1

Likelihood based population viability analysis in the presence of observation error

Khurram Nadeem and Subhash R. Lele

K. Nadeem (knadeem@ualberta.ca) and S. R. Lele, Dept of Mathematical and Statistical Sciences, Univ. of Alberta, Edmonton, AB T6G 2G1, Canada.

Population viability analysis (PVA) entails calculation of extinction risk, as defi ned by various extinction metrics, for a study population. Th ese calculations strongly depend on the form of the population growth model and inclusion of demographic and/or environmental stochasticity. Form of the model and its parameters are determined based on observed population time series data. A typical population time series, consisting of estimated population sizes, inevitably has some observation error and likely has missing observations. In this paper, we present a likelihood based PVA in the presence of observation error and missing data. We illustrate the importance of incorporation of observation error in PVA by reanalyzing the population time series of song sparrow Melospiza melodia on Mandarte Island, British Columbia, Canada from 1975 – 1998. Using Akaike information criterion we show that model with observation error fi ts the data bet-ter than the one without observation error. Th e extinction risks predicted by with and without observation error models are quite diff erent. Further analysis of possible causes for observation error revealed that some component of the observation error might be due to unreported dispersal. A complete analysis of such data, thus, would require explicit spatial models and data on dispersal along with observation error. Our conclusions are, therefore, two-fold: 1) observation errors in PVA matter and 2) integrating these errors in PVA is not always enough and can still lead to important biases in parameter estimates if other processes such as dispersal are ignored.

Since the pioneering work of Shaff er (1981), population viability analysis (PVA) has become a key tool in wildlife management and conservation (Beissinger 2002). It is a procedure that uses population abundance data and popula-tion growth models to estimate the probability that a popu-lation will persist for a specifi ed time into the future (Mills 2008). A typical PVA constitutes data collection, model formulation, model estimation and validation, and estima-tion of the extinction risk (Ralls et al. 2002). Th e last two decades have experienced a sea change in both the scale and complexity of PVA as population growth models have grown from modeling single population to spatially explicit metapopulations and beyond (Beissinger 2002). One of the major changes is the inclusion of environmental and demo-graphic stochasticity (Dennis et al. 1991).

Recently, state – space models have been used to accom-modate observation error (also called measurement error) and missing values (McGowan et al. 2011) in ecological analyses. A state – space model consists of two components: a stochastic model for unobserved population abundances and a stochastic model for the observation error (de Valpine and Hastings 2002). State – space models provide a fl exible framework for estimating parameters of the population growth models in the presence of process variation and observation error (de Valpine and Hastings 2002, Clark

and Bj ø rnstad 2004, Staples et al. 2004, Dennis et al. 2006, Lele 2006, Newman et al. 2006, S æ ther et al. 2007). Maximum-likelihood estimation in linear Gaussian state – space models can be conveniently obtained by using the Kalman fi lter (Harvey 1993, Schnute 1994). However, for nonlinear state – space models likelihood based statistical inference is extremely diffi cult. Evaluation of the likelihood function involves computationally intensive high dimen-sional numerical integration (Kitagawa 1987, de Valpine and Hastings 2002, de Valpine 2002). See Pederson et al. (2011) for a review of methods of estimation for state – space population time series models.

Lele et al. (2007) introduced a new method, called data cloning, for the likelihood analysis of general nonlinear state – space population time series models. Data cloning is used to obtain maximum likelihood estimates and associ-ated standard errors for any general hierarchical model. It can also be used to conduct information based model selec-tion (Ponciano et al. 2009) and to predict unobserved states in a state – space model (Lele et al. 2010). Handling missing data, higher order Markov models or spatial data is diffi cult with most existing methods of inference for hierarchical models. For example, Kitagawa ’ s algorithm (de Valpine and Hastings 2002) involves two or higher dimensional numeri-cal integration if there are missing data or when the process

Oikos 000: 001–009, 2012

doi: 10.1111/j.1600-0706.2011.20010.x

© 2011 Th e Authors. Oikos © 2012 Nordic Society Oikos

Subject Editor: Stefano Allesina. Accepted 20 October 2011

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model involves delayed density dependence, making it less practical for such cases. On the other hand, the Bayesian approach (Clark and Bj ø rnstad 2004) based on the Markovchain Monte Carlo algorithm handles such situations without signifi cant computational problems. Because data cloning uses the Bayesian computational machinery, it inherits all the computational advantages of the Bayesian approach at the same time avoiding the pitfalls of having the inference depend on the choice of the prior distribution (Lele et al. 2007).

