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On Λ-Fleming–Viot processes with general frequency-dependent selection

Part of: Stochastic processes Special processes

Published online by Cambridge University Press:  23 November 2020

Adrian Gonzalez Casanova  and
Charline Smadi
Adrian Gonzalez Casanova*
Affiliation:
Universidad Nacional Autónoma de México
Charline Smadi*
Affiliation:
Université Grenoble Alpes, INRAE, LESSEM and CNRS, Institut Fourier
*
*Postal address: Área de la Investigación Científica, Circuito Exterior, Ciudad Universitaria, 04510Coyoacan, CDMX, México. Email address: adriangcs@matem.unam.mx
**Postal address: 38610Gières, France. Email address: charline.smadi@inrae.fr
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Abstract

We construct a multitype constant-size population model allowing for general selective interactions as well as extreme reproductive events. Our multidimensional model aims for the generality of adaptive dynamics and the tractability of population genetics. It generalises the idea of Krone and Neuhauser [39] and González Casanova and Spanò [29], who represented the selection by allowing individuals to sample several potential parents in the previous generation before choosing the ‘strongest’ one, by allowing individuals to use any rule to choose their parent. The type of the newborn can even not be one of the types of the potential parents, which allows modelling mutations. Via a large population limit, we obtain a generalisation of $\Lambda$ -Fleming–Viot processes, with a diffusion term and a general frequency-dependent selection, which allows for non-transitive interactions between the different types present in the population. We provide some properties of these processes related to extinction and fixation events, and give conditions for them to be realised as unique strong solutions of multidimensional stochastic differential equations with jumps. Finally, we illustrate the generality of our model with applications to some classical biological interactions. This framework provides a natural bridge between two of the most prominent modelling frameworks of biological evolution: population genetics and eco-evolutionary models.

Keywords

Λ-Fleming–Viot processfrequency-dependent selectionpopulation geneticsancestral processmultidimensional SDEs with jumps

MSC classification

Primary: 60G99: None of the above, but in this section
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory 92D10: Genetics
Type
Research Papers
Information
Journal of Applied Probability , Volume 57 , Issue 4 , December 2020 , pp. 1162 - 1197
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Copyright
© Applied Probability Trust 2020

