Large scale behaviour of the spatial $𝛬$-Fleming-Viot process

Berestycki, N.; Etheridge, A. M.; Véber, A.

Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) no. 2, pp. 374-401.

On étudie le processus $𝛬$-Fleming-Viot spatial (Electron. J. Probab. 15 (2010) 162-216) modélisant les fréquences locales de types génétiques dans une population évoluant dans ${ℝ}^d$. On considère le cas particulier où il n’y a que deux types possibles, notés $0$ et $1$. Initialement, tous les individus présents dans le demi-espace des points dont la première coordonnée est négative sont de type $1$, tandis que les individus présents dans le demi-espace complémentaire sont de type $0$. On s’intéresse au comportement des fréquences locales sur des échelles de temps et d’espace très grandes. On considère deux cas : dans le premier, l’évolution du processus est due uniquement à des événements ‘locaux’ ; dans le second, on incorpore des événements d’extinction et recolonisation de grande ampleur. On choisit la fréquence de ces événements de sorte qu’après une renormalisation spatiale et temporelle appropriée, la lignée ancestrale d’un individu de la population converge vers un processus $\alpha$-stable symétrique, d’indice $\alpha \in (1,2]$ (où $\alpha =2$ correspond au mouvement brownien). On étudie l’évolution du processus des fréquences alléliques aux mêmes échelles spatio-temporelles. Lorsque $\alpha =2$ et $d\ge 2$, celui-ci converge vers un processus déterministe. Dans tous les autres cas, le processus limite est aléatoire et on l’identifie comme la fonction indicatrice d’un ensemble aléatoire évoluant au cours du temps. En particulier, les deux types ne coexistent pas à la limite. On caractérise chaque ensemble en termes d’un processus dual constitué de mouvements stables symétriques coalescents ayant un intérêt en eux-mêmes. La géométrie complexe des ensembles limites est illustrée par des simulations.

We consider the spatial $𝛬$-Fleming-Viot process model (Electron. J. Probab. 15 (2010) 162-216) for frequencies of genetic types in a population living in ${ℝ}^d$, in the special case in which there are just two types of individuals, labelled $0$ and $1$. At time zero, everyone in a given half-space has type 1, whereas everyone in the complementary half-space has type $0$. We are concerned with patterns of frequencies of the two types at large space and time scales. We consider two cases, one in which the dynamics of the process are driven by purely ‘local’ events and one incorporating large-scale extinction recolonisation events. We choose the frequency of these events in such a way that, under a suitable rescaling of space and time, the ancestry of a single individual in the population converges to a symmetric stable process of index $\alpha \in (1,2]$ (with $\alpha =2$ corresponding to Brownian motion). We consider the behaviour of the process of allele frequencies under the same space and time rescaling. For $\alpha =2$, and $d\ge 2$ it converges to a deterministic limit. In all other cases the limit is random and we identify it as the indicator function of a random set. In particular, there is no local coexistence of types in the limit. We characterise the set in terms of a dual process of coalescing symmetric stable processes, which is of interest in its own right. The complex geometry of the random set is illustrated through simulations.

[1] N. H. Barton, A. M. Etheridge and A. Véber. A new model for evolution in a spatial continuum. Electron. J. Probab. 15 (2010) 162-216. | MR | Zbl

[2] N. H. Barton, J. Kelleher and A. M. Etheridge. A new model for extinction and recolonization in two dimensions: Quantifying phylogeography. Evolution 64 (2010) 2701-2715.

[3] J. Bertoin and J.-F. Le Gall. Stochastic flows associated to coalescent processes. Probab. Theory Related Fields 126 (2003) 261-288. | MR | Zbl

[4] P. Billingsley. Probability and Measure. Wiley, New York, 1995. | MR | Zbl

[5] P. J. Donnelly and T. G. Kurtz. Particle representations for measure-valued population models. Ann. Probab. 27 (1999) 166-205. | MR | Zbl

[6] A. M. Etheridge. Drift, draft and structure: Some mathematical models of evolution. Banach Center Publ. 80 (2008) 121-144. | MR | Zbl

[7] A. M. Etheridge. Some Mathematical Models from Population Genetics. Springer, Berlin, 2011. | MR

[8] A. M. Etheridge and A. Véber. The spatial Lambda-Fleming-Viot process on a large torus: Genealogies in the presence of recombination. Ann. Appl. Probab. 22 (2012) 2165-2209. | MR | Zbl

[9] S. N. Ethier and T. G. Kurtz. Markov Processes: Characterization and Convergence. Wiley, New York, 1986. | MR | Zbl

[10] S. N. Evans. Coalescing Markov labelled partitions and a continuous sites genetics model with infinitely many types. Ann. Inst. Henri Poincaré Probab. Stat. 33 (1997) 339-358. | EuDML | Numdam | MR | Zbl

[11] J. Felsenstein. A pain in the torus: Some difficulties with the model of isolation by distance. Amer. Nat. 109 (1975) 359-368.

[12] M. Kimura. Stepping stone model of population. Ann. Rep. Nat. Inst. Genetics Japan 3 (1953) 62-63.

[13] J. Pitman. Coalescents with multiple collisions. Ann. Probab. 27 (1999) 1870-1902. | MR | Zbl

[14] H. Saadi. $𝛬$-Fleming-Viot processes and their spatial extensions. Ph.D. thesis, University of Oxford, 2011.

[15] S. Sagitov. The general coalescent with asynchronous mergers of ancestral lines. J. Appl. Probab. 26 (1999) 1116-1125. | MR | Zbl

[16] A. Véber and A. Wakolbinger. A flow representation for the spatial $𝛬$-Fleming-Viot process, 2011. To appear.