extensions (diploidy, phenotypic plasticity, changing environments)—and is aimed at master’s students with a solid background in Markov processes.

Significance. The notes fill a genuine pedagogical gap: they collect, in a single coherent narrative, a body of technically demanding results that previously existed only as a scattered sequence of research papers (Champagnat, Méléard, Fournier, Bovier et al.). By emphasising the interplay of the three scaling parameters and by making the underlying martingale, branching-process and large-deviation arguments explicit, they render the field accessible to a new generation of probabilists and mathematical biologists. The derivations are standard and already peer-reviewed; the value lies in the unified presentation, the careful statement of hypotheses (e.g. Assumption 5.1), and the inclusion of more recent extensions. No new theorems are claimed, but the synthesis itself is a substantial contribution to the literature.

minor comments (4)
  1. The title on the arXiv page and in the manuscript header reads “Stochastic individual-bases models”; this is almost certainly a typographical error for “individual-based” and should be corrected before any formal publication.
  2. In several places (e.g. the statement of Theorem 5.2 and the discussion surrounding Assumption 5.1) the text refers to “the polymorphic evolution sequence” without immediately recalling that its well-definedness as a pure jump process rests on the global attractivity hypothesis. A short parenthetical reminder would help the student reader.
  3. Chapter 6 ends with a series of conjectures on evolutionary branching under simultaneous limits; while clearly labelled as such, a brief pointer to the most recent progress (or lack thereof) would be useful for readers who wish to pursue the open problems.
  4. A few figures (e.g. the broken-line pictures in Chapters 7–9) are described only in the text; if the notes are to be used as a course text, the actual plots should be included or a repository link provided.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: lecture notes derive scaling limits from the microscopic Markov generator via standard weak-convergence, branching-process and large-deviation arguments; target objects (PES, CEAD, metastable jumps) are not inserted by definition.

full rationale

The manuscript is an explicit pedagogical synthesis of existing rigorous results (primarily Champagnat–Méléard and collaborators, with some co-authored papers). The individual-based process is constructed from its generator (2.4); all subsequent limits (LLN to Lotka–Volterra, PES under µ_K ≪ 1/(K ln K), CEAD under simultaneous K,µ,σ o limits, escape from ESC via fitness valleys, etc.) are obtained by coupling to branching processes, Wentzell–Freidlin exit estimates, and tightness arguments that do not presuppose the macroscopic objects. Self-citations appear only as references to prior proofs that are independent of the present exposition; they are not load-bearing uniqueness theorems that force the conclusions. Assumption 5.1 (global attractivity of the (d+1)-dimensional competitive LV system) is stated as a hypothesis under which the PES is well-defined as a pure jump process, not derived circularly. No parameter is fitted to data and then “predicted,” no ansatz is smuggled via self-citation, and no known empirical pattern is merely renamed. Circularity burden is therefore minimal (score 1 for ordinary self-citation of the authors’ earlier technical papers).

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

As lecture notes the text inherits the standard axioms of continuous-time Markov processes on measure spaces and the classical competitive Lotka–Volterra theory. No free parameters are fitted; the only modelling choices are the usual boundedness and positivity assumptions on birth, death and competition rates that guarantee non-explosion. No new physical entities are postulated.

assumptions (3)
  • standard math The individual-based process is a pure-jump Markov process on the space of finite point measures whose generator is given by the birth, death and mutation rates of Assumption 2.2.
    Standard construction of continuous-time Markov chains; existence and uniqueness follow from bounded rates and the non-explosion lemma (Lemma 2.6).
  • domain assumption Competitive Lotka–Volterra systems of any finite dimension possess a unique locally strictly stable equilibrium whenever the invasion-fitness conditions of Definition 3.7 hold (Assumption 5.1).
    Required for the polymorphic evolution sequence to be a well-defined pure-jump process; not true for arbitrary competition kernels, but standard in the adaptive-dynamics literature.
  • domain assumption Mutation kernels have bounded support and the trait space is a Polish space (or a finite graph in later chapters).
    Technical hypothesis that keeps the state space manageable and allows tightness arguments.

