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Cambridge University Press
0521830451 - The Theoretical Biologist’s Toolbox - Quantitative Methods for Ecology and Evolutionary Biology - by Marc Mangel
Index
Index
AIC (Aikaike Information Criterion) 131–132
AIDS
evolution of drug resistance in HIV 182
viral dynamics 208
Alabama beach mouse (Peromyscus polionotus ammobates) 298, 299
algebra, two prey diet choice problem 3–5, 18
Allee effects 290
Anderson’s theory of vitality 314–316
Anisopteromalus calandre (parasitoid) 137
Anopheles spp. (mosquitoes), vector of malaria 188–193, 194, 207
Aphytis lingnanensis (parasitoid) 134, 150, 151
arithmetic mean (arithmetic average) 31–35
Asobara tabida (parasitoid) 137
asymptotic expansion 113–114, 129
asymptotic normal theory 127
asymptotic size 25–27
Atlantic cod (Gadus morhua) stocks 216–217
Atlantic salmon (Salmo salar), egg size and parent–offspring conflict 8–10, 18
backward equations 268–272, 276–279, 284
backward iteration 155, 166
Bayes, Thomas 125–127
Bayes’s Theorem 82, 83–84
Bayesian methods 91–95
fishery stock assessment analysis 100–101
statistical analysis 125–127, 128–129
updating of t-distribution parameter 130
behavior and population dynamics combined 155–159, 160, 166–167
Bernoulli, Daniel 73–74, 88–92
Bernoulli trials 88–92
beta density 94–95, 123–124, 125
conjugate prior for the binomial parameter 128–129
beta function 127
Beverton, Ray 29, 30, 73
Beverton Holt stock–recruitment relationship 212, 213–215, 239–241
BIC (Bayesian Information Criterion) 131–132
bifurcations 40–48, 74–76
binomial coefficient 88–92
binomial distribution 88–95
Poisson limit of the binomial 100
biodemography of survival 311–314, 319–320
bioeconomics and overfishing 218–224, 241–242
bioinformatics 168–169
blue noise 283
box model (Freedman) for appropriate probability model 101, 102
Brownian motion 77–78, 251–254, 260–264, 282
BSE (mad-cow disease) 208
calculus
egg size in Atlantic salmon (Salmo salar) 8–10, 18
extraordinary sex ratios 10–12, 18
stochastic calculi 282, 318–319
Callosobruchus chinesis (bruchid beetle) 137
catastrophe theory 45–48, 74–76
catastrophic changes in population size 294–296, 318
Cauchy distribution 120–121
chaos theory and complexity 40–43, 74
Chapman–Kolmogorov equation (Master Equation) 270, 272
chi-square distribution 115–116
cholera 208
coefficient of variation 87–88
conditional probability 81–84, 85–86
confidence intervals 93, 94
conjugate priors 112, 128–129
contagion and virulence 176–178, 182, 183
continuous random variables 84–85, 86–88
CPUE (Catch Per Unit Effort) 217–218, 229–231, 240–241
Creutzfeldt–Jacob disease (CJD) 208
cusp bifurcation 45–48, 74–76
cusp catastrophe 45–48, 74–76
Darwinian fitness see fitness measures
Darwinian gradualism, challenges to 309–311
data sampling and appropriate probability model 101, 102
Dawson’s integral 307–309
delay differential equations 164
delay differential models, host–parasitoid dynamics 164
deterministic chaos 40–43, 74
differential equations
bifurcations 40–48, 74–76
classification of steady states 48–58, 74–76
diffusion and exponential growth 64–69, 78
diffusion and logistic growth 69–73, 79
diffusion as a random walk 58–64, 77–78
discrete logistic map 38–43, 74
individual growth 23–29, 30, 73
life history invariants 29, 30, 73
linear and nonlinear diffusion 79
logistic equation 36–38, 74
measures of fitness in fluctuating environments 31–36
population growth in fluctuating environments 31–36, 73–74
predation and random search 20–23, 24
two-dimensional 48–58
diffusion
and exponential growth 64–69, 78
and logistic growth 69–73, 79
as a random walk 58–64, 77–78
in a bounded region 62–64, 78
in an unbounded region 60–62, 78
linear and nonlinear 79
model of the process 58–64, 77–78
