1
Extinction risk depends strongly on factors contributing
to stochasticity
Brett A. Melbourne1 & Alan Hastings2
1
Department of Ecology and Evolutionary Biology, University of Colorado, Boulder CO
80309, USA
2
Department of Environmental Science and Policy, University of California, Davis CA
95616, USA
Extinction risk in natural populations depends on stochastic factors that affect
individuals, and is estimated by incorporating such factors into stochastic models1-9.
Stochasticity can be put into four categories, including the probabilistic nature of
birth and death at the level of individuals (demographic stochasticity2), variation in
population-level birth and death rates among times or locations (environmental
stochasticity1,3), the sex of individuals6,8, and variation in vital rates among individuals
within a population (demographic heterogeneity7,9). Mechanistic stochastic models
that include all of these factors have not previously been developed to examine their
combined effects on extinction risk. Here we derive a family of stochastic Ricker
models with different combinations of all these stochastic factors, and show that
extinction risk depends strongly on the combination of factors that contribute to
stochasticity. Further, we show that only with the full stochastic model can the relative
importance of environmental and demographic variability, and therefore extinction
risk, be correctly determined from data. Using the full model we find that
demographic sources of stochasticity are the prominent cause of variability in a
laboratory population of Tribolium, while using only the standard simpler models
2
would lead to the erroneous conclusion that environmental variability dominates. Our
results demonstrate that current estimates of extinction risk for natural populations
could be underestimated by orders of magnitude because variability has mistakenly
been attributed to the environment rather than the demographic factors described
here that entail much higher extinction risk for the same variability level.
An essential question in ecology and conservation biology is the determination of the
likelihood of extinction in a biological system10. The likelihood of extinction clearly
depends on understanding the relative importance of different processes that affect the
stochastic dynamics of biological populations, and how these interact with density
dependent and density independent processes5,6. Ecologists have long sought simple
approaches to the question of predicting the likelihood of extinction11,12. In conservation
biology, the simple idea of a population level that determines which kind of forces might
lead to extinction has been appealing4,13-15. However, a more detailed and more mechanistic
approach is clearly needed to answer these questions more carefully in a way that uses
available data.
Models that incorporate stochasticity to examine its effect on population growth and
extinction have a long history1-6,13,16-21. The first stochastic models showed that populations
could go extinct even if deterministic models concluded they would persist indefinitely16.
Early results also showed that the variance of population fluctuations and the probability of
extinction depend on which biological processes are subject to stochasticity, and that the
long term growth rate of a stochastic population differs from an equivalent population with
deterministic dynamics16,17. These general results have proved to be robust, and later
studies have concentrated on how different sources of stochasticity in the life history of
organisms affect population growth and extinction.
3
There are many sources of stochasticity that contribute to variance in population
growth and thus contribute to the risk of stochastic extinction. Two broad classes are
recognised most commonly6. Demographic stochasticity occurs because the birth or death
of an individual is a random event, such that individuals identical in their probability
distributions for reproduction or longevity nevertheless differ by chance in how many
offspring they produce or when they die2,20. Environmental stochasticity occurs because
fluctuations in exogenous environmental factors such as temperature and rainfall drive
population-level fluctuations in birth and death rates3,20. In small populations, demographic
stochasticity increases extinction risk because of unfortunate coincidences in the fate of
individuals, which are cancelled out in larger populations. In contrast, environmental
stochasticity increases extinction risk over a larger range of population sizes because the
whole population is affected simultaneously.
Two further sources of stochasticity have long been recognised17 but only recently
analysed, namely stochastic sex determination6,22,23 and demographic heterogeneity7,9, with
the former in some sense an extreme form of the latter. These can be viewed as components
of demographic stochasticity6,7 although we separate them here because they are
fundamentally different to randomness in births and deaths. In sexually reproducing
species, the sex of an offspring is often randomly determined, giving rise to a stochastically
fluctuating sex ratio in the population. Most current models of extinction risk include only
females. However a stochastic sex ratio can increase the variance in population growth and
extinction risk over and above the effects of demographic stochasticity on females alone
because males contribute to density dependent regulation or because the lack of males
reduces female mating success 8,23,24.
4
Demographic heterogeneity refers to variation in birth or death rates among
individuals within a population, such as might occur among individuals of different size7,9.
This contrasts with demographic stochasticity, which in its original definition and
subsequent application concerns chance events assuming a fixed value of the birth or death
rate of an individual2,20. Demographic stochasticity, sex ratio stochasticity, and
demographic heterogeneity all contribute to the total demographic variance. Demographic
heterogeneity can either increase or decrease the demographic variance, depending on the
details of the stochastic process, and so can either increase or decrease the extinction risk7.
A problem that remains is how to combine the various sources of stochasticity into an
analytically tractable model. Many current approaches begin by assuming a deterministic
skeleton to which noise terms are added, where the statistical distribution of the noise is
chosen to reflect a broad class of stochasticity6,25. Among other models, the Ricker model26
has often been used as a deterministic skeleton25,27. In contrast, here we incorporate
stochasticity directly into the birth and death processes, allowing the mean and variance of
population growth to arise mechanistically from the underlying process assumptions. Our
models are for discrete individuals. We derive our stochastic models from Ricker's
assumptions but extend these by specifying the stochastic mechanisms at different stages in
the life history of an individual and scaling up to the population level (Supplementary
Methods). Ricker's assumptions26 lead to the Poisson-Ricker model, which contains
demographic stochasticity arising from the number of eggs laid by individuals and survival
of individual eggs from predation by adults. To this basic model we add environmental
stochasticity and demographic heterogeneity in the number of offspring, and stochasticity
in the sex of offspring. We focus on births because variability in births has greater or equal
effects than mortality, but our models extend generally to mortality variation
5
(Supplementary Discussion). We use different combinations of the various stochastic
sources to derive a family of nested stochastic Ricker models (Fig. 1).