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Pith. "Pith review of Stochastic individual-bases models." pith.science (2026). https://pith.science/paper/PABX4KQT

@misc{pith2026260711520,
  author       = {Pith},
  title        = {Pith review of: Stochastic individual-bases models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PABX4KQT}},
  note         = {Machine review of arXiv:2607.11520}
}
read the original abstract

These are lecture notes for an advanced topics course in the master's programme at Bonn University. It aims to give a concise review of some of the work that has been done around the topic of adaptive dynamics from a rigorous stochastic point of view over the last 25 years and to make this area accessible to students with a good knowledge of probability in general, and the theory of Markov processes in particular. Our emphasis is on the issue of emerging scaling limits, where scaling parameters are time, population size, mutation rates, and mutation step size. These allow to exhibit within a fairly simple class of models a variety of biologically relevant phenomena. These notes are organised in three parts. Part 1 presents some historical background as well as the mathematical setting and main tools. Part 2 is the core of the notes and discusses the various scaling regimes and scaling limits. Part 3 looks at some extensions on the basic model, notably diploid models, phenotypic plasticity, and effects of environmental changes over time.

Figures

Figures reproduced from arXiv: 2607.11520 by the authors.

Figure 6.4
Figure 6.4. Thus, we expect the stochastic system to stay close to this invariant manifold [PITH_FULL_IMAGE:figures/full_fig_p078_6_4.png] view at source ↗
Figure 9.1
Figure 9.1. 0 1 2 3 4 α β0(t) β1(t) β2(t) β3(t) 1 1 − 1/α 1 − 2/α0 O  1/Kµ4 K  O (ln K) β4(t) [PITH_FULL_IMAGE:figures/full_fig_p106_9_1.png] view at source ↗
Figure 9
Figure 9. visualises the mutation spreading neighbourhood (blue) for a set of coex [PITH_FULL_IMAGE:figures/full_fig_p109_9.png] view at source ↗
Figures from the paper (3 more)
Figure 9
Figure 9. Figure 9: depicts the directed trait graph and individual growth rates [PITH_FULL_IMAGE:figures/full_fig_p114_9.png]
Figure 9.6
Figure 9.6. Figure 9.6: {2} {4a} {4b} {6a} {6b} {7a} {9b} {9a} GESC {2} {4a} {4b} {6a} {6b} {7a} {9b} {9a} G 2 {4b} {6a} {6b} {9b} {9a} G 3 2 2 2 2 2 3 3 3 [PITH_FULL_IMAGE:figures/full_fig_p116_9_6.png]
Figure 10.5
Figure 10.5. Figure 10.5: Theorem 10.9. Consider the dynamical system (10.87) started with initial conditions (10.89). Suppose the following Assumptions E on the parameters hold: (E1) ∆ is sufficiently small, (E2) b is sufficiently large, (E3) 0 ≤ η < c/2. Then the system converges to the fi…

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Works this paper leans on

121 extracted references · 2 linked inside Pith

  1. [1]

    Aldous,Stopping times and tightness, Ann

    D. Aldous,Stopping times and tightness, Ann. Probability, 6 (1978), pp. 335–340

    1978

  2. [2]

    V . I. Arnold,Ordinary differential equations, Springer Textbook, Springer-Verlag, Berlin, 1992

    1992

  3. [3]

    K. B. Athrey a,Some results on multitype continuous time Markov branching processes, Ann. Math. Statist., 39 (1968), pp. 347–357

    1968

  4. [4]

    K. B. Athrey a andP. E. Ney,Branching processes, Springer-Verlag, New York-Heidelberg, 1972. Die Grundlehren der mathematischen Wissenschaften, Band 196

    1972

  5. [5]

    Baar andA

    M. Baar andA. Bovier,The polymorphic evolution sequence for populations with phenotypic plasticity, Electron. J. Probab., 23 (2018), pp. 1–27

    2018

  6. [6]

    M. Baar, A. Bovier,andN. Champagnat,From stochastic, individual-based models to the canonical equation of adaptive dynamics in one step, Ann. Appl. Probab., 27 (2017), pp. 1093–1170

    2017

  7. [7]

    M. Baar, L. Coquille, H. Mayer, M. H ¨olzel, M. Roga v a, T. T¨uting,andA. Bovier,A stochastic model for immunotherapy of cancer, Scientific Reports, 6 (2016), p. 24169

    2016

  8. [8]

