9780521539852

Branching Processes: Variation, Growth, and Extinction of Populations

by
  • ISBN13:

    9780521539852

  • ISBN10:

    0521539854

  • Edition: 1st
  • Format: Paperback
  • Copyright: 2007-11-05
  • Publisher: Cambridge University Press
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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

Authorsp. ix
Acknowledgmentsp. x
Notational Standardsp. xii
Generalitiesp. 1
The Role of Modelsp. 2
The Role of Randomnessp. 4
Branching Processes: Some First Wordsp. 6
Stochastic and Deterministic Modeling: An Illustrationp. 7
Structure of the Bookp. 10
Discrete-Time Branching Processesp. 11
The Basic Processp. 13
Basic Properties and Two Benchmark Processesp. 16
Several Typesp. 21
Generation Overlapp. 31
State Dependencep. 36
Dependence on the Population Itselfp. 38
Interaction Between Individualsp. 42
Sexual Reproductionp. 43
Varying Environmentsp. 46
Migrationp. 52
Branching in Continuous Timep. 56
Generations in Real Timep. 56
Reproducing Only Oncep. 59
General Branching Processesp. 66
Age-distribution and Other Composition Mattersp. 79
Interaction, Dependence upon Resources, Varying Environment, and Populationp. 81
Large Populationsp. 82
Approximations of Branching Processesp. 82
Discrete-Time Dynamical Systems as Population Modelsp. 88
Branching Processes and Structured Population Dynamicsp. 94
Extinctionp. 107
The Role of Extinction in Evolutionp. 107
Extinction or Explosion: The Merciless Dichotomyp. 108
Extinction and Generating Functionsp. 110
Time to Extinction in Simple Processesp. 115
Multi-type Processesp. 122
Slightly Supercritical Populationsp. 124
Accounting for Time Being Continuousp. 130
Population Size Dependent Processesp. 133
Effects of Sexual Reproductionp. 135
Environmental Variation Revisitedp. 145
Development of Populationsp. 153
Exponential Growthp. 154
Asymptotic Composition and Mass Growthp. 161
Reproductive Valuep. 165
Populations Bound for Extinctionp. 167
Interaction and Dependencep. 170
Growth of Populations with Sexual Reproductionp. 177
Immigration in Subcritical Populationsp. 179
Quasi-stationarity: General Remarksp. 183
Quasi-stationary Behavior in a Simple Discrete-time Modelp. 190
Specific Modelsp. 200
Coalescent Processes: Reversed Branchingp. 200
Ancestral Inference in Branching Processesp. 208
The Cell Cyclep. 218
Telomere Shortening: An Overviewp. 225
The Polymerase Chain Reactionp. 231
Modeling Measles Outbreaksp. 236
Metapopulationsp. 249
Multi-type Branching Processes and Adaptive Dynamics of Structured Populationsp. 266
Appendixp. 278
Expectation and Variancep. 278
Useful Equalities and Inequalitiesp. 280
Conditioningp. 283
Distributions and Their Transformsp. 284
Convergencep. 290
The Perron-Frobenius Theoremp. 293
Referencesp. 295
Indexp. 307
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