K. B. Athreya and P. E. Ney, Branching processes, 2004.

S. Billiard, P. Collet, R. Ferrì-ere, S. Méléard, and V. C. Tran, The effect of horizontal trait inheritance on competitive exclusion and coexistence. arXiv preprint

S. Billiard and C. Smadi, The interplay of two mutations in a population of varying size: a stochastic eco-evolutionary model for clonal interference Stochastic Processes and their Applications, 2016.

S. Billiard and C. Smadi, Beyond clonal interference: Scrutinizing the complexity of the dynamics of three competing clones arXiv preprint, p.2016

S. Baigent, Lotka?Volterra Dynamics?An introduction Preprint, 2010.

R. Brink-spalink, Stochastic Models in Population Genetics: The Impact of Selection and Recombination, p.2015

R. Brink-spalink and C. Smadi, Abstract, Advances in Applied Probability, vol.123, issue.01, 2017.
DOI : 10.1016/0040-5809(92)90045-U

G. Cardano, Ars magna or the rules of algebra, 1968.

N. Champagnat, A microscopic interpretation for adaptive dynamics trait substitution sequence models, Stochastic Processes and their Applications, pp.1127-1160, 2006.
DOI : 10.1016/j.spa.2006.01.004

URL : https://hal.archives-ouvertes.fr/hal-00015130

N. Champagnat, P. Jabin, and S. Méléard, Adaptation in a stochastic multi-resources chemostat model, Journal de Math??matiques Pures et Appliqu??es, vol.101, issue.6, pp.755-788, 2014.
DOI : 10.1016/j.matpur.2013.10.003

URL : https://hal.archives-ouvertes.fr/hal-00784166

N. Champagnat and S. Méléard, Polymorphic evolution sequence and evolutionary branching. Probability Theory and Related Fields, pp.45-94, 2011.
DOI : 10.1007/s00440-010-0292-9

URL : https://hal.archives-ouvertes.fr/inria-00345399

Y. F. Chan, M. E. Marks, F. C. Jones, G. Villarreal, M. D. Shapiro et al., Adaptive Evolution of Pelvic Reduction in Sticklebacks by Recurrent Deletion of a Pitx1 Enhancer, Science, vol.6, issue.3, pp.327302-305, 2010.
DOI : 10.1016/j.cell.2008.01.014

J. Coville and F. Fabre, Convergence to the equilibrium in a lotka-volterra ode competition system with mutations, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00781200

W. A. Cresko, A. Amores, C. Wilson, J. Murphy, M. Currey et al., Parallel genetic basis for repeated evolution of armor loss in Alaskan threespine stickleback populations, Proceedings of the National Academy of Sciences of the United States of America, pp.6050-6055, 2004.
DOI : 10.1073/pnas.97.9.4530

J. Cuevas, R. Geller, R. Garijo, J. López-aldeguer, and R. Sanjuán, Extremely High Mutation Rate of HIV-1 In Vivo, PLOS Biology, vol.8, issue.9, p.2015
DOI : 10.1371/journal.pbio.1002251.s011

O. Diekmann, A beginner's guide to adaptive dynamics, Mathematical Modelling of Population Dynamics, pp.47-86, 2004.
DOI : 10.4064/bc63-0-2

J. W. Drake, Rates of spontaneous mutation among RNA viruses., Proceedings of the National Academy of Sciences, vol.90, issue.9, pp.4171-4175, 1993.
DOI : 10.1073/pnas.90.9.4171

S. Duffy, L. A. Shackelton, and E. C. Holmes, Rates of evolutionary change in viruses: patterns and determinants, Nature Reviews Genetics, vol.79, issue.4, pp.267-276, 2008.
DOI : 10.1099/0022-1317-79-12-3091

F. Dumortier, J. Llibre, and J. C. Artés, Qualitative theory of planar differential systems, 2006.

H. Erkko, B. Xia, J. Nikkilä, J. Schleutker, K. Syrjäkoski et al., A recurrent mutation in PALB2 in Finnish cancer families, Nature, vol.95, issue.7133, pp.446316-319, 2007.
DOI : 10.1038/nature05609