    Bansaye andS

    V . Bansaye andS. M ´el´eard,Stochastic models for structured populations, vol. 1 of Mathematical Bio- sciences Institute Lecture Series. Stochastics in Biological Systems, Springer, Cham; MBI Mathematical Biosciences Institute, Ohio State University, Columbus, OH, 2015. Scaling limits and long time behavior

    2015

Show all 121 references
  1. [9]

    H. J. E. Beaumont, J. Gallie, C. Kost, G. C. Ferguson,andP. B. Rainey,Experimental evolution of bet hedging, Nature, 462 (2009), pp. 90–93

    2009

  2. [10]

    Berestycki, E

    J. Berestycki, E. Brunet,andZ. Shi,The number of accessible paths in the hypercube, Bernoulli, 22 (2016), pp. 653–680

    2016

  3. [11]

    ,Accessibility percolation with backsteps, ALEA Lat. Am. J. Probab. Math. Stat., 14 (2017), pp. 45–62

    2017

  4. [12]

    Bolker andS

    B. Bolker andS. W. Pacala,Using moment equations to understand stochastically driven spatial pattern formation in ecological systems, Theor. Popul. Biol., 52 (1997), pp. 179–197

    1997

  5. [13]

    B. M. Bolker andS. W. Pacala,Spatial moment equations for plant competition: understanding spatial strategies and the advantages of short dispersal, Am. Nat., 153 (1999), pp. 575–602

    1999

  6. [14]

    Bovier, L

    A. Bovier, L. Coquille,andR. Neukirch,The recovery of a recessive allele in a Mendelian diploid model, J. Math. Biol., 77 (2018), pp. 971–1033

    2018

  7. [15]

    Bovier, L

    A. Bovier, L. Coquille,andC. Smadi,Crossing a fitness valley as a metastable transition in a stochastic population model, Ann. Appl. Probab., 29 (2019), pp. 3541–3589

    2019

  8. [16]

    Bovier andF.denHollander,Metastability: A potential-theoretic approach, vol

    A. Bovier andF.denHollander,Metastability: A potential-theoretic approach, vol. 351 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer, Cham, 2015

    2015

  9. [17]

    Bovier andS.-D

    A. Bovier andS.-D. Wang,Trait substitution trees on two time scales analysis, Markov. Proc. Rel. Fields, 19 (2013), pp. 607–642

    2013

  10. [18]

    Bramson,Convergence of solutions of the Kolmogorov equation to travelling waves, Mem

    M. Bramson,Convergence of solutions of the Kolmogorov equation to travelling waves, Mem. Amer. Math. Soc., 44 (1983), pp. iv+190

    1983

  11. [19]

    B ¨urger,The mathematical theory of selection, recombination, and mutation, Wiley Series in Mathe- matical and Computational Biology, John Wiley & Sons, Ltd., Chichester, 2000

    R. B ¨urger,The mathematical theory of selection, recombination, and mutation, Wiley Series in Mathe- matical and Computational Biology, John Wiley & Sons, Ltd., Chichester, 2000

    2000

  12. [20]

    Champagnat,A microscopic interpretation for adaptive dynamics trait substitution sequence models, Stochastic Processes Appl., 116 (2006), pp

    N. Champagnat,A microscopic interpretation for adaptive dynamics trait substitution sequence models, Stochastic Processes Appl., 116 (2006), pp. 1127–1160

    2006

  13. [21]

    Champagnat, R

    N. Champagnat, R. Ferri`ere,andG. BenArous,The canonical equation of adaptive dynamics: A math- ematical view, Selection, 2 (2001), pp. 73–83

    2001

  14. [22]

    Champagnat, R

    N. Champagnat, R. Ferri`ere,andS. M ´el´eard,From individual stochastic processes to macroscopic mod- els in adaptive evolution, Stochastic Models, 24 (2008), pp. 2–44

    2008

  15. [23]

    Champagnat andV

    N. Champagnat andV . Hass,Convergence of population processes with small and frequent mutations to the canonical equation of adaptive dynamics, Ann. Appl. Probab., 35 (2025), pp. 1–63. 154 12. EVOLUTION IN A CHANGING ENVIRONMENT

    2025

  16. [24]

    Champagnat andS

    N. Champagnat andS. M ´el´eard,Polymorphic evolution sequence and evolutionary branching, Probab. Theor. Rel. Fields, 151 (2011), pp. 45–94

    2011

  17. [25]

