Raphaël Cerf – författare

This monograph studies a series of mathematical models of the evolution of a population under mutation and selection. Its starting point is the quasispecies equation, a general non-linear equation which describes the mutation-selection equilibrium in Manfred Eigen’s famous quasispecies model. A detailed analysis of this equation is given under the assumptions of finite genotype space, sharp peak landscape, and class-dependent fitness landscapes. Different probabilistic representation formulae are derived for its solution, involving classical combinatorial quantities like Stirling and Euler numbers.It is shown how quasispecies and error threshold phenomena emerge in finite population models, and full mathematical proofs are provided in the case of the Wright–Fisher model. Along the way, exact formulas are obtained for the quasispecies distribution in the long chain regime, on the sharp peak landscape and on class-dependent fitness landscapes.Finally, several other classical population models are analyzed, with a focus on their dynamical behavior and their links to the quasispecies equation. This book will be of interest to mathematicians and theoretical ecologists/biologists working with finite population models.

This monograph studies a series of mathematical models of the evolution of a population under mutation and selection. Its starting point is the quasispecies equation, a general non-linear equation which describes the mutation-selection equilibrium in Manfred Eigen’s famous quasispecies model. A detailed analysis of this equation is given under the assumptions of finite genotype space, sharp peak landscape, and class-dependent fitness landscapes. Different probabilistic representation formulae are derived for its solution, involving classical combinatorial quantities like Stirling and Euler numbers.

It is shown how quasispecies and error threshold phenomena emerge in finite population models, and full mathematical proofs are provided in the case of the Wright–Fisher model. Along the way, exact formulas are obtained for the quasispecies distribution in the long chain regime, on the sharp peak landscape and on class-dependent fitness landscapes.

Finally, several other classical population models are analyzed, with a focus on their dynamical behavior and their links to the quasispecies equation. 

This book will be of interest to mathematicians and theoretical ecologists/biologists working with finite population models.

 

This monograph studies a series of mathematical models of the evolution of a population under mutation and selection. Its starting point is the quasispecies equation, a general non-linear equation which describes the mutation-selection equilibrium in Manfred Eigen’s famous quasispecies model. A detailed analysis of this equation is given under the assumptions of finite genotype space, sharp peak landscape, and class-dependent fitness landscapes. Different probabilistic representation formulae are derived for its solution, involving classical combinatorial quantities like Stirling and Euler numbers.It is shown how quasispecies and error threshold phenomena emerge in finite population models, and full mathematical proofs are provided in the case of the Wright–Fisher model. Along the way, exact formulas are obtained for the quasispecies distribution in the long chain regime, on the sharp peak landscape and on class-dependent fitness landscapes.Finally, several other classical population models are analyzed, with a focus on their dynamical behavior and their links to the quasispecies equation. This book will be of interest to mathematicians and theoretical ecologists/biologists working with finite population models.

Three series of lectures were given at the 34th Probability Summer School in Saint-Flour (July 6–24, 2004), by the Professors Cerf, Lyons and Slade. We have decided to publish these courses separately. This volume contains the course of Professor Cerf. We cordially thank the author for his performance at the summer school, and for the redaction of these notes. 69 participants have attended this school. 35 of them have given a short lecture. The lists of participants and of short lectures are enclosed at the end of the volume. The Saint-Flour Probability Summer School was founded in 1971. Here are the references of Springer volumes which have been published prior to this one. All numbers refer to theLecture Notes in Mathematics series, except S-50 which refers to volume 50 of the Lecture Notes in Statistics series. 1971: vol 307 1980: vol 929 1990: vol 1527 1997: vol 1717 1973: vol 390 1981: vol 976 1991: vol 1541 1998: vol 1738 1974: vol 480 1982: vol 1097 1992: vol 1581 1999: vol 1781 1975: vol 539 1983: vol 1117 1993: vol 1608 2000: vol 1816 1976: vol 598 1984: vol 1180 1994: vol 1648 2001: vol 1837 & 1851 1977: vol 678 1985/86/87: vol 1362 & S-50 2002: vol 1840 1978: vol 774 1988: vol 1427 1995: vol 1690 2003: vol 1869 1979: vol 876 1989: vol 1464 1996: vol 1665 2004: vol 1878 & 1879 Further details can be found on the summer school web site