Probabilistic Models of Population Evolution

1. Front Matter

   Pages i-viii

   PDF

2. Introduction

   • Étienne Pardoux

   Pages 1-3

3. Branching Processes

   • Étienne Pardoux

   Pages 5-11

4. Convergence to a Continuous State Branching Process

   • Étienne Pardoux

   Pages 13-21

5. Continuous State Branching Process (CSBP)

   • Étienne Pardoux

   Pages 23-44

6. Genealogies

   • Étienne Pardoux

   Pages 45-61

7. Models of Finite Population with Interaction

   • Étienne Pardoux

   Pages 63-82

8. Convergence to a Continuous State Model

   • Étienne Pardoux

   Pages 83-98

9. Continuous Model with Interaction

   • Étienne Pardoux

   Pages 99-108

10. Back Matter

    Pages 109-125

    PDF

This 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.

• Evolutionary Models
• Markov Chains
• Population Models
• Bienaymé-Galton-Watson Processes
• Continuous State Branching Processes
• Populations with Interactions

“This book presents the mathematical description of evolutionary models of populations subject to interactions within the population. … For readers’ convenience, the book contains an appendix summarizing the necessary backgrounds and technical results on stochastic calculus. The materials are presented with clarity, brevity and styles. … The book provides nice supplementary material for courses in stochastic processes and stochastic calculus.” (Chao Zhu, Mathematical Reviews, April, 2017)



“The main originality of the book is the fact that it describes the evolution of a population where the birth or death rates of the various individuals are affected by the size of the population. … The book is mainly intended to readers with some basic knowledge of stochastic processes and stochastic calculus. All in all, this is a serious piece of work.” (Marius Iosifescu, zbMATH 1351.92003, 2017)

• Institut de Math´ematiques de Marseille, Aix-Marseille Universit´e, Marseilles, France

  Étienne Pardoux

Étienne Pardoux is Professor at Aix-Marseille University, working in the field of Stochastic Analysis, in particular Stochastic partial differential equations. He obtained his PhD in 1975 at University of Paris-Sud.

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