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Modelling Biological Populations in Space and Time : Cambridge Studies in Mathematical Biology - Eric Renshaw

Modelling Biological Populations in Space and Time

Cambridge Studies in Mathematical Biology

Paperback Published: 25th October 1993
ISBN: 9780521448550
Number Of Pages: 424

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This volume develops a unifying approach to population studies that emphasizes the interplay between modeling and experimentation and that will provide mathematicians and biologists with a framework within which population dynamics can be fully explored and understood. A unique feature of the book is that deterministic and stochastic models are considered together; spatial effects are investigated by developing models that highlight the consequences that geographical restriction and species mobility may have on population development. Model-based simulations of processes are used to explore hitherto unforeseen features and thereby suggest further profitable lines of both experimentation and theoretical study. Most aspects of population dynamics are covered, including birth-death and logistic processes, competition and predator-prey relationships, chaos, reaction time delays, fluctuating environments, spatial systems, velocities of spread, epidemics, and spatial branching structures.

"...an excellent introduction to the world of deterministic and stochastic models for population growth, both for students of applied mathematics and statistics at the undergraduate and postgraduate level and for advanced students of theoretical biology." Andris Abakuks, Mathematical Reviews "An excellent reference for the class of mathematical models that it considers...The quality of the book is high...As an applied statistician with an interest--and some background--in the models discussed, I found the book easy to read, extremely useful, and its perspective sensible. It is well worth its price." Judy Zeh, Journal of the American Statistical Association

