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.
Keywords: population dynamics, selection model, measure solutions, semigroup, asymptotic behaviour, simulations
Martin, Hugo 1
CC-BY 4.0
@article{MSIA_2023__12_1_155_0,
author = {Martin, Hugo},
title = {Measure framework for the pure selection equation: global well posedness and numerical investigations},
journal = {MathematicS In Action},
pages = {155--173},
year = {2023},
publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
volume = {12},
number = {1},
doi = {10.5802/msia.36},
language = {en},
url = {https://www.numdam.org/articles/10.5802/msia.36/}
}
TY - JOUR AU - Martin, Hugo TI - Measure framework for the pure selection equation: global well posedness and numerical investigations JO - MathematicS In Action PY - 2023 SP - 155 EP - 173 VL - 12 IS - 1 PB - Société de Mathématiques Appliquées et Industrielles UR - https://www.numdam.org/articles/10.5802/msia.36/ DO - 10.5802/msia.36 LA - en ID - MSIA_2023__12_1_155_0 ER -
%0 Journal Article %A Martin, Hugo %T Measure framework for the pure selection equation: global well posedness and numerical investigations %J MathematicS In Action %D 2023 %P 155-173 %V 12 %N 1 %I Société de Mathématiques Appliquées et Industrielles %U https://www.numdam.org/articles/10.5802/msia.36/ %R 10.5802/msia.36 %G en %F MSIA_2023__12_1_155_0
Martin, Hugo. Measure framework for the pure selection equation: global well posedness and numerical investigations. MathematicS In Action, Maths Bio, Tome 12 (2023) no. 1, pp. 155-173. doi: 10.5802/msia.36
[1] Selection-mutation process of RNA viruses, Phys. Rev. E, Volume 57 (1998) no. 2, pp. 2471-2474 | DOI
[2] 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 | Zbl | MR | DOI
[3] 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 | Zbl | MR | DOI
[4] Survival of the fittest in a generalized logistic model, Math. Models Methods Appl. Sci., Volume 09 (1999) no. 09, pp. 1379-1391 | Zbl | MR | DOI
[5] 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 | Zbl | MR | DOI
[6] Evolutionary dynamics of competing phenotype-structured populations in periodically fluctuating environments, J. Math. Biol., Volume 80 (2019) no. 3, pp. 775-807 | Zbl | MR | DOI
[7] Concentration phenomenon in some non-local equation, Discrete Contin. Dyn. Syst., Ser. B, Volume 22 (2017) no. 3, pp. 763-781 | Zbl | MR | DOI
[8] Mathematical Theory of Selection, John Wiley & Sons, 2000, 420 pages https://www.ebook.de/...
[9] 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 | Zbl | MR | DOI
[10] Mass concentration in a nonlocal model of clonal selection, J. Math. Biol., Volume 73 (2016) no. 4, pp. 1001-1033 | Zbl | MR | DOI
[11] Asymptotic stability of equilibria of selection-mutation equations, J. Math. Biol., Volume 54 (2006) no. 4, pp. 489-511 | Zbl | MR | DOI
[12] Horizontal gene transfer: numerical comparison between stochastic and deterministic approaches, ESAIM, Proc. Surv., Volume 67 (2020), pp. 135-160 | Zbl | MR | DOI
[13] Measure Solutions for Some Models in Population Dynamics, Acta Appl. Math., Volume 123 (2012) no. 1, pp. 141-156 | Zbl | MR | DOI
[14] 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 | Zbl | MR | DOI
[15] A microscopic interpretation for adaptive dynamics trait substitution sequence models, Stochastic Processes Appl., Volume 116 (2006) no. 8, pp. 1127-1160 | Zbl | MR | DOI
[16] The Canonical Equation of Adaptive Dynamics: A Mathematical View, Selection, Volume 2 (2002) no. 1-2, pp. 73-83 | DOI
[17] From Individual Stochastic Processes to Macroscopic Models in Adaptive Evolution, Stoch. Models, Volume 24 (2008), pp. 2-44 | Zbl | MR | DOI
