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Critical Population and Error Threshold on the Sharp Peak Landscape for a Moran Model
Raphaël Cerf, Université Paris Sud, Orsay, France

Memoirs of the American Mathematical Society
2015; 87 pp; softcover
Volume: 233
ISBN-10: 1-4704-0967-4
ISBN-13: 978-1-4704-0967-8
List Price: US$71
Individual Members: US$42.60
Institutional Members: US$56.80
Order Code: MEMO/233/1096
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The goal of this work is to propose a finite population counterpart to Eigen's model, which incorporates stochastic effects. The author considers a Moran model describing the evolution of a population of size \(m\) of chromosomes of length \(\ell\) over an alphabet of cardinality \(\kappa\). The mutation probability per locus is \(q\). He deals only with the sharp peak landscape: the replication rate is \(\sigma>1\) for the master sequence and \(1\) for the other sequences. He studies the equilibrium distribution of the process in the regime where \[\ell\to +\infty,\qquad m\to +\infty,\qquad q\to 0,\] \[{\ell q} \to a\in ]0,+\infty[, \qquad\frac{m}{\ell}\to\alpha\in [0,+\infty].\]

Table of Contents

  • Introduction
  • The model
  • Main results
  • Coupling
  • Normalized model
  • Lumping
  • Monotonicity
  • Stochastic bounds
  • Birth and death processes
  • The neutral phase
  • Synthesis
  • Appendix on Markov chain
  • Bibliography
  • Index
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