    Champagnat, S

    N. Champagnat, S. M ´el´eard,andV . C. Tran,Stochastic analysis of emergence of evolutionary cyclic behavior in population dynamics with transfer, Ann. Appl. Probab., 31 (2021), pp. 1820–1867

    2021

  18. [26]

    Collet, S

    P. Collet, S. M´el´eard,andJ. A. J. Metz,A rigorous model study of the adaptive dynamics of Mendelian diploids, J. Math. Biol., 67 (2013), pp. 569–607

    2013

  19. [27]

    Coquille, A

    L. Coquille, A. Kraut,andC. Smadi,Stochastic individual-based models with power law mutation rate on a general finite trait space, Electron. J. Probab., 26 (2021), pp. Paper No. 123, 37

    2021

  20. [28]

    Coron,Stochastic modeling of density-dependent diploid populations and the extinction vortex, Adv

    C. Coron,Stochastic modeling of density-dependent diploid populations and the extinction vortex, Adv. Appl. Probab., 46 (2014), pp. 446–477

    2014

  21. [29]

    ,Slow-fast stochastic diffusion dynamics and quasi-stationarity for diploid populations with vary- ing size, J. Math. Biol., 72(1-2) (2016), pp. 171–202

    2016

  22. [30]

    Coron, S

    C. Coron, S. M ´el´eard, E. Porcher,andA. Robert,Quantifying the mutational meltdown in diploid populations, Am. Nat., 181 (2013), pp. 623–636

    2013

  23. [31]

    M. C. Cowperthw aite, J. J. Bull,andL. A. Meyers,From bad to good: Fitness reversals and the ascent of deleterious mutations, PLoS Comp. Biol., 2 (2006), p. e141

    2006

  24. [32]

    J. F. Crow, M. Kimura,andOthers,An introduction to population genetics theory, New York, Evanston and London: Harper & Row, Publishers, 1970

    1970

  25. [33]

    Darwin,The Origin of Species, John Murray, London, 1859

    C. Darwin,The Origin of Species, John Murray, London, 1859

  26. [34]

    Darwin,Temple of Nature; or, The Origin of Society, John W

    E. Darwin,Temple of Nature; or, The Origin of Society, John W. Butler, and Bonsal and Niles, Baltimore, 1804

  27. [35]

    D. A. Dawson,Introductory Lectures on Stochastic Population Systems, arXiv 1705.03781, 2017

    2017 arXiv

  28. [36]

    Versuche und Beobachtungen über die Entstehung von Arten im Pflanzenreich, Veit & Comp., Leipzig, 1901-03

    H.deVries,Die Mutationstheorie. Versuche und Beobachtungen über die Entstehung von Arten im Pflanzenreich, Veit & Comp., Leipzig, 1901-03

    1901

  29. [37]

    D. L. DeAngelis andV . Grimm,Individual-based models in ecology after four decades, F1000Prime Reports, 6 (2014)

    2014

  30. [38]

    Dieckmann andR

    U. Dieckmann andR. Law,The dynamical theory of coevolution: a derivation from stochastic ecological processes, J. Math. Biol., 34 (1996), pp. 579–612

    1996

  31. [39]

    Dieckmann, R

    ,Moment approximations of individual-based models, in The geometry of ecological interactions: simplifying spatial complexity, U. Dieckmann, R. Law, and J. A. J. Metz, eds., Cambridge University Press, 2000, pp. 252–270

    2000

  32. [40]

    C. R. Doering, C. Mueller,andP. Smereka,Interacting particles, the stochastic Fisher-Kolmogorov- Petrovsky-Piscounov equation, and duality, Phys. A, 325 (2003), pp. 243–259. Stochastic systems: from randomness to complexity (Erice, 2002)

    2003

  33. [41]

    Dupuis andR

    P. Dupuis andR. S. Ellis,A weak convergence approach to the theory of large deviations, Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons, Inc., New York, 1997. A Wiley-Interscience Publication

    1997

  34. [42]

    Esser andA

    M. Esser andA. Kraut,Crossing a fitness valley in a changing environment: With and without pit stop. arXiv:2503.19766, 2025

    2025 arXiv

  35. [43]

    Esser andA

    M. Esser andA. Kraut,Effective growth rates in a periodically changing environment: from mutation to invasion, Stochastic Process. Appl., 184 (2025), pp. Paper No. 104598, 36