In this paper, we use data cloning to conduct PVA in the presence of process variation and observation error. Th is includes model estimation, model selection and forecasting of future population trajectories to compute various extinc-tion metrics used in PVA. Propagation of uncertainty in parameter estimation in forecasting future trajectories is an important issue. Not accounting for this uncertainty leads to inappropriate estimation of the extinction risk (Taylor 1995, Ak ç akaya and Raphael 1998, Ludwig 1999). To incorporate the estimation uncertainty in forecasting, Dennis and Otten (2000) and S æ ther et al. (2000) integrate over the bootstrap distribution of the parameter estimators. In this paper we account for estimation uncertainty by integrating over the asymptotic normal distribution of the parameter estimates (Lele et al. 2010).

We illustrate our methodology using time series of songsparrow Melospiza melodia population abundances on Mandarte Island, British Columbia, Canada from 1975 – 1998 reported in S æ ther et al. (2000). Th e song sparrow popula-tion data are particularly interesting. Th e original analysis by S æ ther et al. (2000) considered theta-logistic and logistic growth models (Gilpin and Ayala 1973, Morris and Doak 2002). Based on the estimated parameters, they chose thelogistic growth model to describe the data. Th roughout their analysis, they assumed that observation error was not present by arguing that virtually all birds were banded and low shrub vegetation on the island allowed them to be enumerated accurately. In our analysis, we tested the assumption of no observation error and based on the Akaike information criterion (AIC) found that observation error indeed is present and that theta-logistic model fi ts the data better than the logistic model. Th e extinction risk under the

theta-logistic model is substantially diff erent than under the logistic model. Th is illustrates that presence of observa-tion error can obscure the form of density dependence, and thereby, drastically altering the PVA results. While trying to understand why our analysis, in contrast to S æ ther et al. (2000), suggested presence of observation error, we found out that the data included a small number of immigrant song sparrows (1.6 birds per year on an average, Smith et al. 2006) but were not recorded as ‘ immigrants ’ . It seems that disper-sal could be an important component of the observation error. Unfortunately we do not have enough information to decompose dispersal from observation error. Our conclu-sions are, therefore, two-fold: 1) observation errors in PVA matter and 2) integrating these errors in PVA is not always enough and can still lead to important biases in parameter estimates if other processes such as dispersal are ignored.

Observation error, estimation error and prediction of future trajectories

Th e PVA consists of two components. First component deals with fi tting a specifi ed population growth model to observed population time series in the presence of pro-cess variation and observation error. Diff erent population growth models have diff erent extinction properties (Pascual et al. 1997, Henle et al. 2004) and hence an important step after estimation is model selection. We use data cloning to obtain maximum likelihood estimates of the parameters and to conduct model selection. Th e second component of PVA consists of ‘ risk estimation ’ , that is computing diff erent extinction metrics using the fi tted model.

A summary of various extinction metrics used in the literature on PVA is in Table 1 with detailed description given in Supplementary material Appendix 1. Th e com-putational algorithms for their estimation are described in Supplementary material Appendices 2 – 3. To facilitate thedescription, fi rst we introduce some notation. Let X t represent log-population abundance, ln( N t ), at time t, and n � ( n 0 , n 1 , n 2 , … , n q ) denote a time series of abundances up to time q . Th roughout this paper we represent random variables by capital letters and their realizations by small let-ters. For instance, x t represents a realization of the random

Table 1. A brief description of extinction metrics used in PVA.

Extinction metric Notation Description

Population prediction interval (PPI) [N(r)α, �] This is defi ned as the lower (1 � α ) prediction interval for a future population

abundance N q � r at time q � r .Conditional time to quasi-extinction T

∼Conditional on the last observed population size n q � n e and all the subsequent

sample paths of the population process that reach the quasi-extinction treshold n e within a given time t , T

∼ is defi ned as the fi rst passage time of

population abundance to reach n e .Conditional mean time to quasi-extinction t∼ This is defi ned as the mean of the random variable T

∼ .

Conditional median time to quasi-extinction ξ∼(.5) This is defi ned as the median of T∼ .

Conditional probability of quasi-extinction π ( n e ,t ) This is defi ned as the proportion of all sample paths hitting n e within time t .Probability of hitting the lower threshold fi rst π [e,v] Assuming n v � n e to be an upper policy-set population threshold, π [e,v] is

defi ned as the proportion all sample paths that reach n e before n v within a given time t .

Probability of hitting the upper threshold fi rst π [v,e] This is defi ned as the proportion all sample paths that reach n v before n e within time t .

Probability of recovering from quasi-extinction π [s,1] Given that the population abundance has fallen below a ‘ warning threshold ’ of n s , π [ s ,1] is the probability that the abundance level will exceed n s within a specifi ed time interval.