[18] Adaptation in a stochastic multi-resources chemostat model, J. Math. Pures Appl., Volume 101 (2014) no. 6, pp. 755-788 | Zbl | MR | DOI
[19] 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 | Zbl | Numdam | MR | DOI
[20] Polymorphic evolution sequence and evolutionary branching, Probab. Theory Relat. Fields, Volume 151 (2010) no. 1-2, pp. 45-94 | Zbl | MR | DOI
[21] Evolutionary game theory on measure space, Ph. D. Thesis, Universityof Louisiana at Lafayette (2009) | MR
[22] Long-Time Behavior of a PDE Replicator Equation for Multilevel Selection in Group-Structured Populations, J. Math. Biol., Volume 85 (2022), 12 | Zbl | MR
[23] Stochastic individual-based models with power law mutation rate on a general finite trait space, Electron. J. Probab., Volume 26 (2021), 123, 37 pages | Zbl | MR | DOI
[24] 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 | Zbl | MR | DOI
[25] 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 | Zbl | DOI
[26] On selection dynamics for continuous structured populations, Commun. Math. Sci., Volume 6 (2008) no. 3, pp. 729-747 | MR | Zbl | DOI
[27] The dynamical theory of coevolution: a derivation from stochastic ecological processes, J. Math. Biol., Volume 34 (1996) no. 5-6, pp. 579-612 | Zbl | MR | DOI
[28] Spaces of Measures and their Applications to Structured Population Models, Cambridge University Press, 2021 | DOI
[29] Cheating and the evolutionary stability of mutualisms, Proc. R. Soc. Lond., Ser. B, Volume 269 (2002) no. 1493, pp. 773-780 | DOI
[30] Systems of Differential Equations Which Are Competitive or Cooperative: I. Limit Sets, SIAM J. Math. Anal., Volume 13 (1982) no. 2, pp. 167-179 | Zbl | MR | DOI
[31] 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 | Zbl | MR | DOI
[32] Systems of differential equations which are competitive or cooperative: III. Competing species, Nonlinearity, Volume 1 (1988) no. 1, pp. 51-71 | Zbl | MR | DOI
[33] Small populations corrections for selection-mutation models, Netw. Heterog. Media, Volume 7 (2012) no. 4, pp. 805-836 | Zbl | MR | DOI
[34] On a non-local selection–mutation model with a gradient flow structure, Nonlinearity, Volume 30 (2017) no. 11, pp. 4220-4238 | Zbl | MR | DOI
[35] On selection dynamics for competitive interactions, J. Math. Biol., Volume 63 (2010) no. 3, pp. 493-517 | Zbl | MR | DOI
[36] Selection-Mutation dynamics with spatial dependence (2016) (https://arxiv.org/abs/1601.04553)
[37] From adaptive dynamics to adaptive walks, J. Math. Biol., Volume 79 (2019) no. 5, pp. 1699-1747 | Zbl | MR | DOI
[38] 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 (Mondaini, Rubem P., ed.), Springer, 2020, pp. 359-380 | Zbl | DOI
[39] Asymptotic analysis of selection-mutation models in the presence of multiple fitness peaks, Nonlinearity, Volume 33 (2020) no. 11, pp. 5791-5816 | Zbl | MR | DOI
[40] Nonlinear Aspects of Competition Between Three Species, SIAM J. Appl. Math., Volume 29 (1975) no. 2, pp. 243-253 | Zbl | MR | DOI
[41] Transport Equations in Biology, Birkhäuser, 2007, ix+198 pages https://www.ebook.de/... | Zbl
[42] 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 | Zbl | MR | DOI
[43] On the differential equations of species in competition, J. Math. Biol., Volume 3 (1976) no. 1, pp. 5-7 | MR | DOI
[44] Evolutionary Dynamics in Vascularised Tumours under Chemotherapy: Mathematical Modelling, Asymptotic Analysis and Numerical Simulations, Vietnam J. Math., Volume 49 (2021), pp. 143-167 | Zbl | MR | DOI
Cité par Sources :
