Murray J.D. Mathematical Biology: I. An Introduction
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546 Index
Marriage model, equations, 155
Martiel, J.-L., 219, 406
Masland, R.H., 485, 499
Maternity function, 132
Maximum
economic yield (harvesting), 67
growth rate, 31
sustainable yield, 30, 64
May, R.M., 15, 16, 30, 33, 99, 100, 102, 109, 118,
318, 340
McFadden, D., 142
McKean, H.P., 443, 467
McKendrick, A.G., 123, 318, 320, 325
McLean, A.R., 336, 344
McLeod, J.B., 461
McNease, L., 123
Measles, 327
Measurements at different magnification, 486
Medaka, 437, 467, 468
eggs, 467
Meinhardt, H., 206
Menten, M.I., 175
Metabolic control mechanism, 222
Metabolic feedback, 201
Metz, J.A.J., 36
Metzen, W.D., 123
Meyer, E.R., 119
MFD (male determining factor), 142
Michaelis constant, 177, 187
Michaelis, L., 175
Michaelis–Menten
reaction, 175
uptake, 186, 187
Milton, J.G., 21, 28
Mira, C., 62, 109
Mirollo, R.E., 296
Mitchell, A.R., 443
Mitchell, P.J., 401
Mitchie, C.A., 344
Mittler, J.E., 350
Miura, R.M., 511
Mogilner, A., 405
Mollison, D., 442
Monk, A., 219, 406
Monod, J., 221
Montague, P., 485
Morales, M., 119
Morphogenesis, chemical theory, 401
Morse, P., 408
Mosquito swarm, 404
Motoneuron, 423
mRNA (messenger ribonucleic acid), 218, 221
Murray, J.D., 20, 72, 121, 123, 136, 137, 146, 175,
180, 182, 185, 198, 236, 246, 260, 261, 265,
269, 277, 365, 378, 392, 396, 401, 405, 408,
413, 416, 417, 438, 444, 454, 455, 459, 465,
482
Mushroom (reaction kinetics), 208–210, 212
Muskrat, 478
Mutualism, 79, 99
Myers, J.H., 17
Myoglobin, 401
Nagumo equation, 467
Nagumo, J.S., 239, 240
Namias, V., 511
Natural selection, chaos, 103
Neal, E.G., 369
Neanderthal extinction, 115
Negative feedback loop (biological control), 222
Nelson, P.W., 336, 341, 350
Nerve action potential, 218, 467
Neu, J., 293, 429, 430
Neubert, M.G., 402
Neural
activity oscillation, 218
bursting activity, 423
Hodgkin–Huxley theory, 239
signalling, 244
Neural model
Hodgkin–Huxley (nerve membrane), 239
Neuron, 239, 289, 422
periodic firing, 242
Neurotransmitter, 425
Newman, W.I., 452
Nichols, J.D., 122, 123
Nicholson, A.J., 15, 16, 71, 84
Nijhout, H.F., 438
Nile perch (Latis niloticus), 104
Nisbet, R.M., 16, 88, 136
Nondimensionalisation, 7
Nonlinear maps, 45
Nonlocal effects, 408
Normal (Gaussian) probability distribution, 396
Notochord, 423
Nowak, M.A., 336
Noyes, R.M., 258, 259, 261, 276
Obervational coding (marital interaction), 149
Odell, G.M., 465, 501
Odum, E.P., 84
Oestrogen
breast cancer, 252
Okubo, A., 399, 402, 404, 405
Oldstone, M., 316
Onchocersiasis (river blindness), 361
Oregonator (BZ) model (oscillating reaction), 260
Oscillation, 275
Oscillator
annihilation, 292
Index 547
biological, 218, 226
black holes, 278, 293
BZ (Belousov–Zhabotinskii) model, 260
chemical (BZ reaction), 258
coupled, 278, 293
determination of parameter space, 235
detuning, 433
independent, 419
λ − ω system, 238
neural, 424
Oregonator, 260
perturbed, 278, 282
phase-coupled, 430, 431
relaxation, 231
simple example, 282
stability, 435
two-species models, 234
weak coupling, 427
Oster, G.F., 465
Othmer, H.G., 21, 30, 201, 219, 222, 224, 225,
233, 406, 410
Pacemaker, periodic (heart), 278
Panico, J., 487
Parameter space
linear stability, 90
parametric method, 235
two-species oscillations, 235, 236
Parasite
blood fluke, 360
experimental observations, 353
immunological response, 360
infection