S. Ethier and T. Kurtz, Markov processes: Characterization and convergence, 1986.
DOI : 10.1002/9780470316658

N. Fournier and S. Méléard, A microscopic probabilistic description of a locally regulated population and macroscopic approximations, The Annals of Applied Probability, vol.14, issue.4, pp.1880-1919, 2004.
DOI : 10.1214/105051604000000882

S. Gago, S. F. Elena, R. Flores, and R. Sanjuán, Extremely high mutation rate of a hammerhead viroid Science, pp.1308-1308, 2009.

R. C. Griffiths, Lines of descent in the diffusion approximation of neutral Wright-Fisher models, Theoretical population biology, pp.37-50, 1980.

J. Haldane, The Part Played by Recurrent Mutation in Evolution, The American Naturalist, vol.67, issue.708, pp.5-19, 1933.
DOI : 10.1086/280465

J. Hermisson and P. S. Pennings, Soft Sweeps: Molecular Population Genetics of Adaptation From Standing Genetic Variation, Genetics, vol.169, issue.4, pp.2335-2352, 2005.
DOI : 10.1534/genetics.104.036947

J. Hermisson and P. Pfaffelhuber, The pattern of genetic hitchhiking under recurrent mutation, Electronic Journal of Probability, vol.13, issue.0, pp.2069-2106, 2008.
DOI : 10.1214/EJP.v13-577

N. Ikeda and S. Watanabe, Stochastic differential equations and diffusion processes, 1989.

P. W. Messer and D. A. Petrov, Population genomics of rapid adaptation by soft selective sweeps. Trends in ecology & evolution, pp.659-669, 2013.

J. A. Metz, S. A. Geritz, G. Meszéna, F. J. Jacobs, and J. Van-heerwaarden, Adaptive dynamics, a geometrical study of the consequences of nearly faithful reproduction. Stochastic and spatial structures of dynamical systems, pp.183-231, 1996.

A. Nohturfft, X. Hua, M. S. Brown, and J. L. Goldstein, Recurrent G-to-A substitution in a single codon of SREBP cleavage-activating protein causes sterol resistance in three mutant Chinese hamster ovary cell lines, Proceedings of the National Academy of Sciences, pp.9313709-13714, 1996.
DOI : 10.1016/0165-1110(89)90019-5

M. Kimura and T. Ohta, Stepwise mutation model and distribution of allelic frequencies in a finite population., Proceedings of the National Academy of Sciences, vol.75, issue.6, pp.2868-2872, 1978.
DOI : 10.1073/pnas.75.6.2868

A. Moya, E. C. Holmes, and F. González-candelas, The population genetics and evolutionary epidemiology of RNA viruses, Nature Reviews Microbiology, vol.13, issue.4, pp.279-288, 2004.
DOI : 10.1093/oxfordjournals.molbev.a025664

T. Nagylaki, Evolution of a finite population under gene conversion., Proceedings of the National Academy of Sciences, vol.80, issue.20, pp.6278-6281, 1983.
DOI : 10.1073/pnas.80.20.6278

S. Nair, D. Nash, D. Sudimack, A. Jaidee, M. Barends et al., Recurrent Gene Amplification and Soft Selective Sweeps during Evolution of Multidrug Resistance in Malaria Parasites, Molecular Biology and Evolution, vol.24, issue.2, pp.562-573, 2007.
DOI : 10.1093/molbev/msl185

P. S. Pennings and J. Hermisson, Soft Sweeps III: The Signature of Positive Selection from Recurrent Mutation, PLoS Genetics, vol.4, issue.12, p.186, 2006.
DOI : 10.1371/journal.pgen.0020186.g009

P. S. Pennings and J. Hermisson, Soft Sweeps II???Molecular Population Genetics of Adaptation from Recurrent Mutation or Migration, Molecular Biology and Evolution, vol.23, issue.5, pp.1076-1084, 2006.
DOI : 10.1093/molbev/msh192

C. Pokalyuk, The effect of recurrent mutation on the linkage disequilibrium under a selective sweep, Journal of Mathematical Biology, vol.39, issue.4, pp.291-317, 2012.
DOI : 10.1038/ng1946

M. Prezeworski, G. Coop, and J. D. Wall, The signature of positive selection on standing genetic variation Evolution, pp.2312-2323, 2005.