The stochastic models are true Ricker models because they all have conditional mean
Nt+1 equal to the deterministic Ricker model26, that is, E[Nt+1]=RNtexp(-αNt), where Nt is
the population size in generation t, R is the density independent mean per capita growth rate
(finite rate of growth), and α is a measure of density dependent effects (Supplementary
Methods). However, the various stochastic models have different distributions of numbers
next year as a function of numbers this year (Supplementary Table 1) and so differ
substantially in their variance characteristics for the number of individuals in a subsequent
generation (Fig. 2, Supplementary Fig. 1). As expected, the variance in the number of
individuals in the next generation increases as more sources of stochasticity are included in
the models. The Poisson Ricker model, a model of pure demographic stochasticity, has the
smallest variance (Fig. 2).
When the total variance is held at the same value (Supplementary Methods), there is
an important difference between models of environmental stochasticity and demographic
heterogeneity in the variance for the number of individuals the following generation (Fig.
2). For environmental stochasticity the variance in numbers peaks at the stationary point of
the deterministic Ricker function, whereas for demographic heterogeneity the variance is
concentrated at low abundance to the left of the stationary point. This is because
environmental stochasticity results in a density-independent variance parameter, whereas
demographic heterogeneity generates one that is density-dependent (Supplementary
Methods). As a result, demographic heterogeneity entails a greater risk of extinction than
environmental stochasticity for the same total variance (Fig 3). As we highlight below, the
6
similarities in the two variance functions allow these processes to be easily confused, yet
their differences have large effects on extinction risk.
The stochastic sex ratio increases the variance at low to intermediate initial
abundance, and substantially so at abundances less than the stationary point of the Ricker
model (Fig. 2). The effect of the sex ratio is greatest in the demographic models (Fig. 2,
compare P with PB and NBd with NBBd). The combined variance of demographic
stochasticity, environmental stochasticity, demographic heterogeneity, and stochastic sex
ratio is higher than in models of their individual effects and is additive (Fig. 2).
Extinction risk for the stochastic Ricker models differs substantially depending on the
combination of factors in the lifecycle that contribute to stochasticity (Fig. 3). The lowest
extinction risk is for the Poisson Ricker model, which includes only demographic
stochasticity, while the highest extinction risk is for the model that includes all sources of
stochasticity. Significantly, for the same total variance, extinction risk is enhanced more by
demographic heterogeneity or a stochastic sex ratio than by environmental stochasticity.
Extinction risk also depends on the finite rate of growth, R (Fig. 3). Increasing R from 1
initially promotes higher persistence times but increasing R also increases the contribution
of nonlinear dynamics to the variance in population fluctuations, causing persistence times
to eventually decrease. For populations with growth rates R larger than the value (7.4)
producing the first bifurcation in the Ricker model, fluctuations due to nonlinear dynamics
increase and persistence times rapidly drop below those of populations with R equal to 1
(the minimum R required for persistence in the absence of fluctuations).
The characteristic probability mass functions (Supplementary Table 1) of the
different stochastic Ricker models provide an opportunity to distinguish between models by
7
fitting them to data. Using likelihood approaches and information criteria28, we fitted the
models to data from a laboratory experiment on Tribolium castaneum growing in discrete
time cultures in temperature controlled incubators. As in Ricker's fish (Fig. 1), cannibalism
by adults on eggs is the main density regulating process in laboratory populations of T.
castaneum in discrete time cultures29. The best fitting model was the negative binomialbinomial-gamma model, which includes all four sources of stochasticity (Table 1; the fitted
model is shown in Supplementary Fig. 2). No other model fitted nearly as well (Table 1)
and the experimental design provided a robust distinction between models (Supplementary
Discussion). Moreover, the second best model (also by a substantial amount) was the
negative-binomial-gamma model, which left out only the stochastic sex ratio which is then
partly absorbed by the demographic heterogeneity parameter (Table 1).
The likelihood analysis revealed several important features of the stochastic system.
First, the Poisson model was the worst model by a large margin (Table 1, ∆AIC = 336),
suggesting that the most basic assumptions of demographic stochasticity in births, densitydependent, and density-independent survival are completely unable to describe the variance
in abundance even when environmental variability is tightly controlled in the laboratory.
Second, the estimated vital rates of the population were not very different among the
models but the estimates of the stochastic parameters were very sensitive to which
stochastic factors were included in the fitted model (Table 1), highlighting the importance
of a full model specification for correctly identifying the important stochastic factors, and
therefore correctly estimating extinction risk. Strikingly, the full model revealed that
demographic heterogeneity was much more important than environmental stochasticity,
whereas simpler models without demographic heterogeneity erroneously suggest that
environmental variability dominates because any demographic heterogeneity is absorbed by
the environmental variance parameter (Table 1).
8
These results show that many species currently viewed as at risk of extinction from
environmental stochasticity could instead be at much higher risk from undetected
demographic variance. This demographic variance is driven by sex ratio variation and
demographic heterogeneity that has been mistakenly attributed to environmental
stochasticity. The increased extinction risk is a consequence of the fact that, for the same
overall level of variance in abundance for one generational step, sex ratio stochasticity and
demographic heterogeneity give rise to greater variance than environmental stochasticity
when population sizes are small and vulnerable. Thus, identifying the relative contribution
of different stochastic processes is key to understanding fluctuations and estimating
extinction risk because variability is different at different population levels for different
processes. Since natural populations are likely to have greater demographic heterogeneity
than our laboratory stock of Tribolium, the effect we uncover here will be larger in natural
populations. Suitable data could include time series of population abundance using the
methods we develop here, or individual level data, with effort especially needed to
encompass a range of population density to capture the density dependent nature of the
variance in abundance. With field data, care will also be needed to factor in measurement
error because such error will further hide the importance of demographic heterogeneity
relative to environmental stochasticity (Supplemental Discussion). We suggest that
extinction risk for many populations of conservation concern needs urgently to be reevaluated with full consideration of all factors contributing to stochasticity.