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variable X t. Vector valued random variables are denoted by bold-faced letters.

Data cloning algorithm for parameter estimation and model selection for population time series models is well described in various publications (Lele et al. 2007, 2010, Ponciano et al. 2009, Solymos 2010). However, algorithm for estimation of the extinction metrics is new to PVA; hence we describe it in detail here.

Th e basic nonlinear state – space model for population time series analysis (Dennis et al. 2006, Lele 2006, de Valpine and Hastings 2002) is:

Process model: X t � m ( X t � 1, h ) � E t (1a)

Observation model: Y t ∼ f ( y t ; X t , ψ ) (1b)

where E t ∼ N (0, σ 2 ) represents process variation and f (.) denotes observation error distribution that depends on an unknown parameter ψ . Diff erent forms of the growth func-tion m ( X t � 1 ) lead to diff erent density-dependent growth models. For instance, m ( X t � 1 ) � X t � 1 � a � bX t � 1 corre-sponds to the stochastic Gompertz model (Gompertz 1825, Dennis and Taper 1994) and m ( X t � 1 ) � X t � 1 � a � bN t � 1 corresponds to the stochastic Ricker model (Ricker 1954). Th e parameter vector is j � ( h , σ 2 , ψ ).

Let j � ( h , σ 2 , ψ ). denote the maximum likelihood estimators (MLEs). In the following, we describe how to predict future population states given the model and the MLEs of the parameters. Diff erent extinction metrics are functions of the predicted future states and are computed using the predicted future trajectories.

If the true parameter values are known, prediction of the future states of the process is based on the conditional distribution of the future states given the observed popula-tion abundances. Let X � ( X 0 , X 1 , X 2 , … , X q ) and Y � ( Y 0 , Y 1 , Y 2 , … , Y q ) represent vectors of unobserved and estimated log population abundances respectively and let X (f ) � ( X q � 1 , X q � 2 , … , X q � f ) be that of future abundances. If the growth process is a fi rst order Markov process, the conditional distribution of ( X , X (f ) ) is

hg X g X m X f Y X

ft t

t

q f

t tt( , | )

( ; , ) ( ; ( ; ), ) ( ; , )( )X X Y

ο ψ21

2

1∏

1

q

C

∏

( ; )Y ψ

ˆ

ˆ

where g(.; μ , σ 2 ) indicates the density function for N( μ , σ 2 ) and C ( Y ; ψ ) is the normalizing constant. Substitution of known parameter values by their estimates leads to pre-diction intervals that have lower than nominal coverage. To account for the uncertainty in the parameter estimates, we can integrate over the bootstrap distribution (Harris 1989, S æ ther et al. 2000, Dennis and Otten 2000) or the asymp-totic Normal distribution (Hamilton 1986, Lele et al. 2010) of the parameter estimates. Simulation results in Torabi et al. (2011) show that actual coverage of such prediction inter-vals is close to nominal in many cases. Th us, for predicting future states, we use the following prediction distribution:

(2));(

ˆ)ˆ(),;()ˆ),ˆ;(;()ˆ,ˆ;( 22

ψ

ρψ

YC

dXYfXmXgXgh(X, X(f)|Y)

q

t�0tt

fq

t�1t−1t∫ ∏∏

+

�o

where r(j) denotes the asymptotic normal distribution of the estimators j � ( h , σ 2 , ψ ). We prefer data cloning algo-rithm because it gives MLE and the associated variance cova-riance matrix simultaneously. Although other computational algorithms (de Valpine and Hastings 2002) may also be used to get the MLE, they require additional computation to esti-mate the variance – covariance matrix of the MLE.

Th e MCMC algorithm (Robert and Casella 2005) can be used to generate random numbers from the distribution in Eq. 2 without evaluating the high-dimensional integral. Th e full description of the algorithm is available in the Supplementary material Appendix 2.1. Notice that missing data pose no special diffi culties in the state – space formulation (Eq. 1). Let S be the index set of those years for which popu-lations counts are available. Th en, in Eq. 2 we simply replace

f Y Xt tt

q

( ; , )1

∏ ψ by f Y Xt tt S

( ; , )∈∏ ψ . Otherwise, everything else

including the computational eff ort remains exactly the same. Although for pedagogical reasons we consider only the fi rst order density dependent growth models, the method ology can deal with higher-order density regulation models. For example, for the second order Gompertz delayed density dependence model m ( X t � 1 , X t � 2 ) � a � b 1 X t � 1 � b 2 X t � 2 , in Eq. 2 we only need to replace the joint distribution

of hidden states, g Xt

q f

( ; ,ο h 2

1∏ˆ g X m Xt t) ( ; ( ; ), )h1

2ˆ ˆˆ , by

g X g X m X g X m X Xt t tt

( ; , ) ( ; ( ; ), ) ( ; ( , ; ), )ο2

1 02

1 22

2h h

qq f

∏ˆh ˆ ˆ ˆˆ .