    2025

  36. [44]

    A. M. Etheridge,Survival and extinction in a locally regulated population, Ann. Appl. Probab., 14 (2004), pp. 188–214

    2004

  37. [45]

    S. N. Ethier andT. G. Kurtz,Markov processes, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1986

    1986

  38. [46]

    S. N. Ethier andM. F. Norman,Error estimate for the diffusion approximation of the Wright–Fisher model, Proc. Natl. Acad. Sci., 74 (1977), pp. 5096–8. REFERENCES 155

    1977

  39. [47]

    W. J. Ewens,Mathematical Population Genetics. I Theoretical Introduction, vol. 27 of Interdisciplinary Applied Mathematics, Springer-Verlag, New York, second ed., 2004

    2004

  40. [48]

    R. A. Fisher,The correlation between relatives on the supposition of Mendelian inheritance, Trans. Roy. Soc. Edinb., 42 (1918), pp. 399–433

    1918

  41. [49]

    Eugenics, 7 (1937), pp

    ,The wave of advance of an advantageous gene, Ann. Eugenics, 7 (1937), pp. 355–369

    1937

  42. [50]

    Fournier andS

    N. Fournier andS. M ´el´eard,A microscopic probabilistic description of a locally regulated population and macroscopic approximations, Ann. Appl. Probab., 14 (2004), pp. 1880–1919

    2004

  43. [51]

    R. E. Franklin andG. R.G.,Molecular configuration in sodium thymonucleate, Nature, 171 (1953), pp. 740–741

    1953

  44. [52]

    M. I. Freidlin andA. D. Wentzell,Random perturbations of dynamical systems, vol. 260 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer, Hei- delberg, third ed., 2012

    2012

  45. [53]

    Fusco andA

    G. Fusco andA. Minelli,Phenotypic plasticity in development and evolution: facts and concepts, Phil. Trans. Royal Soc. London B: Biol. Sci., 365 (2010), pp. 547–556

    2010

  46. [54]

    Giachetti andJ

    C. Giachetti andJ. J. Holland,Altered replicase specificity is responsible for resistance to defective interfering particle interference of an sdi-mutant of vesicular stomatitis virus, Jour. Virol., 62 (1988), pp. 3614–3621

    1988

  47. [55]

    D. T. Gillespie,A general method for numerically simulating the stochastic time evolution of coupled chemical reactions, J. Computational Phys., 22 (1976), pp. 403–434

    1976

  48. [56]

    J. H. Gillespie,Molecular evolution over the mutational landscape, Evolution, 38 (1984), pp. 1116–1129

    1984

  49. [57]

    Grimm andS

    V . Grimm andS. F. Railsback,Individual-based modeling and ecology, Princeton Series in Theoretical and Computational Biology, Princeton University Press, Princeton, NJ, 2005

    2005

  50. [58]

    J. B. Haldane,A mathematical theory of natural and artificial selection. Part I, Trans. Camb. Phil. Soc., 23 (1924), pp. 19–41

    1924

  51. [59]

    Part II, Trans

    ,A mathematical theory of natural and artificial selection. Part II, Trans. Camb. Phil. Soc., Biol, Sci., 1 (1924), pp. 158–163

    1924

  52. [60]

    G. H. Hardy,Mendelian proportions in a mixed population, Science, 28 (1908), pp. 49–50

    1908

  53. [61]

    Hofbauer andK

    J. Hofbauer andK. Sigmund,Adaptive dynamics and evolutionary stability, Appl Math Lett, 3 (1990), pp. 75–79

    1990

  54. [62]

    ,Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, 1998

    1998

  55. [63]

    H ¨olzel, A

    M. H ¨olzel, A. Bovier,andT. T ¨uting,Plasticity of tumour and immune cells: a source of heterogeneity and a cause for therapy resistance?, Nat. Rev. Cancer, 13 (2013), pp. 365–376

    2013

  56. [64]

    Houchmandzadeh andM

    B. Houchmandzadeh andM. Vallade,Fisher waves: An individual-based stochastic model, Phys Rev E., 96 (2017), p. 012414

    2017

  57. [65]

    Jabin andG

    P.-E. Jabin andG. Raoul,On selection dynamics for competitive interactions, J. Math. Biol., 63 (2011), pp. 493–517

