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**Hierarchically interacting Fleming-Viot processes with selection and mutation: Multiple space time scale analysis and quasi-equilibria.**
*(English)*
Zbl 0920.92016

Summary: Genetic models incorporating resampling and migration are now fairly well-understood. Problems arise in the analysis, if both selection and mutation are incorporated. This paper addresses some aspects of this problem, in particular the analysis of the long-time behaviour before the equilibrium is reached (quasi-equilibrium, which is the time range of interest in most applications).

The first model we use is a countable system of Fleming-Viot processes with selection and interaction between components via migration. Types are in principle described by the set [0,1] or a hierarchically structured countable subset, but we relabel the countable subset occuring in our universe during the evolution via lexicographical ordering by the natural numbers and each component of the system takes values in the probability measure on the natural numbers and represents the frequency of genetic types which occur in this colony. Furthermore we discuss in a second model the effect of adding mutation and recombination to the system already incorporating selection.

Of particular interest in such evolutionary models is the nonequilibrium behaviour. The latter can be studied rigorously by exhibiting quasi-equilibria. These are obtained by letting a parameter in the evolution tend to infinity and observing the system in various different time scales expanding with this parameter in different orders of magnitude. In these limits the analysis simplifies and the behaviour of the space-time rescaled system is described by a sequence of time dependent Markov chains with values in probability measures on the natural numbers and initial points, which vary with the time scale used. The properties of these collections of chains correspond to properties in the longtime behaviour of the original system. The resulting nonequilibrium behaviour displays a very rich structure, we focus on the following phenomena: influence of the migration on the quasi-equilibria, the increased variety of species under selection, the influence of migration on the speed of selection, and the role of mutation.

Finally, in this paper we set up the framework needed to formulate on a rigorous level a number of unresolved issues in evolutionary models which will be discussed in future work. At the same time the analysis provides an interesting example of a rigorous renormalisation analysis.

The first model we use is a countable system of Fleming-Viot processes with selection and interaction between components via migration. Types are in principle described by the set [0,1] or a hierarchically structured countable subset, but we relabel the countable subset occuring in our universe during the evolution via lexicographical ordering by the natural numbers and each component of the system takes values in the probability measure on the natural numbers and represents the frequency of genetic types which occur in this colony. Furthermore we discuss in a second model the effect of adding mutation and recombination to the system already incorporating selection.

Of particular interest in such evolutionary models is the nonequilibrium behaviour. The latter can be studied rigorously by exhibiting quasi-equilibria. These are obtained by letting a parameter in the evolution tend to infinity and observing the system in various different time scales expanding with this parameter in different orders of magnitude. In these limits the analysis simplifies and the behaviour of the space-time rescaled system is described by a sequence of time dependent Markov chains with values in probability measures on the natural numbers and initial points, which vary with the time scale used. The properties of these collections of chains correspond to properties in the longtime behaviour of the original system. The resulting nonequilibrium behaviour displays a very rich structure, we focus on the following phenomena: influence of the migration on the quasi-equilibria, the increased variety of species under selection, the influence of migration on the speed of selection, and the role of mutation.

Finally, in this paper we set up the framework needed to formulate on a rigorous level a number of unresolved issues in evolutionary models which will be discussed in future work. At the same time the analysis provides an interesting example of a rigorous renormalisation analysis.

### MSC:

92D15 | Problems related to evolution |

60J70 | Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) |

60J20 | Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) |