Methods.
We placed adult Tribolium castaneum into 4 cm x 4 cm x 6 cm acrylic containers with 20 g
of standard medium (95% flour, 5% brewer's yeast) to lay eggs for 24 hours, after which
time the adults were removed. We set up 60 separate containers with adult numbers ranging
9
from 2 to 1000. Containers were kept in a constant temperature incubator at 31ºC for the
full beetle lifecycle and their positions within the incubator were randomised weekly. The
24 hour egg-laying period was followed by a further 34 days during which individuals
passed through the egg, larval, and pupal stages. The number of adults emerging at the end
of the 35 day life cycle was recorded for each container. The stochastic Ricker models were
fitted to the emergence data by maximum likelihood28.
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Roughgarden, J. A simple model for population dynamics in stochastic
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Tuljapurkar, S. An uncertain life: demography in random environments. Theor.
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Gabriel, W. & Burger, R. Survival of small populations under demographic
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Engen, S., Lande, R., & Saether, B. E. Demographic stochasticity and Allee effects
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Legendre, S., Clobert, J., Moller, A. P., & Sorci, G. Demographic stochasticity and
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Dennis, B. et al. Estimating chaos and complex dynamics in an insect population.
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Ricker, W. E. Stock and recruitment. J. Fish. Board Can. 11, 559-623 (1954).
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12
Acknowledgements We thank Michelle Gibson, Dylan Hodgkiss, Claire Koenig, Tom McCabe, Devan
Paulus, David Smith, Nancy Tcheou, Roselia Villalobos, and Motoki Wu for assistance. This study was
funded by the National Science Foundation.
Author Information Correspondence and requests for materials should be addressed to B.M.
(brett.melbourne@colorado.edu).
Figure 1. A family of stochastic Ricker models based on Ricker's26 assumptions
about the lifecycle of a fish species that cannibalises its eggs. The stochastic
models incorporate stochasticity in various parts of the lifecycle, including gamma
variation in environmentally determined birth rates, gamma variation in birth rates
between individuals, Poisson variation in birth rates within individuals, Bernoulli
variation in mortality within individuals, and Bernoulli variation in the sex of an
individual at birth.
Figure 2. Variance in the number of individuals in the next generation Nt+1 as a
function of the number of individuals in the current generation Nt for the stochastic
Ricker models. Model parameters: R = 5, α = 0.05, kD = 0.5, kE = 10. The
stochastic parameters (kD, kE) were set so that the total variance due to
demographic heterogeneity was equal to the total variance due to environmental
stochasticity. The vertical bar indicates the position of the stationary point in the
Ricker production function. Abbreviations identify the models listed in Fig. 1.
Figure 3. Intrinsic mean time to extinction30, Tm, for the stochastic Ricker models as
a function of the finite rate of increase, R. Model parameters: kD = 0.5; kE was
adjusted so that the total variance due to demographic heterogeneity was equal to
13
the total variance due to environmental stochasticity; α was adjusted to hold the
equilibrium density at 30 individuals. Abbreviations identify the models listed in Fig.
1.
14
Table 1. Fit of stochastic Ricker models to T. castaneum data.
R
α
Poisson
2.526
0.003636
Negative binomial
2.638
0.003744
2.706
0.003800
Negative binomial-gamma
2.598
0.003727
Poisson-binomial
2.697
0.003753
NB-binomial (demographic)
2.621
0.003731
NB-binomial (environmental)
2.770
0.003831
NB-binomial-gamma
2.613
0.003731
Model
kD
kE
L
∆AIC
-406.5
336
-246.3
18
1.9913
-265.3
56
29.2262
-238.9
5
-282.0
87
-245.8
17
13.1014
-242.6
10
26.6221*
-236.4
0
0.1463
(demographic)
Negative binomial
(environmental)
0.2610
0.3876
1.1475*
The models were fitted to the data by maximising the log likelihood, L, calculated from the
probability mass function of each stochastic Ricker model (Supplementary Table 1). The estimated
parameters were: R the density independent mean per capita growth rate; α the density dependent
parameter; kD and kE the variance parameters for demographic heterogeneity and environmental
stochasticity respectively (small values indicate large variance). The difference in the Akaike
information criterion, ∆AIC, was used to compare models28. *Bias corrected estimates for kD and kE
were 1.07 and 17.62 respectively (see Supplementary Discussion).
600
NBBg
400
NBBd
σ2N t +1
NBg
NBBe
200
NBd
NBe
PB
0
P
0
30
60
Nt
90
60,000
22,000
8,100
NBe
Pois
PB
10
NBd
8
3,000
NBBe
NBg
6
400
150
NBBd
55
4
NBBg
20
2
7
2
5
10
R
15
20
log(Tm)
Tm
1,100
This file contains Supplementary Figures 1-4, Supplementary Methods, Supplementary
Table 1, Supplementary Discussion, and Supplementary Notes.
The Supplementary Figures show stochastic realisations of the models (Fig. S1), the best
model fitted to the Tribolium data (Fig. S2), extensions to the models (Fig. S3) , and
measurement error bias (Fig. S4). The Supplementary Methods provide a detailed
derivation of the stochastic Ricker models, and equations to equate the total variance for
environmental stochasticity and demographic heterogeneity. Supplementary Table 1
provides pmfs for the stochastic Ricker models. The Supplementary Discussion considers
extensions to the stochastic Ricker models, the robustness of the model fit, and
measurement error bias. The Supplementary Notes include additional references.