reaction-diffusion equations 79
see also stochastic population dynamics; stochastic population theory (ecological applications)
diffusion approximation 318–319
diffusion equation definition 59–60, 77–78
Dirac, Paul 61–62, 78
Dirac delta function 61–62, 78
discounting (bioeconomics) 221–224, 241–242
discrete logistic map 38–43, 74
discrete random variables 84–85, 86–88
disease see population biology of disease
disease transmission models 171–173
distribution function 84–85
domains of attraction 49–50, 51
escape from 285–287, 317, 319
Drosophila subobscura (fruit fly) 134–137
ecological applications of stochastic differential equations 283–284
see also stochastic population theory (ecological applications)
ecological aspects of disease models 207
ecosystem-based fisheries management 244–246
Ecosystem Advisory Panel Report 244–246
egg size in Atlantic salmon, parent–offspring conflict 8–10, 18
eigenvalues 53–58, 71–73
eigenvectors 53–58, 71–73
Einstein, Albert 251, 282–283
Eldredge, Niles 309–311
error distribution, normal (Gaussian) distribution 114–116
errors in variables 130
escape from a domain of attraction 285–287, 317, 319
ESS (Evolutionarily Stable Strategy) 11–12, 18–19, 182–184
ESY (Ecologically Sustainable Yield) 238
Euler–Lotka equation of population demography 311–314, 319–320
events 81–84, 85
evolutionary theory
biodemography 311–314, 319–320
escape from a domain of attraction 285–287, 317, 319
punctuated equilibrium 309–311, 319
transitions between adaptive peaks 302–311
exercises, importance of 4–5
expectation 86–88
experiments 81–82
exponential distribution function 85–86
extinction times 130
connecting models and data 319
density independent diffusion approximation 292, 297–301
escape from a domain of attraction 285–287, 317
general density dependent case 301–302
MacArthur–Wilson theory of 287–293, 317–318
role of a ceiling on population size 293–296, 318
extraordinary sex ratios 10–12, 18
Feller, William 268–269
Feynman, Richard 282
Feynman–Kac formula 276–278, 282
financial engineering 320–322
fish stock assessment 94–95
Fisher, R. A. 10–12, 18, 69–73, 125–127
Fisher equation 69–73, 79
fisheries
as an agent of selection 244
ecosystem-based approach to management 244–246
fishery system 210–212, 238
optimal age at maturity 27–28
relative size at maturity 29, 30, 73
salmon life histories 227, 242–243
sustainability issues 210
fisheries models
age structure 224–227, 239–241
Atlantic cod (Gadus morhua) stocks 216–217
Bayesian methods in stock assessment and management 241–242
behavior of fishermen 239
Beverton Holt stock–recruitment relationship 212, 213–215, 239–241
bioeconomics and overfishing 218–224, 241–242
Catch Per Unit Effort (CPUE) 217–218, 229–231, 240–241
discounting (bioeconomics) 218, 221–224, 241–242
Ecologically Sustainable Yield (ESY) 238
hake (Merluccius capensis and M. paradoxus) 229–231
marine reserves model 231–236, 243–244
Maximum Net Productivity (MNP) 216, 218
Maximum Sustainable Yield (MSY) 216, 218, 241
open and closed populations 214–215
Optimal Sustainable Population size (OSP) 217–218
process uncertainty and observation error 228–231, 238, 243
Ricker stock–recruitment relationship 212, 213, 239–241
risk analysis in decision making 236–237, 246–247
salmon fisheries management 227–228, 242–243
Schaefer model and its extensions 215–218, 220–221
stochastic models 228–231, 233–236
stock–recruitment relationships 212–215, 239–241
targets, thresholds and reference points 241
use for management 238
yield per recruit 225–227
fishery science
Ricker map 39–40
Ricker recruitment function 74
fitness measures
energy acquisition 3–5
number of grand offspring 11–12
per capita growth rate 31–36
fluctuation and dissipation 282–283
focus (steady state) 54–55, 71–73
foraging in patchy environments 2–8, 18
