Branching Processes: Variation, Growth, and Extinction of Populations

by Patsy Haccou , Peter Jagers , Vladimir A. Vatutin

• ISBN13: 9780521539852
• ISBN10: 0521539854
• Format: Paperback
• Copyright: 2007-11-05
• Publisher: Cambridge University Press

Summary

Biology takes a special place among the other natural sciences because biological units, be they pieces of DNA, cells, or organisms, reproduce more or less faithfully. As for any other biological processes, reproduction has a large random component. The theory of branching processes was developed especially as a mathematical counterpart to this most fundamental of biological processes. This active and rich research area allows us to make predictions about both extinction risks and the development of population composition, and also uncovers aspects of a population's history from its current genetic composition. Branching processes play an increasingly important role in models of genetics, molecular biology, microbiology, ecology, and evolutionary theory. This book presents this body of mathematical ideas for a biological audience, but should also be enjoyable to mathematicians.

Author Biography

Vladimir A. Vatutin is Leading Researcher at the Steklov Mathematical Institute, Moscow, Russia.

Table of Contents

Authors	p. ix
Acknowledgments	p. x
Notational Standards	p. xii
Generalities	p. 1
The Role of Models	p. 2
The Role of Randomness	p. 4
Branching Processes: Some First Words	p. 6
Stochastic and Deterministic Modeling: An Illustration	p. 7
Structure of the Book	p. 10
Discrete-Time Branching Processes	p. 11
The Basic Process	p. 13
Basic Properties and Two Benchmark Processes	p. 16
Several Types	p. 21
Generation Overlap	p. 31
State Dependence	p. 36
Dependence on the Population Itself	p. 38
Interaction Between Individuals	p. 42
Sexual Reproduction	p. 43
Varying Environments	p. 46
Migration	p. 52
Branching in Continuous Time	p. 56
Generations in Real Time	p. 56
Reproducing Only Once	p. 59
General Branching Processes	p. 66
Age-distribution and Other Composition Matters	p. 79
Interaction, Dependence upon Resources, Varying Environment, and Population	p. 81
Large Populations	p. 82
Approximations of Branching Processes	p. 82
Discrete-Time Dynamical Systems as Population Models	p. 88
Branching Processes and Structured Population Dynamics	p. 94
Extinction	p. 107
The Role of Extinction in Evolution	p. 107
Extinction or Explosion: The Merciless Dichotomy	p. 108
Extinction and Generating Functions	p. 110
Time to Extinction in Simple Processes	p. 115
Multi-type Processes	p. 122
Slightly Supercritical Populations	p. 124
Accounting for Time Being Continuous	p. 130
Population Size Dependent Processes	p. 133
Effects of Sexual Reproduction	p. 135
Environmental Variation Revisited	p. 145
Development of Populations	p. 153
Exponential Growth	p. 154
Asymptotic Composition and Mass Growth	p. 161
Reproductive Value	p. 165
Populations Bound for Extinction	p. 167
Interaction and Dependence	p. 170
Growth of Populations with Sexual Reproduction	p. 177
Immigration in Subcritical Populations	p. 179
Quasi-stationarity: General Remarks	p. 183
Quasi-stationary Behavior in a Simple Discrete-time Model	p. 190
Specific Models	p. 200
Coalescent Processes: Reversed Branching	p. 200
Ancestral Inference in Branching Processes	p. 208
The Cell Cycle	p. 218
Telomere Shortening: An Overview	p. 225
The Polymerase Chain Reaction	p. 231
Modeling Measles Outbreaks	p. 236
Metapopulations	p. 249
Multi-type Branching Processes and Adaptive Dynamics of Structured Populations	p. 266
Appendix	p. 278
Expectation and Variance	p. 278
Useful Equalities and Inequalities	p. 280
Conditioning	p. 283
Distributions and Their Transforms	p. 284
Convergence	p. 290
The Perron-Frobenius Theorem	p. 293
References	p. 295
Index	p. 307
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