Branching Processes: Variation, Growth, and Extinction of Populations

by Patsy Haccou , Peter Jagers , Vladimir A. Vatutin

• ISBN13: 9780521832205
• ISBN10: 0521832209
• Edition: 1st
• Format: Hardcover
• Copyright: 2005-06-20
• 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

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

Table of Contents

Authors	ix
Acknowledgments	x
Notational Standards	xii
Generalities	1	(10)
The Role of Models	2	(2)
The Role of Randomness	4	(2)
Branching Processes: Some First Words	6	(1)
Stochastic and Deterministic Modeling: An Illustration	7	(3)
Structure of the Book	10	(1)
Discrete-Time Branching Processes	11	(45)
The Basic Process	13	(3)
Basic Properties and Two Benchmark Processes	16	(5)
Several Types	21	(10)
Generation Overlap	31	(5)
State Dependence	36	(2)
Dependence on the Population Itself	38	(4)
Interaction Between Individuals	42	(1)
Sexual Reproduction	43	(3)
G. Alsmeyer
Varying Environments	46	(6)
Migration	52	(4)
Branching in Continuous Time	56	(26)
Generations in Real Time	56	(3)
Reproducing Only Once	59	(7)
General Branching Processes	66	(13)
Age-distribution and Other Composition Matters	79	(2)
Interaction, Dependence upon Resources, Varying Environment, and Population	81	(1)
Large Populations	82	(25)
Approximations of Branching Processes	82	(6)
T. G. Kurtz
Discrete-Time Dynamical Systems as Population Models	88	(6)
F. C. Klebaner
Branching Processes and Structured Population Dynamics	94	(13)
M. Gyllenberg
P. Jagers
Extinction	107	(46)
The Role of Extinction in Evolution	107	(1)
Extinction or Explosion: The Merciless Dichotomy	108	(2)
Extinction and Generating Functions	110	(5)
Time to Extinction in Simple Processes	115	(7)
Multi-type Processes	122	(2)
Slightly Supercritical Populations	124	(6)
Accounting for Time Being Continuous	130	(3)
Population Size Dependent Processes	133	(2)
F. C. Klebaner
Effects of Sexual Reproduction	135	(10)
G. Alsmeyer
Environmental Variation Revisited	145	(8)
Development of Populations	153	(47)
Exponential Growth	154	(7)
Asymptotic Composition and Mass Growth	161	(4)
Reproductive Value	165	(2)
Populations Bound for Extinction	167	(3)
Interaction and Dependence	170	(7)
Growth of Populations with Sexual Reproduction	177	(2)
G. Alsmeyer
Immigration in Subcritical Populations	179	(4)
Quasi-stationarity: General Remarks	183	(7)
A. D. Barbour
Quasi-stationary Behavior in a Simple Discrete-time Model	190	(10)
G. Hognas
Specific Models	200	(78)
Coalescent Processes: Reversed Branching	200	(8)
P. Jagers
S. Sagitov
Ancestral Inference in Branching Processes	208	(10)
S. Tavare
The Cell Cycle	218	(7)
M. Alexandersson
Telomere Shortening: An Overview	225	(6)
P. Olofsson
M. Kimmel
The Polymerase Chain Reaction	231	(5)
P. Jagers
Modeling Measles Outbreaks	236	(13)
V. A. A. Jansen
N. Stollenwerk
Metapopulations	249	(17)
M. Gyllenberg
Multi-type Branching Processes and Adaptive Dynamics of Structured Populations	266	(12)
M. Durinx
Johan A. J. Metz
Appendix	278	(17)
Expectation and Variance	278	(2)
Useful Equalities and Inequalities	280	(3)
Conditioning	283	(1)
Distributions and Their Transforms	284	(6)
Convergence	290	(3)
The Perron-Frobenius Theorem	293	(2)
References	295	(12)
Index	307