    2011

  58. [66]

    Jain,Evolutionary dynamics of the most populated genotype on rugged fitness landscapes, Phys

    K. Jain,Evolutionary dynamics of the most populated genotype on rugged fitness landscapes, Phys. Rev. E, 76 (2007), p. 031922

    2007

  59. [67]

    Jain andJ

    K. Jain andJ. Krug,Evolutionary trajectories in rugged fitness landscapes, J. Statist. Mech., P04008 (2005)

    2005

  60. [68]

    1275–1288

    ,Deterministic and stochastic regimes of asexual evolution on rugged fitness landscapes, Genetics, 175 (2007), pp. 1275–1288

    2007

  61. [69]

    C.d. L. Jean-BaptistePierreAntoine deMonet,Philosophie Zoologique, Dentu, Paris, 1809

  62. [70]

    Kauffman andS

    S. Kauffman andS. Levin,Towards a general theory of adaptive walks on rugged landscapes, J. Theoret. Biol., 128 (1987), pp. 11–45

    1987

  63. [71]

    S. A. Kauffman,The origins of order: Self-organization and selection in evolution, in Spin glasses and biology, World Scientific, 1992, pp. 61–100. 156 12. EVOLUTION IN A CHANGING ENVIRONMENT

    1992

  64. [72]

    Kesten andB

    H. Kesten andB. P. Stigum,Additional limit theorems for indecomposable multidimensional Galton- Watson processes, Ann. Math. Statist., 37 (1966), pp. 1463–1481

    1966

  65. [73]

    ,A limit theorem for multidimensional Galton-Watson processes, Ann. Math. Statist., 37 (1966), pp. 1211–1223

    1966

  66. [74]

    ,Limit theorems for decomposable multi-dimensional Galton-Watson processes, J. Math. Anal. Appl., 17 (1967), pp. 309–338

    1967

  67. [75]

    H. K. Khalil,Nonlinear systems, Macmillan Publishing Company, New York, 1992

    1992

  68. [76]

    Kimura,Solution of a process of random genetic drift with a continuous model, Proc

    M. Kimura,Solution of a process of random genetic drift with a continuous model, Proc. Natl. Acad. Sci., 41 (1955), pp. 144–150

    1955

  69. [77]

    Kisdi andS

    É. Kisdi andS. A. H. Geritz,Adaptive dynamics in allele space: Evolution of genetic polymorphism by small mutations in a heterogeneous environment, Evolution, 53 (1999), pp. 993–1008

    1999

  70. [78]

    A. N. Kolmogorov, I. G. Petrovsky,andN. S. Piscounov,Etude de l’équation de la diffusion avec crois- sance de la quantité de matière et son application à un problème biologique, Moscow Univ. Math. Bull., 1 (1937), pp. 1–25

    1937

  71. [79]

    Kraut andA

    A. Kraut andA. Bovier,From adaptive dynamics to adaptive walks, J. Math. Biol., 79 (2019), pp. 1699– 1747

    2019

  72. [80]

    Krug,Accessibility percolation in random fitness landscapes, in Probabilistic Structures in Evolution, E

    J. Krug,Accessibility percolation in random fitness landscapes, in Probabilistic Structures in Evolution, E. Baake and A. Wakolbinger, eds., EMS Press, Berlin, 2021, pp. 1–22

    2021

  73. [81]

    Krug andC

    J. Krug andC. Karl,Punctuated evolution for the quasispecies model, Physica A, 318 (2003), pp. 137– 143

    2003

  74. [82]

    T. G. Kurtz,Solutions of ordinary differential equations as limits of pure jump Markov processes, J. Appl. Probability, 7 (1970), pp. 49–58

    1970

  75. [83]

    T. G. Kurtz,Limit theorems for sequences of jump Markov processes approximating ordinary differential processes, J. Appl. Probability, 8 (1971), pp. 344–356

    1971

  76. [84]

    T. G. Kurtz,Strong approximation theorems for density dependent Markov chains, Stochastic Process. Appl., 6 (1977/78), pp. 223–240

    1977

  77. [85]

    Lennon, F.denHollander, M

    J. Lennon, F.denHollander, M. Wilke-Berenguer,andJ. Blath,Principles of seed banks and the emergence of complexity from dormancy, Nat. Comm., 12 (2021), p. 4807

