Measure framework for the pure selection equation: global well posedness and numerical investigations
MathematicS In Action, Maths Bio, Volume 12 (2023) no. 1, pp. 155-173.

We study the classic pure selection integrodifferential equation, stemming from adaptative dynamics, in a measure framework by mean of duality approach. After providing a well posedness result under fairly general assumptions, we focus on the asymptotic behaviour of various cases, illustrated by some numerical simulations.

Published online:
DOI: 10.5802/msia.36
Classification: 45J05, 45M15, 65R20, 92D15, 92D40
Keywords: population dynamics, selection model, measure solutions, semigroup, asymptotic behaviour, simulations
Hugo Martin 1

1 IRMAR, Université de Rennes, CNRS, IRMAR - UMR 6625, 35000 Rennes, France and INRAE, Agrocampus Ouest, Université de Rennes, IGEPP, Le Rheu, France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Hugo Martin. Measure framework for the pure selection equation: global well posedness and numerical investigations. MathematicS In Action, Maths Bio, Volume 12 (2023) no. 1, pp. 155-173. doi : 10.5802/msia.36. https://msia.centre-mersenne.org/articles/10.5802/msia.36/

[1] Amal Aafif; Juan Lin Selection-mutation process of RNA viruses, Phys. Rev. E, Volume 57 (1998) no. 2, pp. 2471-2474 | DOI

[2] Azmy S. Ackleh; John Cleveland; Horst R. Thieme Population dynamics under selection and mutation: Long-time behavior for differential equations in measure spaces, J. Differ. Equations, Volume 261 (2016) no. 2, pp. 1472-1505 | DOI | MR | Zbl

[3] Azmy S. Ackleh; Ben G. Fitzpatrick; Horst R. Thieme Rate distributions and survival of the fittest: a formulation on the space of measures, Discrete Contin. Dyn. Syst., Ser. B, Volume 5 (2005) no. 4, pp. 917-928 | DOI | MR | Zbl

[4] Azmy S. Ackleh; David F. Marshall; Henry E. Heatherly; Ben G. Fitzpatrick Survival of the fittest in a generalized logistic model, Math. Models Methods Appl. Sci., Volume 09 (1999) no. 09, pp. 1379-1391 | DOI | MR | Zbl

[5] Azmy S. Ackleh; Nicolas Saintier Diffusive limit to a selection-mutation equation with small mutation formulated on the space of measures, Discrete Contin. Dyn. Syst., Ser. B, Volume 26 (2021) no. 3, pp. 1469-1497 | DOI | MR | Zbl

[6] Aleksandra Ardaševa; Robert A. Gatenby; Alexander R. A. Anderson; Helen M. Byrne; Philip K. Maini; Tommaso Lorenzi Evolutionary dynamics of competing phenotype-structured populations in periodically fluctuating environments, J. Math. Biol., Volume 80 (2019) no. 3, pp. 775-807 | DOI | MR | Zbl

[7] Olivier Bonnefon; Jérôme Coville; Guillaume Legendre Concentration phenomenon in some non-local equation, Discrete Contin. Dyn. Syst., Ser. B, Volume 22 (2017) no. 3, pp. 763-781 | DOI | MR | Zbl

[8] R. Burger Mathematical Theory of Selection, John Wiley & Sons, 2000, 420 pages https://www.ebook.de/de/product/3601775/burger_mathematical_theory_of_selection.html

[9] Jan-Erik Busse; Sílvia Cuadrado; Anna Marciniak-Czochra Local asymptotic stability of a system of integro-differential equations describing clonal evolution of a self-renewing cell population under mutation, J. Math. Biol., Volume 84 (2022), 10 | DOI | MR | Zbl

[10] Jan-Erik Busse; Piotr Gwiazda; Anna Marciniak-Czochra Mass concentration in a nonlocal model of clonal selection, J. Math. Biol., Volume 73 (2016) no. 4, pp. 1001-1033 | DOI | MR | Zbl

[11] Àngel Calsina; Sílvia Cuadrado Asymptotic stability of equilibria of selection-mutation equations, J. Math. Biol., Volume 54 (2006) no. 4, pp. 489-511 | DOI | MR | Zbl