(PDF file 949KB)
Extinction risk depends strongly on factors contributing to stochasticity
Brett A. Melbourne1 & Alan Hastings2
1
2
Department of Ecology and Evolutionary Biology, University of Colorado, Boulder CO 80309, USA
Department of Environmental Science and Policy, University of California, Davis CA 95616, USA
100
Poisson
NB−environmental
NB−demographic
NB−gamma
Poisson−binomial
NB−binomial−env
NB−binomial−dem
NB−binomial−gamma
80
60
40
N t +1
20
0
100
80
60
40
20
0
0
50
100
150 0
50
100
150 0
50
100
150 0
50
100
150
Nt
Supplementary Figure 1 | Stochastic realisations of the
production functions for a family of stochastic Ricker
models. For each level of initial abundance Nt, 100
realisations of the abundance in the next generation Nt+1
were simulated. Points are jittered along the x-axis for
clarity. The curve shows the theoretical mean of the
distribution. Error bars show the theoretical standard
deviation. Model
100
200
300
Supplementary Figure 2 | Data from the Tribolium
experiment showing the best fitting stochastic Ricker
model. The best fitting model was the negative binomialbinomial-gamma Ricker model, which is a model for the
combined effects of all stochastic sources: demographic
stochasticity, stochastic sex determination, environmental
stochasticity, and demographic heterogeneity. The curve
shows the theoretical mean of the distribution. Error bars
show the theoretical standard deviation. Details of the model
fit are given in Table 1 of the main text.
0
N t +1
parameters: R = 5, α = 0.05, kD = 0.5, kE = 10. The
stochastic parameters (kD, kE) were set so that the total
variance due to demographic heterogeneity was equal to
the total variance due to environmental stochasticity; this
also corresponds to equal variance at the stationary point in
the production function. Abbreviations identify the models
listed in Fig. 1 of the main text.
0
200
400
600
Nt
800
1000
Supplementary Methods
Derivation of stochastic Ricker models. The original
derivation of the deterministic Ricker model was for
fish populations undergoing cannibalism of eggs by
adults26. Ricker assumed first that eggs were laid in a
short discrete event at the beginning of the year. For
the rest of the year, adults were free to cannibalise eggs
and juveniles. Here, we extend Ricker's model to
derive a family of stochastic models that include
stochasticity in various aspects of the lifecycle (see Fig.
1 in the main text).
The foundation of the stochastic models is the
Poisson Ricker model, which we derive first. It
includes three sources of demographic stochasticity in
the lifecycle: births, density-dependent mortality, and
density-independent mortality. To this basic model we
then add either environmental heterogeneity (variation
in the birth rate in time or space), demographic
heterogeneity (variation in the birth rate between
individuals within the population), or both to derive
respectively the negative binomial-environmental
(NBe), the negative binomial-demographic (NBd), and
the negative binomial-gamma (NBg) Ricker models.
We then derive models where sex is determined
stochastically. We first add stochastic sex
determination to the Poisson Ricker model, which
leads to the Poisson-binomial (PB) Ricker model.
Finally, we add environmental heterogeneity,
demographic heterogeneity, or both to the Poissonbinomial Ricker model to derive respectively the
negative binomial-binomial-environmental (NBBe),
the negative binomial-binomial-demographic (NBBd),
and the negative binomial-binomial-gamma (NBBg)
Ricker models. These models form a nested family of
stochastic Ricker models, where the NBBg model is
the full model.
Poisson Ricker model. The Poisson Ricker model is a
basic model of demographic stochasticity. There are Nt
adults in the population at time t, the beginning of the
lifecycle. Let individual adults give birth randomly
according to a Poisson process at a constant rate β in a
short, defined period at the beginning of the lifecycle.
Then, Bi,t, the number of eggs or young produced by
adult i at the beginning of the lifecycle, is a Poisson
random variable (e.g. ref. 31):
Bi ,t ~ Poisson(β ) ,
(S1)
where β is the mean number of births per adult.
To become an adult, each individual offspring
must now survive being cannibalised or dying from
density-independent causes. Ricker assumed that an
individual adult encounters and eats eggs or young
randomly with constant probability and no handling
time26. With these assumptions about the stochastic
search process, the probability ci that an individual
offspring is not eaten by adult i by the end of the period
of exposure to predation is
ci = e − α ,
(S2)
where α is the adult search rate (see e.g. p 53 ref. 32).
The probability c that an individual offspring is not
eaten by any adults is thus
Nt
c = ∏ ci = e −αN t .
(S3)
i
The probability that an individual survives all forms of
mortality during the lifecycle is then
s = (1 − m )c ,
(S4)
where m is the probability of density-independent
mortality.
Summing up survival of all offspring from adult i
gives a binomial distribution for Si,t+1, the number
surviving to the adult stage, given that Bi,t were
produced by that adult. That is
Si ,t +1 ~ Binomial(Bi ,t , s ) .
(S5)
Since Bi,t is Poisson (Eq. S1), Si,t+1 has a compound
binomial-Poisson distribution. By the law of total
probability this compound distribution reduces to a
Poisson distribution (see e.g. ref. 31):
(
)
Si ,t +1 ~ Poisson R e −αN t ,
(S6)
where R = β (1-m) and is immediately identifiable as
the finite rate of population increase of the
deterministic Ricker model.
Finally, we add up the surviving offspring
produced by all of the adults. Since the sum of
independent Poisson random variables is also Poisson
(see e.g. ref. 33), the total offspring surviving to
become adults is:
Nt
(
)
N t +1 = ∑ Si ,t +1 ~ Poisson N t R e −αN t .
