 | Renshaw E. Stochastic population processes: analysis, approximations, simulations. - Oxford; New York: Oxford University Press, 2011. - xii, 652 p.: ill. - Ref.: p.628-646. - Sub. ind.: p.647-652. - ISBN 978-0-19-957531-2
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Preface ........................................................ ix
1 Introduction ................................................. 1
1.1 Some simple stochastic processes ........................ 3
1.2 Single-species population dynamics ...................... 7
1.3 Bivariate populations .................................. 15
1.4 Spatial-temporal processes ............................. 21
2 Simple Markov population processes .......................... 31
2.1 Simple Poisson process ................................. 31
2.1.1 Time to subsequent events ....................... 31
2.1.2 Number of events ................................ 34
2.1.3 General moments ................................. 37
2.1.4 Simulation ...................................... 40
2.1.5 Generalizations ................................. 43
2.1.6 Do moments uniquely define a distribution? ...... 47
2.2 Pure death process ..................................... 50
2.2.1 Stochastic model ................................ 50
2.2.2 Reverse transition probabilities ................ 53
2.2.3 Bridge probabilities ............................ 54
2.3 Pure birth process ..................................... 55
2.3.1 Stochastic model ................................ 56
2.3.2 Laplace transform solution ...................... 57
2.3.3 Solution via the p.g.f. ......................... 59
2.3.4 Time to a given state ........................... 60
2.3.5 Generalized birth process ....................... 63
2.4 Simple birth-death process ............................. 70
2.4.1 Stochastic model ................................ 70
2.4.2 Extinction ...................................... 74
2.4.3 A simple mixture representation ................. 76
2.4.4 The backward equations .......................... 77
2.4.5 Time to a given state ........................... 79
2.4.6 Reverse transition probabilities ................ 80
2.4.7 Simulating the simple birth-death process ....... 81
2.4.8 The dominant leader process ..................... 85
2.5 Simple immigration-birth-death process ................. 86
2.5.1 Stochastic model (λ = 0) ........................ 87
2.5.2 Stochastic model (λ ≥ 0) ........................ 88
2.5.3 Equilibrium probabilities ....................... 90
2.5.4 Perfect simulation .............................. 91
2.6 Simple immigration-emigration process .................. 95
2.6.1 Equilibrium distribution ........................ 96
2.6.2 Time-dependent solutions ........................ 97
2.7 Batch events .......................................... 101
2.7.1 Birth-mass annihilation and immigration
process ........................................ 101
2.7.2 Mass immigration-death process ................. 103
2.7.3 Regenerative phenomena ......................... 104
3 General Markov population processes ........................ 107
3.1 Classification of states .............................. 108
3.2 Equilibrium solutions ................................. 117
3.2.1 Logistic population growth ..................... 123
3.2.2 Simulated realizations ......................... 126
3.2.3 Multiple equilibria ............................ 129
3.2.4 Power-law processes ............................ 132
3.3 Time-dependent solutions .............................. 141
3.3.1 Approximate solutions .......................... 141
3.3.2 Probability of ultimate extinction ............. 143
3.3.3 Mean time to ultimate extinction ............... 145
3.4 Moment closure ........................................ 148
3.4.1 Cumulant equations ............................. 149
3.5 Avoiding the Kolmogorov equations ..................... 153
3.5.1 Generating cumulant equations directly ......... 153
3.5.2 Local approximations ........................... 156
3.5.3 Application to Africanized honey bees .......... 159
3.6 Diffusion approximations .............................. 164
3.6.1 Equilibrium probabilities ...................... 165
3.6.2 Perturbation methods ........................... 168
3.6.3 Additive mass-immigration process .............. 175
3.7 The saddlepoint approximation ......................... 181
3.7.1 Basic derivation ............................... 183
3.7.2 Examples ....................................... 185
3.7.3 Relationship with the Method of Steepest
Descents ....................................... 188
3.7.4 The truncated saddlepoint approximation ........ 190