marginal value theorem (plane geometry) 5–8, 18
two prey diet choice problem 3–5, 18
forward equations 272–276, 278, 284
forward iteration 155, 166
Fourier coefficients 65, 66–69, 78
Fourier series 65, 66–69, 78
Frank, F. C. 56–58, 76–77
frequency dependent model for disease transmission 172, 173
frequentist approach to statistical analysis 125–127
fruit flies, lifestyles 134
gambler’s ruin
in a biased game 257–260
in a fair game 253, 254–257
rare events happen quickly 309
gamma density 103–104, 105–107
conjugate prior for the Poisson process 112, 128–129
in the negative binomial distribution 106, 107–112
gamma function 104–105, 127
Gaussian (normal) distribution 112–116
Gaussian white noise 261–264
Generalized Function 61–62, 78
Generalized Linear Model 80–81
genetic models
diffusion processes in population genetics 284
spread of an advantageous allele 69–73, 79
genomics 168–169
geometric mean (geometric average) 32–35
Gompertz mortality model 319–320
Gould, Stephen J. 309–311
growth equations, individual 23–29, 30, 73
hake (Merluccius capensis and M. paradoxus) 229–231
Halticoptera rosae (parasitoid) 134
Hamilton, W. D. 10–12, 18, 312
helminth worm parasites 193–201, 208
accounting for free-living stages 199
ecological setting for host–parasite dynamics 199–201
underlying host–worm model 194–198
hepatitis C virus spread 169–170
HIV see AIDS
Hopf bifurcation 76
host–parasite interactions see helminth worm parasites
host–parasitoid dynamics see parasitoids
hypothesis testing 113–114
immune system see optimal immune response
independent increments 282
Individual-Based Models (IBMs) 166 see also forward iteration
individual growth equations 23–29, 30, 73
invasion biology 69–73, 79
unbeatable (ESS) level of virulence 182–184
island biogeography, MacArthur–Wilson theory 287–293, 317–318
isocline analysis 56–58, 71–73
iteration, forward and backward 155, 166
Ito calculus 253–254, 282, 318–319
Jensen’s inequality 33–34
Kac, Mark 276–278, 282
Kermack–McKendrick epidemic theorem 174–175, 177
Kimura, M. 306–307
Kolmogorov backward equation 268–272
Kolmogorov forward equation 272–276
law of total probability 83–84
least squares 116–119
Leptopilinia heterotoma (parasitoid) 134
Levin’s patch model 74–76
life history invariants 29, 30, 73
life tables 311–312, 319–320
Lighthill, M. J. (Sir James) 61–62, 78
likelihood 91–95, 116, 127
likelihood ratio 131–132
linear regression 116–118
linear superposition of solutions 52–58
log-likelihood 91–95, 116
log-normal distribution 121–122
logistic equation 36–38, 74
Lotka–Volterra competition equations 48–49, 56–58
Lotka–Volterra mutualistic interaction equations 48–49, 77
Lotka–Volterra predator-prey equations 48–49, 57–58
Lotka’s renewal equation for population growth 311–312, 319–320
Lucilia cuprina (Australian sheep-blowfly) 149–150
MacArthur–Wilson theory of extinction time 287–293, 317–318
macroparasites 168–169
malaria
annual death rate 188–189
caused by parasitic Plasmodium spp. 189–190, 207
cycle of infection 189–190
history of study and fight against 188–189
mosquito vectors (Anopheles spp.) 188–193, 194, 207
standard vector model 190–193, 194, 207
vector-based disease 188–193, 194, 207
marginal value theorem 5–8, 18, 178, 182, 183
marine reserves (marine protected areas) model 231–236, 243–244
Markov process 260–261
mass action model for disease transmission 169–170, 172, 173
Master Equation see Chapman–Kolmogorov equation Chapman–Kolmogorov equation
mathematics, pure and applied 15
mean 86–88
mean-variance power laws 127–128
metapopulation ecology 317–318
microparasites 168–169
MLE (maximum likelihood estimate) 91–95
MNP (Maximum Net Productivity) 36, 37, 216, 218
model selection, via likelihood ratio, AIC and BIC 130–132
model testing, methods 129
moments 86–88