coccidia, 361
Parasite (helminth)
acquired immunity, 351
experiments, 353
Heligmosoides polygyrus, 352
immune threshold, 354
immunological response, 353, 354
infection model, 351
population dynamics, 351
survival, 354
Trichostrongylus retortaeformis, 416
Parasite model
goals, 360
high protein diet, 357
low protein diet, 356
Parry, S., 361
Pasanen, E.G., 142
Paumgartner, D., 486, 494
Paveri-Fontana, S.L., 318
Pearl, R., 3, 4
Peitgen, H.-O., 57, 58, 109, 484, 489
Peng, B.B., 234
Perception, cusp catastrophe, 11
Peregoy, P.L., 11
Perelson, A.S., 180, 336, 337, 341
Period doubling, 30, 55, 103
Periodic
bursting, 232
cell division, 219, 221
changes in enzyme synthesis, 221
chaotic sequences, 276
emergence (eclosion) of fruit flies, 218
emission of cyclic AMP (Dictyostelium), 219
neuron firing, 240
pacemaker, 278
pacemaker cells, 291
solutions of feedback control mechanisms, 224
testosterone (hormone) level, 244
Peskin, C.S., 1, 26, 44
Pest control, 114
Peterman, T.A., 394
Phase, 279, 418
critical, 279
indeterminate, 288
lag, 423
locked, 424
locking, 293, 295, 424, 434
resetting (in oscillators), 280, 282
resetting curves, 282
shift, 15, 279, 283
Phase plane singularities, 502
Phase resetting in oscillators, 280
black hole (singularity), 286
Type 0 curves, 284
Type 1 curves, 282
Pheromone, 405
Phyllotaxis, 46
Physiological diseases, 21
Cheyne–Stokes respiration, 21
regulation of haematopoiesis, 26
Pianka, E.R., 99
PID (pelvic inflammatory disorders), 328
Piot, P., 333
Pitchfork bifurcation, 52
Plague
of Athens, 315
Bombay epidemic, 325
bubonic, 326
pneumonic, 326
Plant, R.E., 30, 244
Poisoning, carbon monoxide, 401
Poole, J.C.F., 315
Pooley, A.C., 119, 122
Population
birth rate, 3
carrying capacity, 3
competition models, 94
548 Index
Population (continued)
crash-back, 70
extinction, 71
France, 5
harvesting, 30
hysteresis, 8
lemming, 16
logistic growth, 3
maximum growth rate, 31
predation, 7
predation threshold, 7
recovery, time, 31
self-regulation, 48
sigmoid growth, 3
U.S., 5
vole, 16
world, 2
Population biology, 1
Population model
age structured, 36
cautionary remarks, 101
conservation equation, 2
continuous interacting species, 79
continuous single species, 1
delay, 13
discrete (interacting species), 109
discrete (single species), 44
general, 101
harvesting, 30
insect outbreak, 7
Kolmogorov, 101
Leslie matrices, 36
mutualism, 99
renewable resources, 1
symbiosis, 99
Porous media equation, 403
Positive feedback loop (biological control),
222
Post-fertilisation (egg) waves, 469
Powell, J.H., 316
Prawda, J., 385
Predator-prey
convective model, 482
density-dependent (discrete) model, 113
discrete growth model, 109
interacting populations, 109
parameter domain of stability, 89
realistic models, 86
Prigogene, I., 253
Principle of Competitive Exclusion, 94
Prions, 391
Prostrate, 219
Protease inhibitor, HIV, 342
Pseudo-steady state hypothesis, 186
Pulmonary blood flow, 486
Raggett, G.F., 326
Rand, R.H., 422, 429
Random walk (diffusion), 395
biased, 399, 416
Range exapnsion, 478
Rapp, P.E., 224, 225
Rate constants, 176
RCISS (marital), 148
scores, 152
Reaction
Belousov–Zhabotinskii (BZ) reaction, 257, 271
bistability, 212
hydrogen peroxide–iodate ion, 219
hysteresis (steady), 208
iodate–arsenous acid, 212
isolas, 212
kinetics, 175
Lotka, 203
matrix (stability), 204
mushroom (steady state), 212
rate, 186
rate constant, 176
rate limiting step, 187