3.7.5 Final comments ................................. 197
4 The random walk ............................................ 199
4.1 The simple unrestricted random walk ................... 200
4.1.1 Normal approximation ........................... 201
4.1.2 Laws of large numbers .......................... 202
4.1.3 Using the 'reflection' principle ............... 203
4.1.4 First passage and return probabilities ......... 206
4.1.5 Probability of long leads: the First Arc Sine
Law ............................................ 208
4.1.6 Simulated illustration ......................... 212
4.2 Absorbing barriers .................................... 214
4.2.1 Probability of absorption at time n ............ 215
4.2.2 One absorbing barrier .......................... 221
4.2.3 Number of steps to absorption .................. 223
4.2.4 Further aspects of the unrestricted random
walk ........................................... 225
4.3 Reflecting barriers ................................... 228
4.3.1 General equilibrium probability distribution ... 228
4.3.2 Relation between reflecting and absorbing
barriers ....................................... 232
4.3.3 Time-dependent probability distribution ........ 234
4.4 The correlated random walk ............................ 238
4.4.1 Occupation probabilities: direct solution ...... 239
4.4.2 Occupation probabilities: p.g.f. solution ...... 241
4.4.3 Application to share trading ................... 246
5 Markov chains .............................................. 252
5.1 Two-state Markov chain ................................ 256
5.1.1 Occupation probabilities ....................... 257
5.1.2 Matrix solution 1 .............................. 258
5.1.3 Matrix solution 2 .............................. 260
5.1.4 The Discrete Telegraph Wave .................... 261
5.1.5 Relation to the continuous-time process ........ 262
5.2 Examples of m-state Markov chains ..................... 265
5.2.1 The Ehrenfest model ............................ 265
5.2.2 The Perron-Frobenius Theorem ................... 272
5.3 First return and passage probabilities ................ 274
5.3.1 Classification of states ....................... 274
5.3.2 Relating first return and passage
probabilities .................................. 276
5.3.3 Closed sets of states .......................... 280
5.3.4 Irreducible chains ............................. 281
5.4 Branching processes ................................... 282
5.4.1 Population size moments ........................ 283
5.4.2 Probability of extinction ...................... 286
5.5 A brief note on martingales ........................... 290
6 Markov processes in continuous time and space .............. 295
6.1 The basic Wiener process .............................. 296
6.1.1 Diffusion equations for the Wiener process ..... 298
6.1.2 Wiener process with reflecting barriers ........ 300
6.1.3 Wiener process with absorbing barriers ......... 304
6.2 The Fokker-Planck diffusion equation .................. 309
6.2.1 Simulation of the simple immigration-death
diffusion process .............................. 312
6.2.2 Equilibrium probability solution ............... 317
6.2.3 Boundary conditions ............................ 321
6.2.4 Time-dependent probability solutions ........... 322
6.2.5 The associated stochastic differential
equation ....................................... 323
6.3 The Ornstein-Uhlenbeck process ........................ 325
6.3.1 The OU process as a time-transformed Wiener
process ........................................ 329
6.3.2 Rapid oscillations of the Wiener and OU
processes ...................................... 329
7 Modelling bivariate processes .............................. 331
7.1 Simple immigration-death-switch process ............... 331
7.1.1 Generating moments ............................. 334
7.2 Count-dependent growth ................................ 337
7.2.1 Stochastic representation ...................... 337
7.3 Bivariate saddlepoint approximation ................... 339
7.3.1 Simple illustrations ........................... 340
7.3.2 Cumulant truncation ............................ 342
7.3.3 A cautionary tale! ............................. 346
7.3.4 A spatial example .............................. 347
7.4 Counting processes .................................... 349
7.4.1 Paired-immigration-death process ............... 350
7.4.2 Single-paired-immigration-death counting
process ........................................ 353
7.4.3 Batch-immigration-death counting process ....... 357
7.4.4 Summary and further developments ............... 364
7.5 Applying MCMC to hidden event times ................... 367
