Branching processes : variation, growth, and extinction of populations

Library	Item Barcode	Call Number	Material Type	Item Category 1	Status
Searching... PSZ JB	30000010141870	QH352 H32 2005	Open Access Book	Book	Searching... Unknown
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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 Notes

Peter Jagers is Professor of Mathematical Statistics at the Chalmers University of Technology and Gothenburg University, Sweden.

G. AlsmeyerT.G. KurtzF.C. KlebanerM. Gyllenberg and P. JagersF.C. KlebanerG. AlsmeyerG. AlsmeyerA.D. BarbourG. HognasP. Jagers and S. SagitovS. TavareM. AlexanderssonP. Olofsson and M. KimmelP. JagersV.A.A. Jansen and N. StollenwerkM. GyllenbergM. Durinx and Johan A.J. Metz

Authors	p. ix
Acknowledgments	p. x
Notational Standards	p. xii
1 Generalities	p. 1
1.1 The Role of Models	p. 2
1.2 The Role of Randomness	p. 4
1.3 Branching Processes: Some First Words	p. 6
1.4 Stochastic and Deterministic Modeling: An Illustration	p. 7
1.5 Structure of the Book	p. 10
2 Discrete-Time Branching Processes	p. 11
2.1 The Basic Process	p. 13
2.2 Basic Properties and Two Benchmark Processes	p. 16
2.3 Several Types	p. 21
2.4 Generation Overlap	p. 31
2.5 State Dependence	p. 36
2.6 Dependence on the Population Itself	p. 38
2.7 Interaction Between Individuals	p. 42
2.8 Sexual Reproduction	p. 43
2.9 Varying Environments	p. 46
2.10 Migration	p. 52
3 Branching in Continuous Time	p. 56
3.1 Generations in Real Time	p. 56
3.2 Reproducing Only Once	p. 59
3.3 General Branching Processes	p. 66
3.4 Age-distribution and Other Composition Matters	p. 79
3.5 Interaction, Dependence upon Resources, Varying Environment, and Population	p. 81
4 Large Populations	p. 82
4.1 Approximations of Branching Processes	p. 82
4.2 Discrete-Time Dynamical Systems as Population Models	p. 88
4.3 Branching Processes and Structured Population Dynamics	p. 94
5 Extinction	p. 107
5.1 The Role of Extinction in Evolution	p. 107
5.2 Extinction or Explosion: The Merciless Dichotomy	p. 108
5.3 Extinction and Generating Functions	p. 110
5.4 Time to Extinction in Simple Processes	p. 115
5.5 Multi-type Processes	p. 122
5.6 Slightly Supercritical Populations	p. 124
5.7 Accounting for Time Being Continuous	p. 130
5.8 Population Size Dependent Processes	p. 133
5.9 Effects of Sexual Reproduction	p. 135
5.10 Environmental Variation Revisited	p. 145
6 Development of Populations	p. 153
6.1 Exponential Growth	p. 154
6.2 Asymptotic Composition and Mass Growth	p. 161
6.3 Reproductive Value	p. 165
6.4 Populations Bound for Extinction	p. 167
6.5 Interaction and Dependence	p. 170
6.6 Growth of Populations with Sexual Reproduction	p. 177
6.7 Immigration in Subcritical Populations	p. 179
6.8 Quasi-stationarity: General Remarks	p. 183
6.9 Quasi-stationary Behavior in a Simple Discrete-time Model	p. 190
7 Specific Models	p. 200
7.1 Coalescent Processes: Reversed Branching	p. 200
7.2 Ancestral Inference in Branching Processes	p. 208
7.3 The Cell Cycle	p. 218
7.4 Telomere Shortening: An Overview	p. 225
7.5 The Polymerase Chain Reaction	p. 231
7.6 Modeling Measles Outbreaks	p. 236
7.7 Metapopulations	p. 249
7.8 Multi-type Branching Processes and Adaptive Dynamics of Structured Populations	p. 266
Appendix	p. 278
A.1 Expectation and Variance	p. 278
A.2 Useful Equalities and Inequalities	p. 280
A.3 Conditioning	p. 283
A.4 Distributions and Their Transforms	p. 284
A.5 Convergence	p. 290
A.6 The Perron-Frobenius Theorem	p. 293
References	p. 295
Index	p. 307

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