Modelling Biological Populations in Space and Time

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Paperback | 25 October 1993

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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.

Industry Reviews

"...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

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Non-Fiction Mathematics Applied Mathematics Mathematical Modelling Earth Sciences, Geography, Environment, Planning The Environment Applied Ecology Science Biology, Life Sciences Life Sciences in General Ecological Science Physics

Preface	p. xiii
A list of symbols and notation	p. xv
Introductory remarks	p. 1
Deterministic or stochastic models?	p. 1
Single-species populations	p. 5
Two-species populations	p. 7
The spatial effect	p. 9
Related topics	p. 11
Simple birth-death processes	p. 15
The pure birth process	p. 16
Deterministic model	p. 17
Stochastic model	p. 17
Simulated model	p. 20
Time to a given state	p. 24
The pure death process	p. 27
Deterministic model	p. 28
Stochastic model	p. 28
Time to extinction	p. 30
Simulation results	p. 31
The simple linear birth and death process	p. 33
Deterministic model	p. 33
Stochastic model	p. 34
Probability of extinction	p. 36
Simulated model	p. 38
Representation as a simple random walk	p. 39
The simple immigration-birth-death process	p. 41
Deterministic model	p. 41
Equilibrium probabilities	p. 42
Appendix	p. 44
General birth--death processes	p. 46
General population growth	p. 46
Probability equations	p. 47
General equilibrium solution	p. 48
Logistic population growth	p. 50
The Verhulst--Pearl equation	p. 51
Growth of a yeast population	p. 53
Growth of two sheep populations	p. 55
Quasi-equilibrium probabilities	p. 58
Simulation of the general population process	p. 59
Simulation results	p. 60
Derivation of inter-event times	p. 63
The Normal approximation to the quasi-equilibrium probabilities	p. 64
Probability of ultimate extinction	p. 65
Mean time to extinction	p. 66
Probability of extinction by time t	p. 68
Derivation of an approximation for p[subscript o](t)	p. 69
Comparison of approximate quasi-equilibrium probability distributions	p. 70
Logistic mean and variance values	p. 71
Derivation of the variance approximation	p. 73
An approximate skewness result	p. 74
A numerical comparison	p. 75
The diffusion approximation	p. 78
Equilibrium probabilities	p. 79
Justifying the Normal approximation	p. 81
Application to the yeast and sheep data	p. 81
Yeast data	p. 81
Tasmanian sheep data	p. 82
South Australian sheep data	p. 83
Appendix	p. 84
Time-lag models of population growth	p. 87
Introduction	p. 88
Reaction time-lag--deterministic analysis	p. 90
Stability conditions	p. 90
Numerical solutions	p. 92
Reaction time-lag--stochastic analysis	p. 94
More general deterministic models	p. 96
Distributed time-delay	p. 97
Periodic and chaotic solutions	p. 100
Three simple deterministic models	p. 100
More general deterministic results	p. 105
Stochastic results	p. 107
Spectral representation	p. 110
Final comments	p. 113
Analysis of field and laboratory data	p. 114
Nicholson's blowflies	p. 116
Description of the data	p. 117
A simple time-delay model	p. 118
Nisbet and Gurney's time-delay model	p. 119
Simulation of the two blowfly models	p. 122
Appendix	p. 125
Competition processes	p. 128
Introduction	p. 129
Experimental background	p. 131
Gause's yeast experiments	p. 131
Birch's grain beetle experiments	p. 135
Stability	p. 137
Local stability	p. 139
Global stability	p. 140
General stability conditions	p. 143
Stochastic behaviour	p. 146
Park's flour beetle experiments	p. 146
Gause's competitive exclusion principle	p. 148
Simulation of two-species competition	p. 149
Probability equations	p. 154
Extinction	p. 156
Extinction probabilities	p. 156
Times to extinction	p. 160
Quasi-equilibrium probabilities	p. 161
Predator-prey processes	p. 166
The Lotka-Volterra model	p. 167
Local deterministic solution	p. 169
Biological investigations	p. 171
Simulation of the stochastic model	p. 173
A trajectory indicator	p. 175
Final comments	p. 176
The Volterra model	p. 176
Deterministic trajectories	p. 177
General local solution	p. 178
Local Volterra solution	p. 180
The coefficient of variation	p. 182
Comparison with simulated runs	p. 185