(S7)
i
Thus, with Ricker's assumptions we find that Nt+1 has a
Poisson distribution with mean equal to the
deterministic Ricker model. The probability mass
function (pmf) for the Poisson Ricker model is given in
Supplementary Table 1. Dennis et al. also derived a
Poisson Ricker model for demographic stochasticity
from similar assumptions25.
Supplementary Table 1. Probability mass functions (pmfs) of stochastic Ricker models with discrete individuals.
Model
pmf
e − μ μ n , μ = n R e −αnt
t
Poisson
nt +1!
t +1
nt +1
Negative binomial environmental
⎛ nt +1 + k E − 1⎞ ⎛ μ ⎞
⎟⎟
⎜⎜
⎟⎟ ⎜⎜
⎝ kE − 1 ⎠ ⎝ kE + μ ⎠
Negative binomial demographic
⎛ nt +1 + nt k D − 1⎞ ⎛
μ ⎞
⎟⎟
⎜⎜
⎟⎟ ⎜⎜
−
n
k
1
n
k
t D
⎝
⎠⎝ t D + μ ⎠
Negative binomial gamma
Poisson-binomial
kE
⎛ k E ⎞ , μ = n R e −αnt
⎜⎜
⎟⎟
t
⎝ kE + μ ⎠
nt +1
⎛ nt k D ⎞
⎜⎜
⎟⎟
⎝ nt k D + μ ⎠
nt k D
, μ = nt R e −αn
t
n
t +1
⎞ ⎛ nt k D ⎞
⎛ nt +1 + nt k D − 1⎞ ⎛
μ
⎜
⎟
⎟⎟
⎜
⎟
(
)
G
R
E ⎜
∫
⎟⎜
⎟ ⎜⎜
⎝ nt k D − 1 ⎠ ⎝ nt k D + μ ⎠ ⎝ nt k D + μ ⎠
RE = 0
nt
− λ nt + 1
⎛ nt ⎞ F
n −F e λ
, λ = F ZR e −αnt
⎜⎜ ⎟⎟ z (1 − z ) t
∑
nt +1!
F =0 ⎝ F ⎠
∞
nt +1
nt k D
, G (R ) = R k
E
E
⎛n ⎞
∑ ⎜⎜ F ⎟⎟ z (1 − z )
⎛ nt +1 + k E − 1⎞ ⎛ λ ⎞
⎟⎟
⎜⎜
⎟⎟ ⎜⎜
⎝ kE − 1 ⎠ ⎝ kE + λ ⎠
Negative-binomialbinomial-demog.
⎛n ⎞
∑ ⎜⎜ F ⎟⎟ z (1 − z )
⎞
⎛ nt +1 + Fk D − 1⎞ ⎛ λ
⎟⎟
⎜⎜
⎟⎟ ⎜⎜
⎝ Fk D − 1 ⎠ ⎝ Fk D + λ ⎠
Negative-binomialbinomial-gamma
+ Fk D − 1⎞ ⎛
⎞
⎛n ⎞
λ
n −F ⎛ n
⎟⎟
⎟⎟ ⎜⎜
G (RE )∑ ⎜⎜ t ⎟⎟ z F (1 − z ) t ⎜⎜ t +1
∫
F =0 ⎝ F ⎠
⎝ Fk D − 1 ⎠ ⎝ Fk D + λ ⎠
RE = 0
λ = F RzE e −αnt
F =0
nt
F =0
t
nt − F
F
⎝ ⎠
t
nt − F
F
⎝ ⎠
∞
nt
(
exp −
RE k E
R
)( )
kE kE
R
1 , μ = n R e −αnt
t E
Γ(k E )
kE
Negative-binomialbinomial-environ.
nt
E −1
⎛ k E ⎞ , λ = F R e −αnt
⎜⎜
⎟⎟
z
⎝ kE + λ ⎠
nt +1
⎛ Fk D ⎞
⎜⎜
⎟⎟
⎝ Fk D + λ ⎠
Fk D
nt + 1
, λ = F Rz e −αn
t
⎛ Fk D ⎞
⎜⎜
⎟⎟
⎝ Fk D + λ ⎠
Fk D
, G (R ) = R k
E
E
E −1
(
exp −
RE k E
R
)( )
kE kE
R
1 ,
Γ(k E )
⎛a⎞
⎜ ⎟
The pmf is the conditional probability Pr{ Nt+1 = nt+1 | Nt = nt }, ⎜⎝ b ⎟⎠ is the binomial coefficient, and Γ(x) is the gamma function. Parameters: R
finite rate of growth, α density dependent parameter (adult search rate), RE finite rate of growth for a particular condition of the environment,
F the number of females, z the probability that an individual is female, kE the shape parameter of the gamma distribution for environmental
stochasticity, kD the shape parameter of the gamma distribution for demographic heterogeneity. Parameters are defined in the
Supplementary Methods.
Negative-binomial-environmental Ricker model.
This is a model for the combined effects of
demographic and environmental stochasticity. We
allow the environment to vary stochastically in time,
space, or both. The model development follows that for
the Poisson Ricker model. The birth rate βt,x now varies
stochastically in time and space (x), representing
environmental stochasticity. The number of births
summed over all adults at a particular time and place is
a sum of Poissons, so is also Poisson:
Bt , x = ∑ Bi ,t , x ~ Poisson(N t β t , x ) .
Nt
(S8)
i
We represent variation in βt,x in time or space as
a gamma random variable with mean β and shape
parameter kE:
β t , x ~ Gamma(β , k E ) .
(S9)
Since the distribution is conditional on Nt,x, the
distribution of Nt,xβt,x is also gamma but with mean
Nt,xβ and shape parameter kE,
N t , x β t , x ~ Gamma(N t , x β , k E ) .