7.5.1 Introducing Markov chain Monte Carlo ........... 367
7.5.2 Fitting the simple immigration-death process
to incomplete observations ..................... 369
7.5.3 Extension to the single/paired-immigration-
death process .................................. 378
7.5.4 A comparison of Metropolis Q- and direct
P-matrix strategies ............................ 381
8 Two-species interaction processes .......................... 389
8.1 Competition processes ................................. 389
8.1.1 Deterministic analysis ......................... 390
8.1.2 Stability ...................................... 393
8.1.3 Stochastic behaviour ........................... 398
8.1.4 Moment equations ............................... 408
8.2 Predator-prey processes ............................... 413
8.2.1 The Lotka-Volterra process ..................... 414
8.2.2 The Volterra process ........................... 419
8.2.3 A model for prey cover ......................... 426
8.2.4 Sustained deterministic and stochastic limit
cycles ......................................... 428
8.3 Epidemic processes .................................... 434
8.3.1 Simple epidemic ................................ 435
8.3.2 General epidemic ............................... 445
8.3.3 Recurrent epidemics ............................ 454
8.4 Cumulative size processes ............................. 464
8.4.1 Deterministic models ........................... 465
8.4.2 Stochastic simulation .......................... 466
8.4.3 Probability solutions .......................... 468
8.4.4 Power-law processes ............................ 471
9 Spatial processes .......................................... 474
9.1 General results ....................................... 474
9.2 Two-site models ....................................... 477
9.2.1 Moments ........................................ 478
9.2.2 Exact probabilities ............................ 481
9.2.3 An approximate stochastic solution ............. 484
9.2.4 Slightly connected processes ................... 488
9.2.5 Sequences of integral equations ................ 490
9.2.6 Riccati representations ........................ 493
9.3 Stepping-stone processes .............................. 496
9.3.1 Birth-death-migration processes on the
infinite line .................................. 498
9.3.2 Birth-death-migration processes on the finite
line ........................................... 508
9.3.3 Basic simulation algorithms .................... 514
9.3.4 Tau-leaping and other extensions ............... 519
9.4 Velocities of propagation ............................. 524
9.4.1 Wave profiles for two-way migration ............ 526
9.4.2 Wave profiles for non-nearest-neighbour
migration ...................................... 533
9.4.3 Travelling waves ............................... 541
9.5 Turing's model for morphogenesis ...................... 549
9.5.1 Solution of the linearized equations ........... 551
9.5.2 An example of wave formation ................... 553
9.5.3 Stability and the 'Stochastic Dynamic' ......... 559
9.6 Markov chain approach ................................. 561
9.6.1 A more refined approximating process ........... 563
9.6.2 Simulating the Markov chain representation ..... 565
9.6.3 Stochastic cellular automata ................... 567
10 Spatial-temporal extensions ................................ 575
10.1 Power-law lattice processes ........................... 575
10.1.1 First' and second-order moments ................ 576
10.1.2 The spectrum ................................... 578
10.1.3 General power-law spectra ...................... 584
10.1.4 The inverse problem ............................ 585
10.1.5 Simulated realizations ......................... 589
10.1.6 An application to sea waves .................... 592
10.2 Space-time marked point processes ..................... 596
10.2.1 The general model .............................. 598
10.2.2 Choosing growth and interaction functions ...... 599
10.2.3 Parameter selection ............................ 603
10.2.4 Simulation algorithm ........................... 604
10.2.5 Convergence issues ............................. 608
10.2.6 Application to forestry ........................ 610
10.2.7 Application to tightly packed particle
systems ........................................ 613
10.2.8 Other stochastic strategies .................... 619
10.2.9 Final comments ................................. 624
References .................................................... 628
Subject Index ................................................. 647
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