Prefacep. xiii
A list of symbols and notationp. xv
Introductory remarksp. 1
Deterministic or stochastic models?p. 1
Single-species populationsp. 5
Two-species populationsp. 7
The spatial effectp. 9
Related topicsp. 11
Simple birth-death processesp. 15
The pure birth processp. 16
Deterministic modelp. 17
Stochastic modelp. 17
Simulated modelp. 20
Time to a given statep. 24
The pure death processp. 27
Deterministic modelp. 28
Stochastic modelp. 28
Time to extinctionp. 30
Simulation resultsp. 31
The simple linear birth and death processp. 33
Deterministic modelp. 33
Stochastic modelp. 34
Probability of extinctionp. 36
Simulated modelp. 38
Representation as a simple random walkp. 39
The simple immigration-birth-death processp. 41
Deterministic modelp. 41
Equilibrium probabilitiesp. 42
Appendixp. 44
General birth--death processesp. 46
General population growthp. 46
Probability equationsp. 47
General equilibrium solutionp. 48
Logistic population growthp. 50
The Verhulst--Pearl equationp. 51
Growth of a yeast populationp. 53
Growth of two sheep populationsp. 55
Quasi-equilibrium probabilitiesp. 58
Simulation of the general population processp. 59
Simulation resultsp. 60
Derivation of inter-event timesp. 63
The Normal approximation to the quasi-equilibrium probabilitiesp. 64
Probability of ultimate extinctionp. 65
Mean time to extinctionp. 66
Probability of extinction by time tp. 68
Derivation of an approximation for p[subscript o](t)p. 69
Comparison of approximate quasi-equilibrium probability distributionsp. 70
Logistic mean and variance valuesp. 71
Derivation of the variance approximationp. 73
An approximate skewness resultp. 74
A numerical comparisonp. 75
The diffusion approximationp. 78
Equilibrium probabilitiesp. 79
Justifying the Normal approximationp. 81
Application to the yeast and sheep datap. 81
Yeast datap. 81
Tasmanian sheep datap. 82
South Australian sheep datap. 83
Appendixp. 84
Time-lag models of population growthp. 87
Introductionp. 88
Reaction time-lag--deterministic analysisp. 90
Stability conditionsp. 90
Numerical solutionsp. 92
Reaction time-lag--stochastic analysisp. 94
More general deterministic modelsp. 96
Distributed time-delayp. 97
Periodic and chaotic solutionsp. 100
Three simple deterministic modelsp. 100
More general deterministic resultsp. 105
Stochastic resultsp. 107
Spectral representationp. 110
Final commentsp. 113
Analysis of field and laboratory datap. 114
Nicholson's blowfliesp. 116
Description of the datap. 117
A simple time-delay modelp. 118
Nisbet and Gurney's time-delay modelp. 119
Simulation of the two blowfly modelsp. 122
Appendixp. 125
Competition processesp. 128
Introductionp. 129
Experimental backgroundp. 131
Gause's yeast experimentsp. 131
Birch's grain beetle experimentsp. 135
Stabilityp. 137
Local stabilityp. 139
Global stabilityp. 140
General stability conditionsp. 143
Stochastic behaviourp. 146
Park's flour beetle experimentsp. 146
Gause's competitive exclusion principlep. 148
Simulation of two-species competitionp. 149
Probability equationsp. 154
Extinctionp. 156
Extinction probabilitiesp. 156
Times to extinctionp. 160
Quasi-equilibrium probabilitiesp. 161
Predator-prey processesp. 166
The Lotka-Volterra modelp. 167
Local deterministic solutionp. 169
Biological investigationsp. 171
Simulation of the stochastic modelp. 173
A trajectory indicatorp. 175
Final commentsp. 176
The Volterra modelp. 176
Deterministic trajectoriesp. 177
General local solutionp. 178
Local Volterra solutionp. 180
The coefficient of variationp. 182
Comparison with simulated runsp. 185
Autocorrelation representationp. 185
Mean time to extinctionp. 189
Cross-correlation representationp. 190
Some stochastic thoughtsp. 191
The Leslie and Gower modelp. 191
The Holling--Tanner modelp. 192
Stability analysisp. 194
Simulation of the stochastic modelp. 197
A model for prey-coverp. 199
Stability analysisp. 199
Deterministic versus stochastic behaviourp. 201
Final commentsp. 203
Spatial predator--prey systemsp. 205
Huffaker's experimentsp. 205
Simulation of the spatial Lotka--Volterra modelp. 208
Spatial simulation approachp. 209
Results for model Ap. 210
Results for model Bp. 213
Follow-up remarksp. 214
Matrix representationp. 214
Four-state representationp. 215
Eight-state representationp. 216
Simulation of the eight-state representationp. 220
Fluctuating environmentsp. 223
Deterministic variabilityp. 223
Examplesp. 225
A simulation runp. 227
Local solutionsp. 228
Conclusionsp. 231
Jillson's flour beetle experimentp. 233
Stochastic behaviour with deterministic variabilityp. 236
The stochastic equationp. 236
Autocovariance resultsp. 238
Mean time to extinctionp. 240
Random environmentsp. 241
Four particular modelsp. 242
The autoregressive model (D)p. 245
Comparison of the autocovariancesp. 248
Coherence timep. 250
Period remembering or period forgetting?p. 252
The Canadian lynx datap. 253
Spatial population dynamicsp. 258
The simple random walkp. 259
Position after n stepsp. 260
Use of the Normal approximationp. 261
Absorbing barriersp. 261
Reflecting barriersp. 263
Brownian motionp. 264
Application of diffusion processesp. 266
Skellam's examplesp. 267
Broadbent and Kendall's examplep. 268
Stepping-stone models (1)p. 272
Equilibrium probabilitiesp. 273
The two-colony modelp. 275
Mean valuesp. 275
Variances and covariancesp. 277
Three special casesp. 278
Approximate probabilitiesp. 279
Conditions for ultimate extinctionp. 281
Simulationp. 281
Stepping-stone models (2)p. 284
Special case of v[subscript 2] = 0p. 285
Velocities for v[subscript 2] ] 0p. 287
Comparison of diffusion and stepping-stone velocitiesp. 287
Further resultsp. 288
An application to the spread of Tribolium confusump. 289
Possible modelsp. 290
A stepping-stone approachp. 291
Discussionp. 293
Simulation of the diffusion and stepping-stone processesp. 295
The diffusion processp. 295
The stepping-stone processp. 297
Comparison of velocitiesp. 298
Spatial predator--prey processes revisitedp. 299
The spatial Volterra modelp. 300
A spatial diffusion modelp. 304
Turing's model for morphogenesisp. 310
Solution of the linearized equationsp. 312
A spatial predator-prey examplep. 314
Types of behaviourp. 314
Application to a spatial Volterra systemp. 317
Simulated stochastic wavesp. 319
Epidemic processesp. 324
Introductionp. 324
Simple epidemicsp. 325
The epidemic curvep. 326
Duration timep. 328
General epidemicsp. 330
The epidemic curvep. 330
The deterministic threshold theoremp. 331
The stochastic threshold theoremp. 332
Stochastic realizationsp. 333
Recurrent epidemicsp. 336
Deterministic analysisp. 338
Stochastic considerationsp. 341
The Lotka-Volterra approachp. 343
An extension to malariap. 344
The basic deterministic modelp. 345
Stochastic commentsp. 348
Spatial modelsp. 350
Deterministic spreadp. 351
Stochastic spreadp. 353
A carcinogenic growth processp. 357
Linear and branching architecturesp. 360
The spatial distribution of Epilobium angustifolium in a recently thinned woodlandp. 362
Two-dimensional spectral analysisp. 362
A simple cosine wave examplep. 363
Analysis of the willow herb datap. 364
Simulation of directional spreadp. 366
Spatial branching models for canopy growthp. 368
Honda's deterministic modelp. 369
Description of measurementsp. 369
Simulation of canopy structurep. 372
Spatial branching models for structural root systemsp. 373
Description of measurementsp. 375
A temporal simulation modelp. 377
A fixed-time simulation modelp. 379
Discussion of the role of simulationp. 380
Referencesp. 385
Author indexp. 394
Subject indexp. 397
Table of Contents provided by Syndetics. All Rights Reserved.

ISBN: 9780521448550
ISBN-10: 0521448557
Series: Cambridge Studies in Mathematical Biology
Audience: Professional
Format: Paperback
Language: English
Number Of Pages: 424
Published: 25th October 1993
Publisher: CAMBRIDGE UNIV PR
Country of Publication: GB
Dimensions (cm): 22.81 x 16.36  x 2.62
Weight (kg): 0.63