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References

Allen, J. (1988). Frequency-dependent selection by predators. Phil. Trans. R. Soc. London B 319 (1196), 485–503.Google ScholarPubMed
Arnason, E. (2004). Mitochondrial cytochrome b DNA variation in the high-fecundity Atlantic cod: trans-Atlantic clines and shallow gene genealogy. Genetics 166 (4), 1871–1885.10.1534/genetics.166.4.1871CrossRefGoogle ScholarPubMed
Baake, E., Lenz, U. and Wakolbinger, A. (2016). The common ancestor type distribution of a $\lambda$ -Wright–Fisher process with selection and mutation. Electron. Commun. Prob. 21, 59, 1–16.10.1214/16-ECP16CrossRefGoogle Scholar
Bansaye, V., Caballero, M.-E. and Méléard, S. (2019). Scaling limits of population and evolution processes in random environment. Electron. J. Prob. 24, 19, 1–38.10.1214/19-EJP262CrossRefGoogle Scholar
Barczy, M., Li, Z. and Pap, G. (2015). Yamada–Watanabe results for stochastic differential equations with jumps. Internat. J. Stoch. Anal. 2015, 460472, 1–23.Google Scholar
Billiard, S. and Smadi, C. (2017). The interplay of two mutations in a population of varying size: a stochastic eco-evolutionary model for clonal interference. Stoch. Process. Appl. 127, 701–748.10.1016/j.spa.2016.06.024CrossRefGoogle Scholar
Birkner, M., Blath, J., Mohle, M., Steinrucken, M. and Tams, J. (2009). A modified lookdown construction for the Xi–Fleming–Viot process with mutation and populations with recurrent bottlenecks. ALEA Lat. Am. J. Prob. Math. Stat. 6, 25–61.Google Scholar
Biswas, N., Etheridge, A. and Klimek, A. (2018). The spatial Lambda-Fleming–Viot process with fluctuating selection. Available at arXiv:1802.08188.Google Scholar
Bovier, A., Coquille, L. and Neukirch, R. (2018). The recovery of a recessive allele in a Mendelian diploid model. J. Math. Biol. 77, 971–1033.10.1007/s00285-018-1240-zCrossRefGoogle Scholar
Bovier, A., Coquille, L. and Smadi, C. (2019). Crossing a fitness valley as a metastable transition in a stochastic population model. Ann. Appl. Prob. 29, 3541–3589.10.1214/19-AAP1487CrossRefGoogle Scholar
Buss, L. and Jackson, J. (1979). Competitive networks: nontransitive competitive relationships in cryptic coral reef environments. Am. Nat. 113 (2), 223–234.10.1086/283381CrossRefGoogle Scholar
Cameron, D. D., White, A. and Antonovics, J. (2009). Parasite–grass–forb interactions and rock–paper–scissor dynamics: predicting the effects of the parasitic plant Rhinanthus minor on host plant communities. J. Ecology 97 (6), 1311–1319.10.1111/j.1365-2745.2009.01568.xCrossRefGoogle Scholar
Champagnat, N. (2006). A microscopic interpretation for adaptive dynamics trait substitution sequence models. Stoch. Process. Appl. 116, 1127–1160.10.1016/j.spa.2006.01.004CrossRefGoogle Scholar
Champagnat, N. and Méléard, S. (2011). Polymorphic evolution sequence and evolutionary branching. Prob. Theory Relat. Fields 151 (1–2), 45–94.10.1007/s00440-010-0292-9CrossRefGoogle Scholar
Champagnat, N., Jabin, P.-E. and Méléard, S. (2014). Adaptation in a stochastic multi-resources chemostat model. J. Math. Pures Appl. 101 (6), 755–788.Google Scholar
Chouteau, M., Arias, M. and Joron, M. (2016). Warning signals are under positive frequency-dependent selection in nature. Proc. Nat. Acad. Sci. 113 (8), 2164–2169.10.1073/pnas.1519216113CrossRefGoogle Scholar
Collet, P., Martínez, S., Méléard, S. and San Martín, J. (2011). Quasi-stationary distributions for structured birth and death processes with mutations. Prob. Theory Relat. Fields 151 (1–2), 191–231.10.1007/s00440-010-0297-4CrossRefGoogle Scholar
Cordero, F., Hummel, S. and Schertzer, E. (2019). General selection models: Bernstein duality and minimal ancestral structures. Available at arXiv:1903.06731.Google Scholar
Coron, C., Méléard, S. and Villemonais, D. (2018). Impact of demography on extinction/fixation events. J. Math. Biol. 78, 549–577.10.1007/s00285-018-1283-1CrossRefGoogle ScholarPubMed
Donnelly, P. and Kurtz, T. (1999). Particle representations for measure-valued population models. Ann. Appl. Prob. 27 (1), 166–205.10.1214/aop/1022677258CrossRefGoogle Scholar
Durrett, R. (2018). Stochastic Calculus: A Practical Introduction. CRC Press.10.1201/9780203738283CrossRefGoogle Scholar
Durrett, R. and Mayberry, J. (2011). Traveling waves of selective sweeps. Ann. Appl. Prob. 21 (2), 699–744.10.1214/10-AAP721CrossRefGoogle Scholar
Durrett, R., Schmidt, D. and Schweinsberg, J. (2009). A waiting time problem arising from the study of multi-stage carcinogenesis. Ann. Appl. Prob. 19 (2), 676–718.10.1214/08-AAP559CrossRefGoogle Scholar
Etheridge, A. and Griffiths, R. (2009). A coalescent dual process in a Moran model with genic selection. Theoret. Pop. Biol. 75 (4), 320–330.10.1016/j.tpb.2009.03.004CrossRefGoogle Scholar
Ewens, W. J. (2004).Mathematical Population Genetics, Vol. I: Theoretical Introduction (Interdisciplinary Applied Mathematics 27). Springer.Google Scholar