mortality
Gompertz mortality model 319–320
see also predation and random search
mortality plateaus 320
MSY (Maximum Sustainable Yield) 216, 218, 241
multinomial distribution 95
mutualism 48–49, 77
Nasionia vitripennis (parasitoid) 144
negative binomial distribution (first form) 102–103
negative binomial distribution (second form) 103–104, 106, 107–112
comparison with Poisson distribution 106, 109, 110, 111
negative binomial model of disease transmission 172–173
Neyman, Jerzy 125–127
Nicholson–Bailey model (population dynamics) 135–137
effects of host refuges 137, 142–145
effects of multiple attacks 143–145
instability 135–140
stabilization 137, 141–145, 164
variation in attack rate 137, 141–143
no-flux boundary conditions 62–64
noise
blue 283
Gaussian white 261–264
red 283
non-informative priors 128–129
non-invadable sex ratio 10–12, 18–19
non-negative measurements 121–122
normal (Gaussian) distribution 112–116
normal probability density function 112–113
observation error and process uncertainty 228–231, 238, 243
open and closed population models 214–215
optical activity, and spontaneous asymmetric synthesis 56–58, 76–77
optimal age at maturity 27–28
optimal foraging theory 317–318
optimal immune response 201–205, 208
T-cell phenotypes in multiple infections 201–203
trade-off with reproduction 203–205
optimal virulence level 176–178, 182, 183
Ornstein–Uhlenbeck process 264–268, 282–283
oscillations 54–55
relaxation oscillations 46, 47–48, 74–76
OSP (Optimal Sustainable Population size) 217–218
parasite burden
costs to an organism 203–205
optimal response to 203–205
parasites see helimith worm parasites
parasitic wasps, sex ratio bias 10–12, 18
parasitoids
behavior and population dynamics combined 155–159, 160, 166–167
classification of life histories 133, 135
delay differential models for host–parasitoid dynamics 164
egg limitation on reproductive effort 159–164
evolution of host choice 150–155, 165
Nicholson–Bailey model 135–137
Nicholson–Bailey model instability 135–140
Nicholson–Bailey model stabilization 137, 141–145, 164
overlapping generations in continuous time 145–150
pheromone marking of hosts 165
reproductive success factors 159–164
spatial aspects of host interaction 164
time limitation on reproductive effort 159–164
typical species 133, 134
wide range of topics for investigation 165
parent–offspring conflict, egg size in Atlantic salmon 8–10, 18
patch leaving 123–124, 125, 165
path integrals 282
per capita growth rate 36–38, 74
as a measure of fitness 31–36
spatial variation 31–32
temporal variation 32–36
persistence time
density independent diffusion approximation 292, 297–301, 299
general density dependent case 301, 301–302
see also extinction times
pest outbreak, relaxation oscillations 46, 47–48, 74–76
phase plane 49–50, 51, 56–58, 76
pheromone marking by parasitoids 165
plane geometry, marginal value theorem 5–8, 18
plankton bloom, relaxation oscillations 46, 47–48, 74–76
Plasmodium spp., cause of malaria 189–190, 207
Poisson distribution 95–100
comparison with negative binomial distribution 106, 109, 110, 111
Poisson increment 279–281
Poisson limit of the binomial 100
Poisson process, gamma density conjugate prior 112
population biology of disease
basic reproductive rate of a disease (R0) 171
cholera 208
complexity of models 169
contagiousness (infectiousness) 176–178, 182, 183
cultural and behavioral effects 207
demographic processes added to models 179–181
ecological aspects of disease models 207
evolution of virulence 178, 182–188, 206
force of infection 169–170
frequency dependent model for transmission 172, 173
general literature 205–206
helminth worms 193–201, 208
hepatitis C virus spread 169–170
Kermack–McKendrick epidemic theorem 174–175, 177
macroparasites 168–169