S. Repping, H. Skaletsky, L. Brown, S. K. Van-daalen, C. M. Korver et al., Polymorphism for a 1.6-Mb deletion of the human Y chromosome persists through balance between recurrent mutation and haploid selection, Nature Genetics, vol.409, issue.3, pp.247-251, 2003.
DOI : 10.1038/35057170

M. Richard, Limit theorems for supercritical age-dependent branching processes with neutral immigration, Advances in Applied Probability, pp.276-300, 2011.
URL : https://hal.archives-ouvertes.fr/hal-00595714

H. Risheg, J. M. Graham, R. D. Clark, R. C. Rogers, J. M. Opitz et al., A recurrent mutation in MED12 leading to R961W causes Opitz-Kaveggia syndrome, Nature Genetics, vol.140, issue.4, pp.451-453, 2007.
DOI : 10.1002/ajmg.a.31302

L. B. Scheinfeldt, S. Biswas, J. Madeoy, C. F. Connelly, E. E. Schadt et al., Population Genomic Analysis of ALMS1 in Humans Reveals a Surprisingly Complex Evolutionary History, Molecular Biology and Evolution, vol.22, issue.6, pp.1357-1367, 2009.
DOI : 10.1093/bioinformatics/btl365

J. Schweinsberg and R. Durrett, Random partitions approximating the coalescence of lineages during a selective sweep, The Annals of Applied Probability, vol.15, issue.3, pp.1591-1651, 2005.
DOI : 10.1214/105051605000000430

E. M. Shore, M. Xu, G. J. Feldman, D. A. Fenstermacher, T. Cho et al., A recurrent mutation in the BMP type I receptor ACVR1 causes inherited and sporadic fibrodysplasia ossificans progressiva, Nature Genetics, vol.103, issue.5, pp.525-527, 2006.
DOI : 10.1016/S0092-8674(00)00121-5

P. Skuce, L. Stenhouse, F. Jackson, V. Hyp?a, and J. Gilleard, Benzimidazole resistance allele haplotype diversity in United Kingdom isolates of Teladorsagia circumcincta supports a hypothesis of multiple origins of resistance by recurrent mutation, International Journal for Parasitology, vol.40, issue.11, pp.401247-1255, 2010.
DOI : 10.1016/j.ijpara.2010.03.016

C. Smadi, An eco-evolutionary approach of adaptation and recombination in a large population of varying size, Stochastic Processes and their Applications, pp.2054-2095, 2015.
DOI : 10.1016/j.spa.2014.12.007

URL : https://hal.archives-ouvertes.fr/hal-00948097

D. Van-ditmarsch, K. E. Boyle, H. Sakhtah, J. E. Oyler, C. D. Nadell et al., Convergent evolution of hyperswarming leads to impaired biofilm formation in pathogenic bacteria Cell reports, pp.697-708, 2013.

B. A. Wilson, D. A. Petrov, and P. W. Messer, Soft Selective Sweeps in Complex Demographic Scenarios, Genetics, vol.198, issue.2, pp.669-684, 2014.
DOI : 10.1534/genetics.114.165571

U. Irstea and L. D. Lisc, Ingénierie des Systèmes Complexes, 9 avenue Blaise Pascal- CS UK E-mail address: charline.smadi@polytechnique.edu Author-produced version of the article published in, Acta Applicandae Mathematicae, vol.1, issue.11, pp.11-51, 20085.