Foucart, C. (2013). The impact of selection in the $\lambda$ -Wright–Fisher model. Electron. Commun. Prob. 18 (4), 1–10.10.1214/ECP.v18-2838CrossRefGoogle Scholar
Gigord, L. D., Macnair, M. R. and Smithson, A. (2001). Negative frequency-dependent selection maintains a dramatic flower color polymorphism in the rewardless orchid Dactylorhiza sambucina (L.) Soò. Proc. Nat. Acad. Sci. 98 (11), 6253–6255.10.1073/pnas.111162598CrossRefGoogle Scholar
Gillespie, J. H. (1984). The neutral theory of molecular evolution. Science 224, 732–734.10.1126/science.224.4650.732CrossRefGoogle ScholarPubMed
González Casanova, A. and Spanò, D. (2018). Duality and fixation in $\xi$ -Wright–Fisher processes with frequency-dependent selection. Ann. Appl. Prob. 28 (1), 250–284.10.1214/17-AAP1305CrossRefGoogle Scholar
Griffiths, R. C. (2014). The $\lambda$ –Fleming–Viot process and a connection with Wright–Fisher diffusion. Adv. Appl. Prob. 46 (4), 1009–1035.10.1017/S0001867800007527CrossRefGoogle Scholar
Hedgecock, D. (1994). Does variance in reproductive success limit effective population sizes of marine organisms. In Genetics and Evolution of Aquatic Organisms, ed. A. R. Beaumont, pp. 122–134. Chapman & Hall.Google Scholar
Hoscheit, P. and Pybus, O. (2019). The multifurcating skyline plot. Virus Evol. 5 (2), vez031, vez040.10.1093/ve/vez031CrossRefGoogle ScholarPubMed
Irwin, K., Laurent, S., Matuszewski, S.et al. (2016). On the importance of skewed offspring distributions and background selection in virus population genetics. Heredity 117 (6), 393–399.10.1038/hdy.2016.58CrossRefGoogle ScholarPubMed
Kallenberg, O. (2006). Foundations of Modern Probability. Springer Science & Business Media.Google Scholar
Kerr, B., Riley, M. A., Feldman, M. W. and Bohannan, B. J. (2002). Local dispersal promotes biodiversity in a real-life game of rock–paper–scissors. Nature 418 (6894), 171.10.1038/nature00823CrossRefGoogle Scholar
Kimura, M. (1968). Evolutionary rate at the molecular level. Nature 217 (5129), 624–626.10.1038/217624a0CrossRefGoogle ScholarPubMed
Kimura, M. and Ohta, T. (1972). Population genetics, molecular biometry, and evolution. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Vol. 5, pp. 43–68. University of California Press.Google Scholar
Kirkup, B. C. and Riley, M. A. (2004). Antibiotic-mediated antagonism leads to a bacterial game of rock–paper–scissors in vivo. Nature 428 (6981), 412–414.Google Scholar
Krone, S. M. and Neuhauser, C. (1997). Ancestral processes with selection. Theoret. Pop. Biol. 51 (3), 210–237.10.1006/tpbi.1997.1299CrossRefGoogle ScholarPubMed
Kurtz, T. (2007). The Yamada–Watanabe–Engelbert theorem for general stochastic equations and inequalities. Electron. J. Prob. 12, 951–965.10.1214/EJP.v12-431CrossRefGoogle Scholar
Kurtz, T. (2014). Weak and strong solutions of general stochastic models. Electron. Commun. Prob. 19, 1–16.10.1214/ECP.v19-2833CrossRefGoogle Scholar
Li, Z. and Pu, F. (2012). Strong solutions of jump-type stochastic equations. Electron. Commun. Prob. 17 (33), 1–13.10.1214/ECP.v17-1915CrossRefGoogle Scholar
Méléard, S. and Tran, V. C. (2009). Trait substitution sequence process and canonical equation for age-structured populations. J. Math. Biol. 58 (6), 881.10.1007/s00285-008-0202-2CrossRefGoogle ScholarPubMed
Nahum, J. R., Harding, B. N. and Kerr, B. (2011). Evolution of restraint in a structured rock–paper–scissors community. Proc. Nat. Acad. Sci. 108 (Suppl. 2), 10831–10838.10.1073/pnas.1100296108CrossRefGoogle Scholar
Pitman, J. (1999). Coalescents with multiple collisions. Ann. Prob. 27 (4), 1870–1902.10.1214/aop/1022874819CrossRefGoogle Scholar
Sagitov, S. (1999). The general coalescent with asynchronous mergers of ancestral lines. J. Appl. Prob. 36 (4), 1116–1125.10.1239/jap/1032374759CrossRefGoogle Scholar
Schweinsberg, J. (2017). Rigorous results for a population model with selection, I: evolution of the fitness distribution. Electron. J. Prob. 22, 37, 1–94.Google Scholar
Schweinsberg, J. (2017). Rigorous results for a population model with selection, II: genealogy of the population. Electron. J. Prob. 22, 38, 1–54.Google Scholar
Sinervo, B. and Lively, C. M. (1996). The rock–paper–scissors game and the evolution of alternative male strategies. Nature 380 (6571), 240–243.10.1038/380240a0CrossRefGoogle Scholar
Taylor, D. R. and Aarssen, L. W. (1990). Complex competitive relationships among genotypes of three perennial grasses: implications for species coexistence. Am. Nat. 136 (3), 305–327.10.1086/285100CrossRefGoogle Scholar
Xi, F. and Zhu, C. (2019). Jump type stochastic differential equations with non-Lipschitz coefficients: non-confluence, Feller and strong Feller properties, and exponential ergodicity. J. Differential Equat. 266, 4668–4711.10.1016/j.jde.2018.10.006CrossRefGoogle Scholar