Autocorrelation representation	p. 185
Mean time to extinction	p. 189
Cross-correlation representation	p. 190
Some stochastic thoughts	p. 191
The Leslie and Gower model	p. 191
The Holling--Tanner model	p. 192
Stability analysis	p. 194
Simulation of the stochastic model	p. 197
A model for prey-cover	p. 199
Stability analysis	p. 199
Deterministic versus stochastic behaviour	p. 201
Final comments	p. 203
Spatial predator--prey systems	p. 205
Huffaker's experiments	p. 205
Simulation of the spatial Lotka--Volterra model	p. 208
Spatial simulation approach	p. 209
Results for model A	p. 210
Results for model B	p. 213
Follow-up remarks	p. 214
Matrix representation	p. 214
Four-state representation	p. 215
Eight-state representation	p. 216
Simulation of the eight-state representation	p. 220
Fluctuating environments	p. 223
Deterministic variability	p. 223
Examples	p. 225
A simulation run	p. 227
Local solutions	p. 228
Conclusions	p. 231
Jillson's flour beetle experiment	p. 233
Stochastic behaviour with deterministic variability	p. 236
The stochastic equation	p. 236
Autocovariance results	p. 238
Mean time to extinction	p. 240
Random environments	p. 241
Four particular models	p. 242
The autoregressive model (D)	p. 245
Comparison of the autocovariances	p. 248
Coherence time	p. 250
Period remembering or period forgetting?	p. 252
The Canadian lynx data	p. 253
Spatial population dynamics	p. 258
The simple random walk	p. 259
Position after n steps	p. 260
Use of the Normal approximation	p. 261
Absorbing barriers	p. 261
Reflecting barriers	p. 263
Brownian motion	p. 264
Application of diffusion processes	p. 266
Skellam's examples	p. 267
Broadbent and Kendall's example	p. 268
Stepping-stone models (1)	p. 272
Equilibrium probabilities	p. 273
The two-colony model	p. 275
Mean values	p. 275
Variances and covariances	p. 277
Three special cases	p. 278
Approximate probabilities	p. 279
Conditions for ultimate extinction	p. 281
Simulation	p. 281
Stepping-stone models (2)	p. 284
Special case of v[subscript 2] = 0	p. 285
Velocities for v[subscript 2] ] 0	p. 287
Comparison of diffusion and stepping-stone velocities	p. 287
Further results	p. 288
An application to the spread of Tribolium confusum	p. 289
Possible models	p. 290
A stepping-stone approach	p. 291
Discussion	p. 293
Simulation of the diffusion and stepping-stone processes	p. 295
The diffusion process	p. 295
The stepping-stone process	p. 297
Comparison of velocities	p. 298
Spatial predator--prey processes revisited	p. 299
The spatial Volterra model	p. 300
A spatial diffusion model	p. 304
Turing's model for morphogenesis	p. 310
Solution of the linearized equations	p. 312
A spatial predator-prey example	p. 314
Types of behaviour	p. 314
Application to a spatial Volterra system	p. 317
Simulated stochastic waves	p. 319
Epidemic processes	p. 324
Introduction	p. 324
Simple epidemics	p. 325
The epidemic curve	p. 326
Duration time	p. 328
General epidemics	p. 330
The epidemic curve	p. 330
The deterministic threshold theorem	p. 331
The stochastic threshold theorem	p. 332
Stochastic realizations	p. 333
Recurrent epidemics	p. 336
Deterministic analysis	p. 338
Stochastic considerations	p. 341
The Lotka-Volterra approach	p. 343
An extension to malaria	p. 344
The basic deterministic model	p. 345
Stochastic comments	p. 348
Spatial models	p. 350
Deterministic spread	p. 351
Stochastic spread	p. 353
A carcinogenic growth process	p. 357
Linear and branching architectures	p. 360
The spatial distribution of Epilobium angustifolium in a recently thinned woodland	p. 362
Two-dimensional spectral analysis	p. 362
A simple cosine wave example	p. 363
Analysis of the willow herb data	p. 364
Simulation of directional spread	p. 366
Spatial branching models for canopy growth	p. 368
Honda's deterministic model	p. 369
Description of measurements	p. 369
Simulation of canopy structure	p. 372
Spatial branching models for structural root systems	p. 373
Description of measurements	p. 375
A temporal simulation model	p. 377
A fixed-time simulation model	p. 379
Discussion of the role of simulation	p. 380
References	p. 385
Author index	p. 394
Subject index	p. 397
Table of Contents provided by Syndetics. All Rights Reserved.

Modelling Biological Populations in Space and Time

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