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We analyse a family of two-types Wright-Fisher models with selection ina random environment. We introduce rare in addition to weak selection tothe model and provide a criterion to quantify the impact of differentshapes of selection on the fate of the weakest allele.
As is common for these models we obtain a dual pair of abranching-coalescing process describing the genealogy and a diffusiondescribing the evolution of the frequency of a selectively weakerallele, with jumps resulting from rare selection. The jumps in the"diffusion" result from coordination in the dual particle system. Thegenerator of the jump component has a representation in terms of thegenerator of the dual of the non-coordinated (weak) selection, which werefer to as Griffiths' representation.
This representation observed and used by Griffiths in the case of$\Lambda$-coalescents the Wright-Fisher diffusion is a key ingredient inthe proof of the criterion for extinction, because it allows for aLyapunov-function-type argument.
This is joint work with A. González Casanova (UNAM) and D. Spanó(Warwick)