malaria 188–193, 194, 207
mass action model for transmission 169–170, 172, 173
microparasites 168–169
negative binomial model of transmission 172–173
optimal immune response 201–205, 208
optimal virulence level for a disease organism 176–178, 182, 183
power model for transmission 172
prion disease kinetics 208
relationship between virulence and contagion 176–178, 182, 183
relevance of 168–169
role of genomics and bioinformatics 168–169
SI model 169–171
SIR model of epidemics 173–178, 179–181, 206, 207
SIRS model of endemic diseases 178–181, 206, 207
spatial aspects of disease transmission 209
standard vector model for malaria 190–193, 194
stochastic epidemics 209
transmission between infected and susceptible individuals 171–173
transmission models 171–173
vector-based diseases 188–193, 194, 207
virulence (infectedness) 176–178, 182, 183
population demography, Euler–Lotka equation 311–314, 319–320
population dynamics and behavior combined 155–159, 160, 166–167
population dynamics models
advanced models 145–150
delay differential models 164
overlapping generations in continuous time 145–150
see also Nicholson–Bailey model
population genetics see genetic models
population growth
deterministic chaos 40–43, 74
in fluctuating environments 31–36, 73–74
rate of 36–38, 74
population oscillations, relaxation oscillations 46, 47–48, 74–76
population size
catastrophic changes in 294–296, 318
ceiling 293–296, 318
MacArthur–Wilson theory of extinction time 287–293, 317–318
power model for disease transmission 172
predation and random search 20–23, 24
predator–prey interactions 48–49, 57–58
prion disease kinetics 208
priors see conjugate priors; non-informative priors
probability density function 84–85
probability distributions
binomial distribution 88–95
chi-square distribution 115–116
log-normal distribution 121–122
multinomial distribution 95
negative binomial distribution (first form) 102–103
negative binomial distribution (second form) 103–104, 106, 107–112
normal (Gaussian) distribution 112–116, 115
Poisson distribution 95–100, 106, 109, 110, 111
t-distribution 80–81, 119–121, 130
probability model, connection between data sample and data source 101, 102
probability theory 81–88
Bayes’s Theorem 82, 83–84
coefficient of variation 87–88
conditional probability 81–84, 85–86
continuous random variables 84–85, 86–88
discrete random variables 84–85, 86–88
distribution function 84–85
events 81–84, 82, 85
expectation 86–88
experiments 81–82
exponential distribution function 85–86
law of total probability 82, 83–84
mean 86–88
moments 86–88
probability density function 84–85
random variables 84–85
sample space 81–82
standard deviation 87–88
variance 86–88
process uncertainty and observation error 228–231, 238, 243
punctuated equilibrium 309–311, 319
random search and predation 20–23, 24
random search with depletion 100–101
random variables 84–85
reaction-diffusion equations 79
red noise 283
reflecting boundary conditions 62–64
relative size at maturity 29, 30, 73
relaxation oscillations 46, 47–48, 74–76
renewal processes 18
resistance, evolution of 182, 206
Rhagoletis basiola (rose hip fly) 134
Rhagoletis completa (walnut husk fly) 134
Ricker map 39–40
Ricker recruitment function 74
Ricker stock–recruitment relationship 212, 213, 239–241
saddle point 49–50, 51, 54, 56–58, 71–73
salmon fisheries management 227–228, 242–243
salmon life histories 227, 242–243
see also Atlantic salmon
sample space 81–82
Schaefer model and its extensions 215–218, 220–221
Seber’s delta method 35–36
separation of variables 62–64, 78
sex ratio bias 10–12, 18
SI model of disease spread in a population 169–171
SIR model of epidemics 173–178, 179–181, 206, 207
SIRS model of endemic diseases 178–181, 206, 207
spatial aspects of disease transmission 209
