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A new model for evolution in a spatial continuum. (English) Zbl 1203.60107
Summary: We investigate a new model for populations evolving in a spatial continuum. This model can be thought of as a spatial version of the Lambda-Fleming-Viot process. It explicitly incorporates both small scale reproduction events and large scale extinction-recolonisation events. The lineages ancestral to a sample from a population evolving according to this model can be described in terms of a spatial version of the Lambda-coalescent. Using a technique of St. N. Evans [Ann. Inst. Henri Poincaré, Probab. Stat. 33, No. 3, 339–358 (1997; Zbl 0884.60096)], we prove existence and uniqueness in law for the model. We then investigate the asymptotic behaviour of the genealogy of a finite number of individuals sampled uniformly at random (or more generally ‘far enough apart’) from a two-dimensional torus of sidelength L as L tends to infinity. Under appropriate conditions (and on a suitable timescale) we can obtain as limiting genealogical processes a Kingman coalescent, a more general Lambda-coalescent or a system of coalescing Brownian motions (with a non-local coalescence mechanism).

MSC:
60J25 Continuous-time Markov processes on general state spaces
92D10 Genetics and epigenetics
92D15 Problems related to evolution
60J75 Jump processes (MSC2010)
Citations:
Zbl 0884.60096
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