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

Part of: Special processes Stochastic 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: [email protected]
**Postal address: 38610Gières, France. Email address: [email protected]
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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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