uptake (velocity), 186, 200
Reaction diffusion chemotaxis
equation, 407
system, 407
Reaction diffusion equations, 411
density-dependent diffusion, 449
exact solution, 449
excitable kinetics, 466
nonlinear convection, 454
scalar, 438
Reaction diffusion mechanism
chemotaxis, 407
Reaction kinetics, 175, 176
activation, 201
activator–inhibitor, 206
autocatalysis, 201
bistability, 212
Brusselator, 253
complex solution behaviour, 231
cooperative phenomena, 197, 224
fast dynamics, 231
first order, 203
gradient system, 217, 254
hysteresis (steady state), 209
inhibition, 201
iodate–arsenous acid, 212
isolas, 209, 210
λ − ω model, 238
Lotka, 203
model autocatalysis, 211
multiple steady state, 208
mushrooms (steady state), 208, 210
Index 549
necessary and sufficient conditions for stability,
226
null clines–steady state local behaviour, 226
periodic bursting, 232
rate limiting step, 187
‘simplest’ (limit cycle), 234
slow dynamics, 232
stability, 205, 226
threshold behaviour, 208
tri-molecular (limit cycle), 234
Red spider mite, 114
Regulation of haematopoiesis, 26
delay model, 26
oscillations, 28
Relaxation oscillator, 231, 268, 276
Belousov–Zhabotinskii (BZ) reaction, 259, 271
model for FKN mechanism, 276
period, 269
period of Field- K
¨
or
¨
os-Noyes (FKN) model for
BZ reaction, 276
Renewal equation, 46
Renewal matrices, 46
Renshaw, E., 402
Rensing, L., 21
Retinal ganglion cell, 485
Rhen, T., 142
Richter, P.H., 57, 58, 109
Ricker curve, 49
Rinzel, J., 218, 231, 233, 239, 241, 296
River blindness (onchocersiasis), 361
Roberts, D.V., 175
Robertson, H.T., 486
Rogers, L.M., 384, 390
R
¨
ossler, O.E., 103
Rotenberg, M., 30
Rothen, F., 47
Roughgarden, J., 44, 402
Routh–Hurwitz conditions, 507
Rubella, 327
Rubin, R., 358
Rubinow, S.I., 175, 198, 401
Sackellares, J.C., 75
Sadler, M.T., 123
Salk, J., 3
Sanchez, D.A., 33
Sarkovskii, A.N., 56
Satsuma, J., 459
Schaap, P., 219, 233
Schaffer, W.M., 83, 103
Schenzle, D., 378
Schierwagen, A.K., 499
Schistosomiasis (Bilharzia), 104, 329
Schmitz, G., 480
Schnakenberg, J., 234
Scott, S.K., 62, 210, 212, 265, 276
Segel, L.A., 7, 175, 178, 219, 406, 407, 437
Self-organisation, spatio-temporal, 257
Sex attractant, 405
Sex determination, TSD versus GSD, 139
Sex ratio, skewed, 122
Sharpe, F.R., 123
SHBG (sex hormone binding globulin), 244
Sheep-blowfly (Lucila cuprina), 15, 16
Shigesada, N., 402–405, 471, 478, 481
Shock solution, 455, 459
Showalter, K., 212–215, 233, 440
SI model
age dependent, 361
criss-cross disease model, 329
Sierpinski fractal, 488
Silk moth (Bombyx mori), 405
SIR (epidemic) models, 320
SIRM (Sterile Insect Release Method) pest control
method, 117
Skellam, J.G., 399, 478
Slater, A.F.G., 352
Slemrod, M., 178
Smallpox, 318
Smith, A.M.A., 122, 123
Smith, D.A., 401
Smith, G.D., 369
Smith, H.L., 102
Smith, P., 20, 505
Smith, W.R., 245, 246
Sneyd, J., 21, 218, 232, 239
Southwood, T.R.E., 71
Sowunmi, C.O.A., 133, 139
Space filling, 496
curve, 497
Sparrow, C., 58, 103
Species invasion, 478
driving force, 480
Sperm entry point, 468
Spinal
cord, 422
transection, 422
Spiro, P.A., 406
Squid (giant), 239, 289
Stability
necessary and sufficient conditions, 91
parameter domain, 91, 92
travelling wave, 447
STD (sexually transmitted disease), 327
contact rate, 331
Stearns, S.C., 133, 141
Stefan, P., 56
Sterling, P., 485–487
Stewart, I.N., 11, 295
Stimson, W.H., 252, 253
550 Index
Stimulus-timing-phase singularity, 280