    2021

  78. [86]

    R. E. Lenski, C. Ofria, R. T. Pennock,andC. Adami,The evolutionary origin of complex features, Nature, 423 (2003), pp. 139–144

    2003

  79. [87]

    A. J. Lotka,Quantitative studies in epidemiology, Nature, 88 (1912), pp. 497–498

    1912

  80. [88]

    Maisnier-Patin, O

    S. Maisnier-Patin, O. G. Berg, L. Liljas,andD. I. Andersson,Compensatory adaptation to the deleteri- ous effect of antibiotic resistance in salmonella typhimurium, Mol. Microbiol., 46 (2002), pp. 355–366

    2002

  81. [89]

    Malthus,An Essay on the Principle of Population as it Affects the Future Improvement of Society, with Remarks on the Speculations of Mr

    T. Malthus,An Essay on the Principle of Population as it Affects the Future Improvement of Society, with Remarks on the Speculations of Mr. Goodwin, M. Condorcet and Other Writers, J. Johnson in St Paul’s Church-yard, London, 1798

  82. [90]

    MaynardSmith, in The Scientist Speculates: an Anthology of Partly-Baked Ideas, I

    J. MaynardSmith, in The Scientist Speculates: an Anthology of Partly-Baked Ideas, I. Good, ed., Basic Books, New York, 1962, pp. 252–256

    1962

  83. [91]

    ,Natural selection and the concept of a protein space, Nature, 225 (1970), pp. 563–564

    1970

  84. [92]

    M ´el´eard,Modèles aléatoires en ecologie et evolution, vol

    S. M ´el´eard,Modèles aléatoires en ecologie et evolution, vol. 77 of Mathématiques & Applications (Berlin) [Mathematics & Applications], Springer-Verlag, Berlin, 2016

    2016

  85. [93]

    Mendel,Versuche über Pflanzen-Hybriden, Verhandlungen des naturforschenden Vereines in Brünn, IV (1865), pp

    G. Mendel,Versuche über Pflanzen-Hybriden, Verhandlungen des naturforschenden Vereines in Brünn, IV (1865), pp. 3–47

  86. [94]

    ,Über einige aus künstlicher Befruchtung gewonnenen Hieracium-Bastarde, Verhandlungen des naturforschenden Vereines in Brünn, VIII (1869), pp. 26–31

  87. [95]

    J. A. Metz,Adaptive dynamics, in Encyclopedia of Theoretical Ecology, Cambridge University Press, Cambridge, 2012. REFERENCES 157

    2012

  88. [96]

    J. A. J. Metz,Invasion fitness, canonical equations, and global invasion criteria for Mendelian popula- tions, in Elements of Adaptive Dynamics, U. Dieckmann and J. A. J. Metz, eds., Cambridge University Press, 2012

    2012

  89. [97]

    J. A. J. Metz, S. A. H. Geritz, G. Mesz ´ena, F. J. A. Jacobs,andJ. S.v anHeerw aarden,Adaptive dy- namics, a geometrical study of the consequences of nearly faithful reproduction, in Stochastic and spatial structures of dynamical systems (Amsterdam, 1995), vol. 45 of Konink. ...

    1995

  90. [98]

    P. A. P. Moran,Random processes in genetics, Math. Proc. Cambridge Phil. Soc., 54 (1958), pp. 60–71

    1958

  91. [99]

    Mukherjee,The Gene, Bodley, London, 2016

    S. Mukherjee,The Gene, Bodley, London, 2016

    2016

  92. [100]

    Nagylaki,Introduction to theoretical population genetics, vol

    T. Nagylaki,Introduction to theoretical population genetics, vol. 21 of Biomathematics, Springer-Verlag, Berlin, 1992

    1992

  93. [101]

    Neidhart andJ

    J. Neidhart andJ. Krug,Adaptive walks and extreme value theory, Phys. Rev. Lett., 107 (2011), p. 178102

    2011

  94. [102]

    Neukirch andA

    R. Neukirch andA. Bovier,Survival of a recessive allele in a Mendelian diploid model, J. Math. Biol., 75 (2017), pp. 145–198

    2017

  95. [103]

    Now ak andJ

    S. Now ak andJ. Krug,Analysis of adaptive walks on NK fitness landscapes with different interaction schemes, J. Stat. Mech. Theory Exp., 2015 (2015), p. P06014