(S10)
The distribution of total offspring Bt,x is then a
gamma mixture of Poissons (Eq. S8), which is one
form of the negative binomial distribution (see e.g. ref.
33):
Bt , x ~ NegBinom(N t , x β , k E ) ,
(S11)
where β is the mean birth rate in time or space.
As for the Poisson Ricker model (Eqs S2-S5),
survival of offspring is binomial:
St +1, x ~ Binomial(Bt , x , s ) .
(S12)
Since Bt,x is negative binomial (Eq. S11), the number of
surviving offspring from time t at location x has a
compound binomial-negative binomial distribution.
This compound distribution reduces to a negative
binomial distribution:
(
N t +1, x ~ NegBinom N t , x R e
−αN t , x
)
, kE ,
(S13)
where again R = β (1-m). The pmf is given in
Supplementary Table 1. The variance parameter, kE, of
this model is not density dependent. That is, the
variance in final abundance Nt+1 varies only with the
final abundance and does not depend on the initial
abundance Nt. Ludwig5 also derived a negative
binomial Ricker model for demographic and
environmental stochasticity but since he did not
incorporate density dependent mortality, the model has
a different parameterisation
Negative-binomial-demographic Ricker model. This
is a model for the combined effects of demographic
stochasticity and demographic heterogeneity. The
model development also follows that for the Poisson
Ricker model. Let adult i give birth randomly at a
constant rate βι specific to that adult. Then, Bi,t, the
number of offspring produced by adult i at the
beginning of the lifecycle, is a Poisson random
variable:
Bi ,t ~ Poisson(β i ) ,
(S14)
where βi is the mean number of births for adult i. The
mean birth rate for a particular adult reflects the
tendency of that individual to produce more or less
offspring than other adults. For example, a large adult
might consistently produce more offspring than a small
one. To capture variation in birth rate between
individuals we assume that βi follows a gamma
distribution, with mean β and variance determined by
the shape parameter kD, and therefore the distribution
of Bi,t is negative binomial (gamma mixture of
Poissons):
Bi ,t ~ NegBinom(β , k D ) .
(S15)
Survival is binomial (Eqs S2-S5),
Si ,t +1 ~ Binomial(Bi ,t , s ) .
(S16)
and so the distribution of Si,t+1 is a compound binomialnegative binomial distribution, which reduces to a
negative binomial (as previously):
(
)
Si ,t +1 ~ NegBinom R e −αN t , k D ,
(S17)
where again R = β (1-m).
Since the sum of independent negative binomials
with the same k parameter is also a negative binomial
(see e.g. ref. 33), the survivors summed over all adults
is:
Nt
(
)
N t +1 = ∑ Si ,t +1 ~ NegBinom N t R e −αN t , k D N t .
(S18)
i
Thus, with demographic heterogeneity in addition to
demographic stochasticity in births and deaths, Nt+1 has
a negative binomial distribution with mean equal to the
deterministic Ricker model. The pmf is given in
Supplementary Table 1. The nature of the variance in
the final abundance for this model is particularly
interesting, since the variance parameter of the
negative binomial, k'=kDNt, is density-dependent; it is a
function of the initial abundance Nt. At low initial
abundance, k' is small, yielding larger variance to mean
ratios for Nt+1, whereas as at high initial abundance k' is
large and the variance approaches the Poisson limit
(Fig. 2 in the main text). This contrasts with the model
for environmental stochasticity (Eq. S13) in which the
variance does not depend on the initial abundance.
Negative-binomial-gamma Ricker model. This is a
model for the combined effects of demographic
stochasticity,
demographic
heterogeneity,
and
environmental stochasticity. We combine the
assumptions of the previous three models, specifically
that the number of offspring produced by an adult is a
Poisson random variable, variation in birth rate
between individuals within a population (demographic
heterogeneity), βi, follows a gamma distribution with
mean βt,x, variation in the population mean at different
times and locations, βt,x, follows a gamma distribution
with mean β, and survival is binomial. With these
assumptions, the distribution of Nt+1,x is a compound
negative binomial-gamma distribution (gamma mixture
of negative binomials) with mean equal to the
deterministic Ricker model and two variance
parameters, kD and kE. The pmf is given in
Supplementary Table 1.
Poisson-binomial Ricker model. This is a model for
the combined effects of demographic stochasticity and
stochastic sex determination. The derivation is similar
to the Poisson Ricker model. The number of offspring
Bi,t from female i at the beginning of the lifecycle, is a
Poisson random variable:
Bi ,t ~ Poisson(β ) ,
(S19)
where β is now the mean number of births per female,
rather than per adult. Survival of offspring is as before
(Eqs S2-S5):
Si ,t +1 ~ Binomial(Bi ,t , s ) ,
(S20)
where, importantly, predation involves both sexes (Eq.
S3). Adding up the surviving offspring produced by all
of the females yields a Poisson distribution:
Ft
(
)
N t +1 = ∑ Si ,t +1 ~ Poisson Ft β (1 − m )e −αN t ,
(S21)
i
where Ft is the number of females at the beginning of
the lifecycle. Since Ft is a binomial random variable,
Ft ~ Binomial(N t , z ) ,
(S22)
where z is the probability that an individual is female,
the distribution of Nt+1 is a Poisson-binomial
distribution (or Bernoulli mixture of Poissons) with
mean equal to the deterministic Ricker model and R =
zβ(1-m). The pmf is given in Supplementary Table 1.
Negative binomial-binomial-environmental Ricker
model. This is a model for the combined effects of
demographic
stochasticity,
stochastic
sex
determination, and environmental stochasticity. The
derivation is substantially the same as the Poissonbinomial and negative binomial-environmental Ricker
models. Adding up the surviving offspring produced by
all of the females yields a negative binomial
distribution:
Ft
(
)
N t +1 = ∑ Si ,t +1 ~ NegBinom Ft β (1 − m )e −αN t , k E . (S23)
i
Since Ft is a binomial random variable, the distribution
of Nt+1 is a negative-binomial-binomial distribution (or
Bernoulli mixture of negative binomials) with mean
equal to the deterministic Ricker model and R = zβ(1m). The pmf is given in Supplementary Table 1.