Stirzaker, D., 16
Strogatz, S.H., 62, 102, 103, 263, 295, 296, 484
Stuart, F.A., 384
Stuchl, I., 293, 294, 313
Substrate, 175
suicide, 188
inhibition, 204, 216
Suicide substrates, 188
Survival reproductive level, 2
Swimming pattern, 424
Switch
biological, 226, 230
hysteresis, 230
Symbiosis, 79, 99
Synchronisation in fruit flies, 295
T-cell recovery (HIV), 344
Taddei-Ferretti, C., 288
Tangent bifurcation, 51, 56
Tatsnami, S., 196
Tauchi, M., 485, 499
Temperature dependent sex determination (TSD)
molecular mechanism, 142
TSD, 121
Temperature sensitive period (TSP), 143
Testosterone, 143
conditions for stability of model’s steady state,
246
control model, 244, 246
Thoenes, D., 419
Thomas
kinetics, 228
mechanism, 204
Thomas, D., 204, 206, 228
Thornley, J.H.M., 47
Threshold
age structured population, 36
FitzHugh–Nagumo (piecewise linear) model,
243
phenomena, 105, 207
phenomenon (epidemic), 321
reaction kinetics, 208, 230
Thucydides, 315
Tilman, D., 402, 438
Titchmarsh, E.C., 448
Topsell, E., 120
Tranquillo, R.T., 406, 408
Travelling wave, 437
form, 439
general results, 454
stability, 447
Trichostrongylus retortaeformis (parasite worm),
416
dispersal model, 416
Trophic levels, 101
Trophic web, 79
TSD (temperature dependent sex
determination), 121
age-structured model, 130
lizards, 123
molecular mechanism, 142
turtles, 123, 142
TSE, 391
Tuberculosis
badgers and cattle, 369
criss-cross infection, 370
human, 317, 334
Tung, K.-K., 151
Turing, A.M., 401
Turtle, snapping, 142
Turtles, TSD, 142
Tyson, J.J., 201, 221, 222, 224, 225,
258–260, 268, 270, 276, 440
Tyson, R., 408
Uppal, A., 210
Vaccination, 318, 322
free ride, 322
herd immunity, 322
van den Driessche, P., 20
Van der Pol equation, 269
V
´
aradi, Z.B., 418
Vegetal pole, 468
Venereal diseases, 327
chlamydia, 327
contact matrix, 331
control model, 332
gonorrhea, 327
syphilis, 327
Ventral root, 423
Verhulst process, 49
Verhulst, P.F., 3
Volterra, V., 79
Von Foerster equation, 37
similarity solution, 39
von Foerster, H., 123
von Koch curve, 487
construction, 488
Waaler, H.T., 384
Waley, S.G., 196
Wall
´
en. P., 422, 423
Walsh, J.A., 351
Waltman, P., 99
Warren, K.S., 351
Watts, S., 316
Watzlawick, P., 164
Index 551
Wave
activation, 468
calcium, 437
stimulated calcium release mechanism, 464
exact solution, 464
exact solution with excitable kinetics, 464
front, 419
gene-culture, 444
kinematic, 418
multi-steady state kinetics, 460
propagating, 439
pseudo, 419
speed, 439
speed dispersion relation, 456
spread of farming, 444
steepness, 446, 451
travelling, 437
variable, 439
Wave solution, exact, 452
Wave speed
few dispersers, 480
Wavefront solution, 440
asymptotic form, 445
excitable kinetics, 466
Fisher–Kolmogoroff equation, 441
stability, 444
Wavespeed
variable diffusion, 471
Weak solution, 459
Webb, G.F., 336, 344
Webb, G.J.W., 122, 123
Weiss, R., 333
Whale, baleen model, 41
Whipworm, 351
Whittaker, R.H., 100
WHO (World Health Organisation), 317
Whorton, 143
Wibbels, T., 144
Willems, J.L., 508
Williamson, M.H., 83, 402
Winfree, A.T., 218, 257, 258, 278, 279, 281, 282,
286, 288, 291, 293, 295
Wittenberg, B.A., 401
Wittenberg, J.B., 401
Woodward, D.E., 123, 136, 137, 408
World Health Organisation (WHO), 317
Wyman, J., 401
Yagil, E., 222
Yagil, G., 222
Yellow fever epidemic, 316
Yorke, J.A., 56, 328, 331, 332
Zeeman, E.C., 11, 13, 147, 437
Zhabotinskii, A.M., 220, 257
Zou, X., 20