[12] Vincent Calvez; Susely Figueroa Iglesias; Hélène Hivert; Sylvie Méléard; Anna Melnykova; Samuel Nordmann Horizontal gene transfer: numerical comparison between stochastic and deterministic approaches, ESAIM, Proc. Surv., Volume 67 (2020), pp. 135-160 | DOI | MR | Zbl

[13] José A. Cañizo; José A. Carrillo; Sílvia Cuadrado Measure Solutions for Some Models in Population Dynamics, Acta Appl. Math., Volume 123 (2012) no. 1, pp. 141-156 | DOI | MR | Zbl

[14] Cécile Carrère; Grégoire Nadin Influence of mutations in phenotypically-structured populations in time periodic environment, Discrete Contin. Dyn. Syst., Ser. B, Volume 25 (2020) no. 9, pp. 3609-3630 | DOI | MR | Zbl

[15] Nicolas Champagnat A microscopic interpretation for adaptive dynamics trait substitution sequence models, Stochastic Processes Appl., Volume 116 (2006) no. 8, pp. 1127-1160 | DOI | MR | Zbl

[16] Nicolas Champagnat; Régis Ferrière; Gerard Ben Arous The Canonical Equation of Adaptive Dynamics: A Mathematical View, Selection, Volume 2 (2002) no. 1-2, pp. 73-83 | DOI

[17] Nicolas Champagnat; Régis Ferrière; Sylvie Méléard From Individual Stochastic Processes to Macroscopic Models in Adaptive Evolution, Stoch. Models, Volume 24 (2008), pp. 2-44 | DOI | MR | Zbl

[18] Nicolas Champagnat; Pierre-Emmanuel Jabin; Sylvie Méléard Adaptation in a stochastic multi-resources chemostat model, J. Math. Pures Appl., Volume 101 (2014) no. 6, pp. 755-788 | DOI | MR | Zbl

[19] Nicolas Champagnat; Pierre-Emmanuel Jabin; Gaël Raoul Convergence to equilibrium in competitive Lotka–Volterra and chemostat systems, C. R. Math. Acad. Sci. Paris, Volume 348 (2010) no. 23-24, pp. 1267-1272 | DOI | Numdam | MR | Zbl

[20] Nicolas Champagnat; Sylvie Méléard Polymorphic evolution sequence and evolutionary branching, Probab. Theory Relat. Fields, Volume 151 (2010) no. 1-2, pp. 45-94 | DOI | MR | Zbl

[21] John Cleveland Evolutionary game theory on measure space, Ph. D. Thesis, Universityof Louisiana at Lafayette (2009) | MR

[22] Daniel B. Cooney; Yoichiro Mori Long-Time Behavior of a PDE Replicator Equation for Multilevel Selection in Group-Structured Populations, J. Math. Biol., Volume 85 (2022), 12 | MR | Zbl

[23] Loren Coquille; Anna Kraut; Charline Smadi Stochastic individual-based models with power law mutation rate on a general finite trait space, Electron. J. Probab., Volume 26 (2021), 123, 37 pages | DOI | MR | Zbl

[24] Manon Costa; Christèle Etchegaray; Sepideh Mirrahimi Survival criterion for a population subject to selection and mutations - Application to temporally piecewise constant environments, Nonlinear Anal., Real World Appl., Volume 59 (2021), 103239 | DOI | MR | Zbl

[25] Ross Cressman; Josef Hofbauer Measure dynamics on a one-dimensional continuous trait space: theoretical foundations for adaptive dynamics, Theor. Popul. Biol., Volume 67 (2005) no. 1, pp. 47-59 | DOI | Zbl

[26] Laurent Desvillettes; Pierre-Emmanuel Jabin; Stéphane Mischler; Gaël Raoul On selection dynamics for continuous structured populations, Commun. Math. Sci., Volume 6 (2008) no. 3, pp. 729-747 | DOI | MR | Zbl

[27] Ulf Dieckmann; Richard Law The dynamical theory of coevolution: a derivation from stochastic ecological processes, J. Math. Biol., Volume 34 (1996) no. 5-6, pp. 579-612 | DOI | MR | Zbl