    2015

  96. [104]

    P. J. O’Hara, S. T. Nichol, F. M. Horodyski,andJ. J. Holland,Vesicular stomatitis virus defective interfering particles can contain extensive genomic sequence rearrangements and base substitutions, Cell, 36 (1984), pp. 915–924

    1984

  97. [105]

    H. A. Orr,A minimum on the mean number of steps taken in adaptive walks, J. Theoret. Biol., 220 (2003), pp. 241–247

    2003

  98. [106]

    S. P ´enisson,Conditional limit theorems for multitype branching processes and illustration in epidemio- logical risk analysis, PhD thesis, Universität Potsdam, Potsdam (Germany), 2010

    2010

  99. [107]

    L. C. G. Rogers andD. Williams,Diffusions, Markov processes, and martingales. Vol. 1, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2000. Foundations, Reprint of the second (1994) edition

    2000

  100. [108]

    ,Diffusions, Markov processes, and martingales. Vol. 2, Cambridge Mathematical Library, Cam- bridge University Press, Cambridge, 2000. Itô calculus, Reprint of the second (1994) edition

    2000

  101. [109]

    Schmiegelt andJ

    B. Schmiegelt andJ. Krug,Evolutionary accessibility of modular fitness landscapes, J. Stat. Phys., 154 (2014), pp. 334–355

    2014

  102. [110]

    S. J. Schrag, V . Perrot,andB. R. Levin,Adaptation to the fitness costs of antibiotic resistance in es- cherichia coli, Proc. Royal Soc. London B: Biological Sciences, 264 (1997), pp. 1287–1291

    1997

  103. [111]

    B. A. Sew astjanow,Verzweigungsprozesse, R. Oldenbourg Verlag, Munich-Vienna, 1975

    1975

  104. [112]

    S. J. Smale,On the differential equations of species in competition, J. Math. Biol., 3 (1976), pp. 5–7

    1976

  105. [113]

    Uchiy ama,The behavior of solutions of some nonlinear diffusion equations for large time, J

    K. Uchiy ama,The behavior of solutions of some nonlinear diffusion equations for large time, J. Math. Kyoto Univ., 18 (1978), pp. 453–508

    1978

  106. [114]

    Volterra,Variations and fluctuations of the number of individuals in animal species living together, J

    V . Volterra,Variations and fluctuations of the number of individuals in animal species living together, J. Conseil Int. Explor. Mer., 3 (1928), pp. 1–51

    1928

  107. [115]

    Wang,Multi-scale analysis of adaptive population dynamics, ph.d

    S.-D. Wang,Multi-scale analysis of adaptive population dynamics, ph.d. thesis, Bonn University, 2011

    2011

  108. [116]

    J. D. Watson andF. H. C. Crick,A structure for deoxyribose nucleic acid, Nature, 171 (1953), pp. 737– 738

    1953

  109. [117]

    Weinberg,Über den Nachweis der Vererbung beim Menschen, Jahreshefte des Vereins für vaterländis- che Naturkunde in Württemberg, 64 (1908), pp

    W. Weinberg,Über den Nachweis der Vererbung beim Menschen, Jahreshefte des Vereins für vaterländis- che Naturkunde in Württemberg, 64 (1908), pp. 368–382

    1908

  110. [118]

    M. H. Wilkins, A. R. Stokes,andH. R. Wilson,Molecular structure of deoxypentose nucleic acids, Nature, 171 (1953), pp. 738–40

    1953

  111. [119]

    S. G. Wright,Evolution in Mendelian populations, Genetics, 16 (1931), pp. 97–157. 158 12. EVOLUTION IN A CHANGING ENVIRONMENT

    1931

  112. [120]

    G. U. Yule,On the theory of inheritance of quantitative compound characters on the basis of Mendel’s laws: a preliminary note, Spottiswoode & Company, Limited, 1907

    1907

  113. [121]

    M. L. Zeeman,Hopf bifurcations in competitive three-dimensional Lotka-Volterra systems, Dynam. Sta- bility Systems, 8 (1993), pp. 189–217. Index adaptive dynamics, 10, 50, 53 adaptive speciation, 11 adaptive walk, 49 greedy, 51 adaptive walks, 11, 44 allele, 6 bet hedging, 131...

    1993

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