spatial aspects of host–parasitoid interaction 164
spatial variation in per capita growth rate 31–32
spiral point (steady state) 54–55, 71–73
spontaneous asymmetric synthesis, and optical activity 56–58, 76–77
stable node 49–50, 51, 54, 56–58, 71–73
standard deviation 87–88
statistical analysis
Bayesian approach 125–127, 128–129
frequentist approach 125–127
steady states
classification 48–58, 71–73, 74–76
determination of stability 137–140
stochastic calculi 282, 318–319
stochastic differential equations 283
stochastic dynamic programming 151–155, 158, 161–164, 165–166
stochastic epidemics 209
stochastic harvesting equation 278–279
stochastic integrals 264–268
stochastic models 228–231, 233–236
stochastic population dynamics
alternative to Brownian motion 279–281
backward equations 268–272, 276–279, 284
Brownian motion 251–254, 260–264, 282
Chapman–Kolmogorov equation (Master Equation) 270, 272
derivative of Brownian motion 261–264
Feynman–Kac formula 276–278, 282
forward equations 272–276, 278, 284
from deterministic to stochastic dynamics 248–251
gambler’s ruin in a biased game 257–260
gambler’s ruin in a fair game 253, 254–257
Gaussian white noise 261–264
general diffusion processes 268–272
independent increments 282
Kolmogorov backward equation 268–272
Kolmogorov forward equation 272–276
Markov process 260–261
nature of stochastic processes 248
Ornstein–Uhlenbeck process 264–268, 282–283
path integrals 282
Poisson increment 279–281
stochastic differential equations 283
stochastic harvesting equation 278–279
stochastic integrals 264–268, 283
thinking along sample paths (trajectories) 248–251, 282
transition density and covariance of Brownian motion 260–261
stochastic population theory (ecological applications) 283–284
Anderson’s theory of vitality 314–316
biodemography of survival 311–314, 319–320
catastrophic changes to population size 294–296
connecting models and data 319
density independent diffusion approximation 292, 297–301
diffusion approximation 318–319
escape from a domain of attraction 285–287, 317
Euler–Lotka equation of population demography 311–314, 319–320
general density dependent case 301–302
life tables 311–312, 319–320
MacArthur–Wilson theory of extinction time 287–293
population genetics applications 284
punctuated equilibrium 309–311
role of a ceiling on population size 293–296
transitions between adaptive peaks 302–311
stock–recruitment relationships 212–215, 239–241
Stratonovich calculus 282, 318–319
Student’s t-random variable 119–120
superparasitism 165
survival
Anderson’s theory of vitality 314–316
biodemography of 311–314, 319–320
Euler–Lotka equation of population demography 311–314, 319–320
life tables 311–312, 319–320
t-distribution 80–81, 119–121, 130
Taylor, L. R. 127–128
temporal variation in per capita growth rate 32–36
theoretical biology
building intuition about biological systems 15–17
gaining mathematical skills 15–17
importance of writing skills 14–15, 19
interaction of mathematics and science 1–2
toolbox metaphor 12–15
total least squares 118–119, 130
traveling waves 69–73
two prey diet choice problem 3–5, 18
unbeatable (uninvadable) sex ratio 10–12, 18–19
unstable node 49–50, 51, 54, 56–58
variance 86–88
vector-based diseases, malaria 188–193, 194, 207
Verdi, Giuseppe 15–17
virulence
AIDS virus 182, 208
and contagion 176–178, 182, 183
coevolution with host response 184–188
drug resistance 182
evolution of 178, 182–188
leader–follower (Stackelberg) game 184–185
optimal level 178, 182, 183
timescale of evolution 182
unbeatable (ESS) level of virulence 182–184
viruses
spread of hepatitis C 169–170
viral dynamics and AIDS 182, 208
von Bertalanffy, Ludwig 23–29, 30
Wiener process 251
writing and the creative process 14–15, 19
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