[28] Christian Düll; Piotr Gwiazda; Anna Marciniak-Czochra; Jakub Skrzeczkowski Spaces of Measures and their Applications to Structured Population Models, Cambridge University Press, 2021 | DOI

[29] Régis Ferrière; Judith L. Bronstein; Sergio Rinaldi; Richard Law; Mathias Gauduchon Cheating and the evolutionary stability of mutualisms, Proc. R. Soc. Lond., Ser. B, Volume 269 (2002) no. 1493, pp. 773-780 | DOI

[30] Morris W. Hirsch Systems of Differential Equations Which Are Competitive or Cooperative: I. Limit Sets, SIAM J. Math. Anal., Volume 13 (1982) no. 2, pp. 167-179 | DOI | MR | Zbl

[31] Morris W. Hirsch Systems of Differential Equations that are Competitive or Cooperative II: Convergence Almost Everywhere, SIAM J. Math. Anal., Volume 16 (1985) no. 3, pp. 423-439 | DOI | MR | Zbl

[32] Morris W. Hirsch Systems of differential equations which are competitive or cooperative: III. Competing species, Nonlinearity, Volume 1 (1988) no. 1, pp. 51-71 | DOI | MR | Zbl

[33] Pierre-Emmanuel Jabin Small populations corrections for selection-mutation models, Netw. Heterog. Media, Volume 7 (2012) no. 4, pp. 805-836 | DOI | MR | Zbl

[34] Pierre-Emmanuel Jabin; Hailiang Liu On a non-local selection–mutation model with a gradient flow structure, Nonlinearity, Volume 30 (2017) no. 11, pp. 4220-4238 | DOI | MR | Zbl

[35] Pierre-Emmanuel Jabin; Gaël Raoul On selection dynamics for competitive interactions, J. Math. Biol., Volume 63 (2010) no. 3, pp. 493-517 | DOI | MR | Zbl

[36] Pierre-Emmanuel Jabin; Raymond Strother Schram Selection-Mutation dynamics with spatial dependence (2016) (https://arxiv.org/abs/1601.04553)

[37] Anna Kraut; Anton Bovier From adaptive dynamics to adaptive walks, J. Math. Biol., Volume 79 (2019) no. 5, pp. 1699-1747 | DOI | MR | Zbl

[38] Tommaso Lorenzi; Fiona R. Macfarlane; Chiara Villa Discrete and continuum models for the evolutionary and spatial dynamics of cancer: a very short introduction through two case studies, Trends in Biomathematics: Modeling Cells, Flows, Epidemics, and the Environment: Selected Works from the BIOMAT Consortium Lectures, Szeged, Hungary, 2019 (Rubem P. Mondaini, ed.), Springer, 2020, pp. 359-380 | DOI | Zbl

[39] Tommaso Lorenzi; Camille Pouchol Asymptotic analysis of selection-mutation models in the presence of multiple fitness peaks, Nonlinearity, Volume 33 (2020) no. 11, pp. 5791-5816 | DOI | MR | Zbl

[40] Robert M. May; Warren J. Leonard Nonlinear Aspects of Competition Between Three Species, SIAM J. Appl. Math., Volume 29 (1975) no. 2, pp. 243-253 | DOI | MR | Zbl

[41] Benoît Perthame Transport Equations in Biology, Birkhäuser, 2007, ix+198 pages https://www.ebook.de/de/product/5951259/benoit_perthame_transport_equations_in_biology.html | Zbl

[42] Camille Pouchol; Emmanuel Trélat Global stability with selection in integro-differential Lotka-Volterra systems modelling trait-structured populations, J. Biol. Dyn., Volume 12 (2018) no. 1, pp. 872-893 | DOI | MR | Zbl

[43] Steve Smale On the differential equations of species in competition, J. Math. Biol., Volume 3 (1976) no. 1, pp. 5-7 | DOI | MR

[44] Chiara Villa; Mark A. J. Chaplain; Tommaso Lorenzi Evolutionary Dynamics in Vascularised Tumours under Chemotherapy: Mathematical Modelling, Asymptotic Analysis and Numerical Simulations, Vietnam J. Math., Volume 49 (2021), pp. 143-167 | DOI | MR | Zbl

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