Again, the rest of the algorithm remains exactly the same.

Effect of observation error on PVA

S æ ther et al. (2000) illustrated the usefulness of vari-ous extinction metrics using the song sparrow Melospiza melodia population counts. Th e data were collected on Mandarte Island, British Columbia, Canada, during 1975 – 1998 (Fig. 1). Population size is the number of territorial females alive each spring (around 30 April). For detailed fi eld methods and population biology of the species, see Smith (1988), Hochachka et al. (1989), Smith and Arcese (1989) and Arcese et al. (1992).

Following S æ ther et al. (2000), we model song sparrowpopulation data using the theta-logistic model. We fi rst consider the model without observation error. Th e theta-logistic model (Eq. 1a) is defi ned by m ( X t � 1 ; h ) � X t � 1 � r[1 – ( N t � 1 / K ) θ ] where h � ( r , K , θ ), r is the specifi c growth rate, K is the carrying capacity and θ represents the theta-logistic type of density dependence. Th e process variance is σ2 � σ2

e � σ2d /Nt�1 where σ2

e and σ2d denote the envi-

ronmental and demographic variances respectively (Engen et al. 1998). Following S æ ther et al. (2000), instead of esti-mating demographic variance from the population data, we simply use the estimate of demographic variance ( σ2

d � 0.66) reported in S æ ther et al. (1998). Th is estimate is based on individual fl uctuations in reproduction and survival of

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error fi ts the data substantially better than both logistic model with observation error and the logistic model with-out observation error. Hence we conclude: 1) song sparrow population counts are subject to observation error, and, 2) as compared to the logistic growth function, theta-logistic provides a better functional form to describe the song sparrow population process.

Interestingly, had we fi xed the logistic model as the correct functional form for population growth, we would have failed to detect the presence of observation error (Table 2) possibly leading to erroneous conclusions. Th is illustrates the importance of fi tting and comparing various biologically plausible growth models both in the presence and absence of observation error. It has been pointed out to us (Mark Taper pers. comm.) that for territorial species such as the song sparrows (Smith and Arcese 1989), the Hassell model (Hassell et al. 1976) or the generalized Beverton – Holt model (Maynard-Smith and Slatkin 1973) might be biologi-cally more plausible than the theta-logistic model. It will be worthwhile to try these additional models along with the meta-population approach.

Th e density regulation parameter θ , when observation error is taken into account (Table 2, Fig. 4), is estimated to be much larger than 1 (θ � 5.38 ) indicating strong den-sity regulation near the carrying capacity K . Th e estimated process variance under the theta-logistic model with obser-vation error is smaller (σ2 � 0.019) than that for the logis-tic model without observation error (σ2 � 0.441 ). Th e large

breeding females. S æ ther et al. (2000) assume that fl uctua-tions in X t are small and use least square method to estimate the parameters. Instead we compute maximum likelihood estimates using data cloning. Th e point estimates (Table 2) are comparable with those of S æ ther et al. (2000). Under the no observation error model, estimate of the density regulation parameter θ is close to 1 suggesting that perhaps logistic model is suffi cient to model the density dependent growth. We use AIC c , the small sample bias-corrected ver-sion of AIC (Burnham and Anderson 2004), to compare logistic and theta-logistic models. Th e AIC c diff erence ( Δ AIC c ) between the theta-logistic and the logistic model is � 2.307. Th us, assuming no observation error, the logis-tic growth model is suffi cient to describe the song sparrow population process. Th is agrees with the conclusion in S æ ther et al. (2000).

Next, instead of simply assuming no observation error, we test if it is present or not. We assume that observation errors are Lognormally distributed. Th at is, in Eq. 1, the observation error distribution f ( y t ; X t , ψ ) is Normal with mean zero and variance τ 2 . Th e maximum likelihood esti-mates of the parameters j � ( h , σ 2 , τ 2 ) are given in Table 2. Using the algorithm in Ponciano et al. (2009), we computed the Δ AIC c values and ranked diff erent models (Table 2). Th e Δ AIC c values are calculated as the diff erence between the AICc value for a given model and the AIC c value of the best-fi tting model. Th e Δ AIC c value for the best fi tting model is, thus, 0. Th e theta-logistic model with observation

Year

Popu

latio

n ab

unda

nce

(a)

1975 1980 1985 1990 1995Year

(b)

1975 1980 1985 1990 1995

40

1

20

40

60

80

1

20

60

80

Figure 1. (a) Comparison of observed population counts (solid black line) of song sparrow from 1975 – 1998, on Mandarte Island, British Columbia, Canada with fi ltered population abundances, N t , from the theta-logistic state – space model (dashed line) and, (b) smooth population trajectories obtained under the logistic model without observation error (solid gray line) and under theta-logistic model with observation error (dashed line). Th eta-logistic model with observation error fi ts the data better than the logistic model without observation error.