Negative binomial-binomial-demographic Ricker
model. This is a model for the combined effects of
demographic
stochasticity,
stochastic
sex
determination, and demographic heterogeneity. The
derivation is substantially the same as the previous
model. The number of surviving offspring from all
females has a negative binomial distribution:
Ft
(
)
N t +1 = ∑ Si ,t +1 ~ NegBinom Ft β (1 − m )e −αN t , k D Ft .
i
(S24)
The distribution of Nt+1 is thus a negative-binomialbinomial distribution with mean equal to the
deterministic Ricker model and R = zβ(1-m). The pmf
is given in Supplementary Table 1. The variance in the
final abundance, Nt+1, depends on the initial abundance,
which contrasts with the previous model for
environmental stochasticity (Eq. S23) in which the
variance does not depend on the initial abundance.
Negative binomial-binomial-gamma Ricker model.
This is a model for the combined effects of all
stochastic
sources:
demographic
stochasticity,
stochastic
sex
determination,
environmental
stochasticity, and demographic heterogeneity. This is
the full model. We combine the assumptions of the
previous three models, which includes the assumptions
for the negative binomial-binomial Ricker model plus
the assumption that the number of females is a
binomial random variable. With these assumptions, the
distribution of Nt+1,x is a compound negative binomialbinomial-gamma distribution (gamma mixture of
negative binomial-binomials) with mean equal to the
deterministic Ricker model and R = zβ(1-m). In
addition to the two variance parameters, kD and kE, the
probability that an individual is female, z, also
influences the variance in Nt+1,x. The pmf is given in
Supplementary Table 1.
Total variance due to environmental stochasticity or
demographic heterogeneity. To compare models
including environmental stochasticity with models
including demographic heterogeneity, we set the
variance parameters of these models (respectively kE
and kD) so that the total variance in Nt+1 was equal. The
total variance is given by integrating the variance
function over Nt, thus
∞
∫σ
2
N t +1 | N t
.dN t ,
(S25)
0
which, when the total variance for environmental
stochasticity and demographic heterogeneity are
equated, yields:
kE =
kD
α
.
(S26)
Supplementary Discussion
Extensions to the models. Several extensions and
variations on our family of models are possible. While
the models described above and in the main text
include demographic stochasticity in all stages of the
lifecycle (births, density independent and density
dependent mortality), we included environmental
stochasticity and demographic heterogeneity only in
births. Mortality is also subject to environmental
stochasticity and demographic heterogeneity. However,
we show here that adding environmental stochasticity
and demographic heterogeneity in mortality to our
mechanistic models does not alter our conclusions,
either because the effect of mortality variance is
negligible compared to variance in births, or its effects
are fully accounted for by the models described in the
main text.
Inclusion of stochasticity in density independent
mortality is straightforward. For example, an
appropriate form for stochastic variation in the
probability of density independent survival (1-m) is the
beta
distribution.
Then,
when
demographic
stochasticity and environmental stochasticity are the
only sources of mortality variation, the resulting model
is the Poisson-beta-environmental Ricker model. Since
the finite rate of increase, R = β(1-m), is the multiple of
births and density independent survival, the effects of
environmental stochasticity in mortality on population
growth and extinction are indistinguishable from the
effects of environmental stochasticity in births
200
(Supplementary Figure 3). The key effect of
environmental stochasticity in either births or mortality
is on the variance of R; the form of the distribution is
not important. Furthermore, because mortality is
bounded between zero and one, variance in mortality is
bounded, reaching a maximum of m(1-m). In contrast,
variance in births is unlimited. Thus, we expect the
greatest potential for stochasticity in R to be
contributed by births. Nevertheless, when mortality
makes an important contribution to stochasticity in R, it
is captured phenomenologically (as demonstrated in
Supplementary Figure 3) by the NBe or NBBe Ricker
models used in the main text, which model variation in
R as a gamma distribution.
m (env)
m (dem)
α (env)
α (dem)
100
50
σ2N t +1
150
NBe
0
P
0
30
60
90
Nt
Supplementary Figure 3 | Variance in the number of
individuals in the next generation Nt+1 as a function of
the number of individuals in the current generation Nt
for models with stochasticity in mortality. Model
parameters: R = 5, α = 0.05, m = 0.9. Density
independent mortality (m) was included as a beta random
variable, for either environmental stochasticity (σ2m = 0.02),
m (env), or demographic heterogeneity (σ2m = 0.08), m
(dem). To model density dependent mortality variation, α
was included as a gamma random variable (kα = 25; σ2α =
0.001) for either environmental stochasticity, α (env), or
demographic heterogeneity, α (dem). All models include
demographic stochasticity, as in the Poisson Ricker model.
Variances in Nt+1 were estimated by simulation. Exact
variances for the Poisson Ricker model (P) and Negativebinomial-environmental Ricker model (NBe; kE = 10, σ2R =
2.5) are shown for reference (black curves). The vertical bar
indicates the position of the stationary point in the Ricker
production function.
The effect of demographic heterogeneity in
density independent mortality is severely restricted, so
that it never increases the variance in abundance7.
When individual mortality is described by a Bernoulli
distribution and the probability of dying varies
independently among individuals, demographic
heterogeneity has no effect on the variance of survival;
the variance in survival is always equal to the variance
due to ordinary demographic stochasticity for the same
mean probability of mortality. The lack of an effect of
demographic heterogeneity is demonstrated in
Supplementary Figure 3 where the variance in m is set
close to the maximum possible, yet the variance in
abundance in the next generation is indistinguishable
from the Poisson Ricker model.
