Branching Processes: Variation, Growth, and Extinction of Populations

Cover......Page 1
Frontmatter......Page 2
Contents......Page 8
Authors......Page 10
Acknowledgments......Page 11
Notational Standards......Page 13
1 - Generalities......Page 14
1.1 The Role of Models......Page 15
1.2 The Role of Randomness......Page 17
1.3 Branching Processes: Some First Words......Page 19
1.4 Stochastic and Deterministic Modeling: An Illustration......Page 20
1.5 Structure of the Book......Page 23
2 - Discrete-Time Branching Processes......Page 24
2.1 The Basic Process......Page 26
2.2 Basic Properties and Two Benchmark Processes......Page 29
2.3 Several Types......Page 34
2.4 Generation Overlap......Page 44
2.5 State Dependence......Page 49
2.6 Dependence on the Population Itself......Page 51
2.7 Interaction Between Individuals......Page 55
2.8 Sexual Reproduction......Page 56
2.9 Varying Environments......Page 59
2.10 Migration......Page 65
3.1 Generations in Real Time......Page 69
3.2 Reproducing Only Once......Page 72
3.3 General Branching Processes......Page 79
3.4 Age-distribution and Other Composition Matters......Page 92
3.5 Interaction, Dependence upon Resources, Varying Environment, and Population......Page 94
4.1 Approximations of Branching Processes......Page 95
4.2 Discrete-Time Dynamical Systems as Population Models......Page 101
4.3 Branching Processes and Structured Population Dynamics......Page 107
5.1 The Role of Extinction in Evolution......Page 120
5.2 Extinction or Explosion: The Merciless Dichotomy......Page 121
5.3 Extinction and Generating Functions......Page 123
5.4 Time to Extinction in Simple Processes......Page 128
5.5 Multi-type Processes......Page 135
5.6 Slightly Supercritical Populations......Page 137
5.7 Accounting for Time Being Continuous......Page 143
5.8 Population Size Dependent Processes......Page 146
5.9 Effects of Sexual Reproduction......Page 148
5.10 Environmental Variation Revisited......Page 158
6 - Development of Populations......Page 166
6.1 Exponential Growth......Page 167
6.2 Asymptotic Composition and Mass Growth......Page 174
6.3 Reproductive Value......Page 178
6.4 Populations Bound for Extinction......Page 180
6.5 Interaction and Dependence......Page 183
6.6 Growth of Populations with Sexual Reproduction......Page 190
6.7 Immigration in Subcritical Populations......Page 192
6.8 Quasi-stationarity: General Remarks......Page 196
6.9 Quasi-stationary Behavior in a Simple Discrete-time Model......Page 203
7.1 Coalescent Processes: Reversed Branching (P. Jagers and S. Sagitov)......Page 213
7.2 Ancestral Inference in Branching Processes (S. Tavaré)......Page 221
7.3 The Cell Cycle (M. Alexandersson)......Page 231
7.4 Telomere Shortening: An Overview (P. Olofsson and M. Kimmel)......Page 238
7.5 The Polymerase Chain Reaction (P. Jagers)......Page 244
7.6 Modeling Measles Outbreaks (V.A.A. Jansen and N. Stollenwerk)......Page 249
7.7 Metapopulations (M. Gyllenberg)......Page 262
7.8 Multi-type Branching Processes and Adaptive Dynamics of Structured Populations (M. Durinx and Johan A.J. Metz)......Page 279
A.1 Expectation and Variance......Page 291
A.2 Useful Equalities and Inequalities......Page 293
A.3 Conditioning......Page 296
A.4 Distributions and Their Transforms......Page 297
A.5 Convergence......Page 303
A.6 The Perron--Frobenius Theorem......Page 306
References......Page 308
Index......Page 320
The International Institute for Applied Systems Analysis......Page 330