Table 2. Maximum likelihood estimates and standard errors of the model parameters. The likelihood function in each case is conditional on the fi rst observed population count. The abbreviations with and without stand for with observation error model and without observation error model respectively.

Theta-logistic (with) Logistic (without) Theta-logistic (without) Logistic (with)

r 0.4662 (0.1527) r 1.1313 (0.3566) r 1.1826 (0.7673) r 1.4274 (0.4376) K 51.1116 (3.8268) K 41.5220 (5.0735) K 41.6922 (6.7235) K 40.6571 (4.1627) θ 5.3802 (2.3421) σ 0.6440 (0.1005) θ 1.0286 (1.6660) σ 0.5080 (0.1648) σ 0.1378 (0.1040) − − σ 0.6446 (0.1003) τ 0.3732 (0.0939) τ 0.2203 (0.0549) − − − − − − Δ AICc 0 28.3124 30.6196 34.6523

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levels in the next fi fteen years. In terms of time to extinction T , under the theta-logistic model, the song sparrow popula-tion is predicted to go extinct after just 4 years ( t 0.1 � 4) with probability 0.9 whereas under the logistic model it is after 18 years ( t 0.1 � 18). Supplementary Appendix 1.2 shows the mathematical relationship between PPI and time to extinc-tion. Estimates of π [s,1] , the probability of recovering from a lower abundance level, are also diff erent under these two models, especially at the lower warning threshold levels (Fig. 3c – d). Th e logistic model gives higher probabilities of surviving from lower abundance levels. In contrast, the theta-logistic model predicts that the odds of song sparrow population recovering from a lower population are relatively small.

At fi rst, the extremely low estimates of π [e,v] under the theta-logistic model seem inconsistent with those of other extinction metrics (Fig. 2a, 3a). However, note that cor-responding to the threshold abundance n e � 3, estimates of the probability of quasi-extinction, π ( n e , 100), and π [v,e] are 0.577 and 0.999 respectively (Table 3). Th us, although the probability of quasi-extinction is large, the population is predicted to reach the carrying capacity before it crashes to extinction with high probability. Th is seemingly anoma-lous behavior is due to the large value of the density regula-tion parameter θ (Table 2, Fig. 4) resulting in strong densityregulation near the carrying capacity (Clark et al. 2010). Th e shape of the smoothed population trajectory (Fig. 1b) shows that the song sparrow population dynamics are intrin-sically cyclical near its carrying capacity, K � 51.1 . Th us, when the population exceeds the carrying capacity, the strong regulatory mechanism kicks in and leads to a cata-strophic decline of the population abundance. Uncertainty in the estimation of θ (Fig. 4) further exacerbates this phe-nomenon. On the other hand, the logistic model yields higher estimates of π [e,v] (Fig. 3b). Fixing the value of θ at 1 results in stable population dynamics.

Th e confi dence bands shown in Fig. 3 are based on the lower and upper 2.5 percentiles of the approximate boot-strap distribution of the corresponding estimated extinction metrics. In the case of π [s,1] (Fig. 3c – d), sampling distribution has large variance resulting in wide confi dence bands. Th isindicates that observed data lack information for the reliable estimation of π [s,1] . However, these confi dence intervals should be interpreted in conjunction with the shape of the corresponding sampling distributions. For instance, the dis-tributions of π [s,1] under the logistic model are left skewed (see Supplementary material Appendix 4 for the approxi-mate bootstrap distributions of the extinction metrics). Th is shows that under the estimated model parameters, most of the future population trajectories that pass the warning thresholds would fail to recover to higher abundance levels. Despite the wide confi dence bands, estimates of π [s,1] impart useful knowledge about the extinction risk. In comparison,the sampling distributions of π[s,1] under the best fi tting theta-logistic model are only slightly left skewed and therefore indicate a smaller chance of recovering from low abundance levels. Furthermore, confi dence intervals for π ( n e ,100) are also wide (Table 3) but the corresponding sampling distributions are left skewed both under the theta-logistic model and the logistic model (Supplementary mate-rial Appendix 4, Fig. 2A).

estimate of observation error variance ( τ2 � 0.049 ) under the theta-logistic model indicates that substantial compo-nent of variation in the population counts is probably due to observation error.