The effect of environmental stochasticity or
demographic heterogeneity in mortality in the density
dependent cannibalism parameter, α, is more complex.
There are similar restrictions on the magnitude of the
effect of stochasticity, as the variance in the density
dependent probability of survival, c, is bounded by c(1c). Variance in α also changes the form of the mean
model, either by modifying the Ricker parameters or
altering the mean model away from the Ricker form.
Using a gamma distribution for α, we show by
simulation that stochasticity in α results in a variance
profile for Nt+1 that peaks well to the right of the
stationary point in the Ricker production function,
which is in dramatic contrast to models with
stochasticity in m or R (Supplementary Figure 3). This
feature is absent from our data, and so we are confident
that this is not important in our laboratory system.
Robustness of the model fit. To examine the
robustness of the model selection process for our data
and experimental design, we generated artificial data
assuming that alternative models to the best fitting
NBBg model were in fact the true model, and then
refitted the models and carried out model selection
using AIC as done with the real data. Each simulation
experiment was initiated with Nt equal to that used in
the laboratory experiment, using the same level of
replication, and with model parameters as estimated
from the data for the alternative models (Table 1 in the
main text). We examined the NBBe and NBBd models
as alternative true models. The simulations show that
our experimental design was excellent for
distinguishing between models, and that our inference
that the NBBg model was the best fitting model is
robust.
When the true model in the simulation was pure
environmental stochasticity (NBBe), it was correctly
identified as the best model (ΔAIC > 2) in 99.1% of
1000 simulation runs when compared to the model for
pure demographic heterogeneity (NBBd). The
probability of wrongly accepting the NBBd model was
extremely low (1 in 1000 simulation runs). The mixed
0.0031
2.4
40
25
0
5
10
20
30
20
15
^
kE
10
^
kD
0
Bias and measurement error. Measurement error in
our laboratory data was negligible (we estimate less
than one percent based on repeat counts) but can be
considerable in field data34 and has a large influence on
estimating extinction risk35. To examine the potential
effect of measurement error on parameter estimates we
generated artificial data, assuming that the NBBg
model was the true model, to which we added
increasing amounts of measurement error noise and
then re-estimated the model parameters. Each
simulation experiment was initiated with Nt equal to
that used in the laboratory experiment, using the same
level of replication, and with "true" model parameters
as estimated from the data for the NBBg model (Table
1 in the main text). We added lognormal error to both
Nt and Nt+1 (rounded to the nearest integer), simulating
the common field situation where the magnitude of the
error variance increases with abundance. For particular
field systems, a mechanistic model of the measurement
process would be desirable to properly account for the
various contributions to the measurement error (e.g. ref
36).
The simulation results show that parameter
estimates were biased by measurement error
(Supplementary Figure 4). In particular, the variance
parameters, which are critical to estimating extinction
risk, were severely biased by measurement error. As
measurement error increased, the apparent contribution
from environmental stochasticity increased relative to
that from demographic heterogeneity (small values of
kD or kE indicate higher variance), so that at high levels
of measurement error their apparent relative
importance was reversed from the true model. This has
important implications for estimating extinction risk
because underestimating the role of demographic
heterogeneity will underestimate the extinction risk.
The simulation results also show that the
maximum likelihood estimates of the biological rate
parameters (R, α) were unbiased, but that the variance
parameters were biased, especially kE (Supplementary
Figure 4; measurement error equal to zero). We
0.0035
0.0033
^
α
2.5
^
R
2.6
2.7
2.8
0.0037
2.9
calculated bias corrected estimates of the variance
parameters by subtracting the relative deviation
(natural logarithm scale) observed in the simulation
study from the maximum likelihood estimate (Table 1).
2.3
model (NBBg) was rarely (2.1%) identified as the best
model when NBBe was the true model. Conversely,
when the true model was pure demographic
heterogeneity (NBBd), it was correctly identified as the
best model in 99.8% of simulation runs when
compared to the model for pure environmental
heterogeneity (NBBe). The probability of wrongly
accepting the NBBe model was extremely low (0 in
1000 simulation runs). The mixed model (NBBg) was
rarely (0.8%) identified as the best model when NBBd
was the true model.
0.0
0.1
0.2
0.3
0.4
0.0
0.1
0.2
0.3
0.4
Measurement error (σ)
Supplementary Figure 4 | The effect of lognormal
measurement error on parameter estimates from the
Negative-binomial-binomial-gamma Ricker model. σ is
the standard deviation of the measurement error on the
natural logarithm scale. Each point is the mean of 850
replicate simulations.
Supplementary Notes
Funding. This study was funded by NSF grant DEB
0516150 to A.H. and B.A.M.
Additional references.
31.
32.
33.
34.
35.
36.
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1998).
Hilborn, R. & Mangel, M. The Ecological Detective:
Confronting Models with Data. (Princeton University
Press, Princeton, New Jersey, 1997).
Johnson, N. L., Kemp, A. W., & Kotz, S. Univariate
Discrete Distributions. (John Wiley and Sons, New
Jersey, 2005).
Dennis, B. et al. Estimating density dependence,
process noise, and observation error. Ecol. Monogr.
76, 323-341 (2006).
Holmes, E. E. Estimating risks in declining
populations with poor data. Proc. Natl. Acad. Sci. U.
S. A. 98, 5072-5077 (2001).
Muhlfeld, C. C., Taper, M. L., Staples, D. F., &
Shepard, B. B. Observer error structure in bull trout
redd counts in Montana streams: Implications for
inference on true redd numbers. Trans. Am. Fish. Soc.
135, 643-654 (2006).