## Probabilistic Models of Population Evolution: Scaling Limits, Genealogies and InteractionsThis expository book presents the mathematical description of evolutionary models of populations subject to interactions (e.g. competition) within the population. The author includes both models of finite populations, and limiting models as the size of the population tends to infinity. The size of the population is described as a random function of time and of the initial population (the ancestors at time 0). The genealogical tree of such a population is given. Most models imply that the population is bound to go extinct in finite time. It is explained when the interaction is strong enough so that the extinction time remains finite, when the ancestral population at time 0 goes to infinity. The material could be used for teaching stochastic processes, together with their applications. Étienne Pardoux is Professor at Aix-Marseille University, working in the field of Stochastic Analysis, stochastic partial differential equations, and probabilistic models in evolutionary biology and population genetics. He obtained his PhD in 1975 at University of Paris-Sud. |

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### Contents

2 | |

5 | |

3 Convergence to a Continuous State Branching Process | 12 |

4 Continuous State Branching Process CSBP | 23 |

5 Genealogies | 45 |

6 Models of Finite Population with Interaction | 62 |

7 Convergence to a Continuous State Model | 83 |

8 Continuous Model with Interaction | 99 |

Appendix | 109 |

121 | |

124 | |

### Other editions - View all

Probabilistic Models of Population Evolution: Scaling Limits, Genealogies ... Étienne Pardoux No preview available - 2016 |

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ancestor binary Biosciences Institute Lecture branching process branching property Brownian motion Chapter conditional law consider continuous function continuous martingale contour process convergence Corollary CSBP deduce define denote exists a constant exponential Fatou's Lemma finite forest of trees genealogical forest Girsanov's theorem Hence Hºdy individuals Institute Lecture Series interaction International Publishing Switzerland jumps Lecture Series 1.6 Lemma 16 limit local martingale Markov process martingale representation theorem Mathematical Biosciences Institute Models of Population monotone convergence theorem Moreover Note occupation times formula offsprings parameter Pardoux Poisson process Population Evolution Probabilistic Models probability space Proposition 37 prove Publishing Switzerland 2016 quadratic variation R-valued random variable Ray—Knight resp result follows satisfies Assumption H1 SDE t t semimartingale sequence ſº solves the SDE Springer International Publishing stochastic sub)critical supercritical tight unique zero Zºds Zºº