Dennis et al. (2006) discuss two methods for state prediction in state – space models: 1) a ‘ fi ltered ’ value of the log-population abundance X t defi ned as E(Xt|Yt � y t , Y t �1 � y t �1, … , Y 0 � y0 ) the mean of the log-population abundance given the previous and the current observations only and, 2) A ‘ smoothed ’ value of X t which is the mean value of X t given all the observations Y 0 , Y 1 , Y 2 , … , Y q , includ-ing those that follow time t . Th e fi ltered and the smooth population trajectories for song sparrow data are shown in Fig. 1 along with the actual observations. Th e observed data seem to fl uctuate around a constant in a systematic fashion. Th e smoothed population trajectory predicted under no observation error logistic model is virtually constant (Fig. 1b) whereas the smoothed populationtrajectory of the best fi tting theta-logistic model shows systematic cycles (Fig. 1b) and follows the observed fl uctua-tions well. Th e fi ltered trajectory, N t , from the theta-logistic state – space model also indicates that the model fi ts the data well (Fig. 1a).

Extinction properties under logistic and theta-logistic models are substantially diff erent. Logistic model without observation error underestimates extinction risk substan-tially. Population prediction intervals (PPIs) for the future abundance of song sparrow population are wider under the theta-logistic model with observation error (Fig. 2). For instance, the 90% prediction interval obtained from the the-ta-logistic model, as compared to the one under the logistic model, predicts that the population will drop to much lower

Theta−logistic model

10

20

30

40

50

60

10 12 14 16 18 20 22 24 26 28 30 32

Low

er le

vel o

f PPI

Year

Logistic model

10

20

30

40

2 4 6 8

2 6 10 14 18 22 26 30 34 38 42 46 50 54 58

Low

er le

vel o

f PPI

Year

(a)

(b)

Figure 2. 95% (solid line), 90% (dashed line), 75% (dotted line) and 50% (dotted-dashed line) lower bounds of prediction intervals for the future population abundance of song sparrows. PPI for theta-logistic model with observation error are lower than for the logistic model without observation error.

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Table 3. Estimates of extinction metrics based on the predicted future trajectories. Numbers in parentheses are 95% confi dence limits. Estimation is based on population forecast up to 100 time points into the future. The estimates correspond to n e � 3 , n v � 60 and n s � 5 .

Extinction metrics Theta-logistic model Logistic model

π(ne, 100) 0.5769 (0.0034, 0.9563) 0.6466 (0.0556, 0.9996)π[e,v] 0.0004 (0.00, 0.2363) 0.0022 (0.00, 0.0654)π[v,e] 0.9988 (0.7516, 1.00) 0.9977 (0.9346, 1.00)π[s,1] 0.4322 (0.0868, 0.9019) 0.6243 (0.0058, 0.9386)t∼ 36.3225 (14.9095, 51.5767) 43.4200 (12.1322, 51.6010)

ξ∼

(.5)13.4200 (1.1925, 41.1521) 5.1902 (3.1533, 13.4767)

0.0

0.2

0.4

0.6

0.8

1.0

ne

π [e, v

]

Theta−logistic model

30 24 18 12

●●

● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

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●

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●

0.0

0.2

0.4

0.6

0.8

1.0

ne

π [e, v

]

Logistic model

30 24 18 12

● ● ● ● ● ●●

●●

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● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

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ns

π [s, 1

]

(c)

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●

●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

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●

0.0

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1.0

ns

π [s, 1

]

(d)

30 24 18 12

8 4 8 4

8 4 8 4

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●

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● ● ● ● ● ● ● ● ● ● ● ● ●●

●

(b)(a)

Figure 3. Profi les of probabilities of population going extinct before reaching a viable level, π[e,v], and of probabilities of recovering from a lower threshold, π[s,1] , under theta-logistic and logistic model for diff erent thresholds along with 95% confi dence intervals.

Th e warning threshold of fi ve in Table 3 was chosen for two reasons: 1) following the usual practice in PVA it is about 10% of the estimated carrying capacity and, 2) it is close to the smallest abundance level of four observed during the years, 1975 – 1998. We chose the quasi-extinction thresh-old, n e , slightly smaller than n s . Of course, one can plot of π ( n e ,100) against various n e values to get a better picture of the extinction risk. Th ese plots for both logistic and theta-logistic models are presented in the Supplementary material Appendix 4, Fig. 1A.

We have implemented the R programs for model estima-tion, model selection and for the computation of extinction metrics in a user-friendly R package PVAClone. Th e pack-age is available to download from the packages section of

the Comprehensive R Archive Network site ( � http://cran.r-project.org/ � ).

Discussion

All population time series data contain observation error to some degree. It is well known that unaccounted for obser-vation error leads to biased estimates of key model para-meters (Freckleton et al. 2006, Barker and Sibly 2008). Barker and Sibly (2008) conducted a simulation study to investigate the eff ect of observation error in estimating the density regulation parameter theta of the theta-logistic model. Th eir results suggest that estimation of theta is subject

7

in estimating θ . If multimodality and likelihood ridges are present, bootstrap distribution, instead of the multivariate normal distribution used in this paper, is probably a better way to incorporate uncertainty in parameter estimates in forecasting future states.

Th e extinction properties of the song sparrow population highlight a few important points. Traditionally, assessment of the extinction risk of a population is based on extinction metrics corresponding to a single quasi-extinction threshold (such as those given in Table 3). A single value for such a threshold is seldom available because it is determined by multitude of factors such as demographic variability and the genetic structure of the population (Morris and Doak 2002). It is therefore better to evaluate the extinction risk based on extinction metrics estimated for a range of quasi-extinction thresholds. Th e estimates for an extinction metric (such as π [e,v] ) and the associated confi dence limits then can be plotted as a function of the threshold abundance levels (e.g. Fig. 3a – b). Th ese plots, called ‘ extinction profi les ’ , also reveal the shape of the distribution of extinction estimates (Fig. 3, Supplementary material Appendix 4 Fig. 2A). Th is is a more useful measure of the precision of the estimate than the associated confi dence interval alone.

Given several extinction metrics, which metrics should one use in practice? Perhaps probability of extinction, if known, best describes the extinction risk of a popula-tion. PPIs are a convenient alternative way of looking at the extinction risk of a population. Both these metrics, however, lack information about the ability of a popula-tion to recover from low abundance levels, as quantifi ed by π[s,1] . Th is metric can be potentially useful for conservation planning as it can be used to rank populations in terms of their ability to recover from low abundance levels. We also notice that risk assessment based on a single extinction metric alone can be misleading. For instance, the estimatesof π [e,v] for the song sparrow population under the theta-logistic model (Fig. 3a) seem to indicate that the popula-tion is highly likely to remain near the carrying capacity when, in fact, the probability of quasi-extinction, π(ne, 100) , is quite large (Table 3). We therefore recommend that the assessment of the extinction risk of a population should not be limited to a single extinction metric. Perhaps a better approach is to gain an overall picture of the extinc-tion risk by obtaining extinction profi les corresponding to various extinction metrics.

Th e distribution of time to extinction is generally right skewed for most stochastic population growth models (Dennis et al. 1991, Grimm and Wissel 2004). Median time to extinction is a better measure of a population ’ s intrinsic ability to persist (Groom and Pascual 1998, Grimm and Wissel 2004) than mean time to extinction. Th e median extinction time is estimated to be higher (13.4 years) under the theta-logistic model than under the logistic model (5.2 years). However, the population prediction intervals (PPIs) indicate that extinction times are likely to be much shorter under the theta-logistic model (Fig. 2). Following S æ ther et al. (2000), we believe that PPIs are a better mea-sure of extinction risk because they deal directly with extinc-tion times. Th e conlusion based on the overall extinction profi le is that extinction risk is predicted to be much greater under the theta-logistic model than the logistic model.

to large bias especially when environmental perturbation is small. Our analysis of the song sparrow population counts also illustrates that incorporating observation error can result in substantially diff erent estimates than when it is not incor-porated. As we show for the song sparrow example, large changes in parameter estimates in turn lead to entirely diff er-ent assessment of the extinction risk. Th ese results highlight the fact that ignoring observation error could be potentially dangerous for conservation decisions. We therefore contend that the hypothesis of no observation error should always be rigorously tested against data.

Although the presence of density regulation parameter theta in the theta-logistic model provides a fl exible descrip-tion of density dependence, a recent simulation study by Clark et al. (2010) suggests that population abundance data generally lack information for reliable estimation of r and θ , especially when the observed population series is stationary, that is, fl uctuating around its carrying capac-ity. Th is lack of information (Polansky et al. 2009, Clark et al. 2010), leads to multimodality and likelihood ridges in the likelihood surface where the best fi tted growth rate curves are frequently biased toward concave fi ts ( θ � 1) but the likelihood ratio confi dence regions include a wide range of models corresponding to more biologically plau-sible convex fi ts ( θ � 1). Clark et al. (2010), in light of these results, conclude that estimation of extinction risk from fi t-ting theta-logistic model is prone to imprecision. However, their analysis also shows that for recovering populations, i.e. when the populations are fl uctuating away from the station-ary equilibrium, model fi tting is not diffi cult. Barker and Sibly (2008) also observe the same phenomenon even when the observation error is present. However, we agree with the dictum that study of likelihood ridges and multimodality should be a part and parcel of any statistical analysis. To detect possible multimodalities, we plotted the profi le likeli-hood for θ using data cloning (Ponciano et al. 2009). Th e profi le likelihood (Fig. 4) shows no signs of multimodality or ridges in the likelihood surface. In fact, the likelihood vanishes virtually to zero over the entire region of concav-ity of the growth rate curves (i.e. for θ � 1). Th us, large fl uctuations in the song sparrow population away from its carrying capacity (Fig. 1) have helped reduce uncertainty

θ

Like

lihoo

d sc

ore

2 4 6 8

0.0

0.2

0.4

0.6

0.8

1.0

θ^

Figure 4. Profi le likelihood (solid black line) for the density regulation parameter θ in the theta-logistic state – space model.

8

Ak ç akaya et al. 2007). As accurate population estimates are seldom available for such fragmented populations, estima-tion of these models ignoring observation error can seri-ously limit their predictive strength. Recent applications of these models have overlooked the issue of observation error mainly due to the computational diffi culties in handling complex hierarchical models (Dennis et al. 1998, Lele et al. 1998, Schtickzelle and Baguette 2004, Jonz é n et al. 2005). Extending the methods presented in this paper to the spa-tially explicit PVA models would therefore be an important further contribution to the PVA tool kit.

Acknowledgements – Th is research was partially funded by the NSERC and by the Higher Education Commission of Pakistan. We are grateful to Peter Arcese and Mark Taper for important dis-cussions. Peter Arcese kindly provided the song sparrow population counts data.

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Th e song sparrow population on the Mandarte Island was, presumably, surveyed very accurately. Field methods indicate that virtually every adult bird was captured and identifi ed with a combination of a numbered metal leg-band and plastic colored leg-bands (Arcese et al. 1992). Th is left us to wonder about the source of observation error detected in our analysis. Freckleton et al. (2006) discuss dispersal and other possible sources of observation error in census data. Th e average number of sparrows immigrating to Mandarte island during the study period was 1.6 whereas number of immigrants arriving after the 1980 and 1989 popula-tion crashes was 2 and 4 respectively (Smith et al. 2006). Th ese immigrants were counted as part of the Mandarte island population and not accounted separately (Arcese et al. 1992), thus providing a possible source of observation error detected in the population counts. Arcese and Marr (2006) employed a balance equation model (Walters 1986) to study the eff ect of such immigration on the probability of extinc-tion. Th eir results revealed that even a few immigrants at low population densities can result in demographic rescue of the population without altering the expected population abun-dance and provide a buff er against the eff ect of environmen-tal stochasticity (see also Stacey and Taper 1992). A better representation of the population process would, therefore, be achieved by explicitly incorporating a dispersal component into the theta-logistic model.

Th e song sparrow population dynamics provide a cau-tionary example of what might happen to PVA if one ignores key process components such as dispersal. We observe that contrary to a very high risk of local extinction, as predicted under the best fi tting theta-logistic model (Fig. 2a), the song sparrow population at the Mandarte Island has not gone extinct after 1998 (Smith et al. 2006). Furthermore, despite the fact that the population did recover twice from very low abundances during the study period (Fig. 1), the estimated recovery probabilities under the theta-logistic model are quite small (Fig. 3c). Clearly, the omission of dispersal process has substantially reduced the predictive power of the best fi tting population growth model. We, therefore, conclude that that the incorporation of observation error although essential to PVA, may not be enough for reliable estimation of the extinction risk if important processes such as dispersal are ignored.

Th e song sparrow population at the Mandarte Island can be viewed as part of a large metapopulation consisting of other small neighbouring island populations (Smith et al. 1996). Th ese small populations are constantly at risk of local extinction due to environmental variation. In fact two of the populations on smaller islands (within 7 km of the Mandarte Island) did go extinct and remained uninhabited for the next two years (Smith et al. 1996). Considering that these islands share similar environmental conditions and that dispersal is an important factor in preventing local extinc-tions via demographic and genetic rescue (Arcese and Marr 2006), a more realistic approach is to quantify the extinction risk at the metapopulation scale.

To summarize, we showed that incorporation of obser-vation error is crucial to PVA even with simple models. Spatially structured metapopulation models have become a key tool for conservation and management of spatially frag-mented populations (Dunning et al. 1995, Ak ç akaya 2000,

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Supplementary material (available online as Appendix O20010 at � www.oikosoffi ce.lu.se/appendix � ). Appendix 1 – 4.