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e c o l o g i c a l m o d e l l i n g 2 0 1 ( 2 0 0 7 ) 19–26

avai lab le at www.sc iencedi rec t .com

journa l homepage: www.e lsev ier .com/ locate /eco lmodel

comparison of models for predicting populationersistence

.J. Cairnsa,b,∗, J.V. Rossb, T. Taimreb

School of Biological Sciences, University of Bristol, Bristol BS8 1UG, UKSchool of Physical Sciences, The University of Queensland, St. Lucia, Qld 4072, Australia

r t i c l e i n f o

rticle history:

ublished on line 6 September 2006

eywords:

xtinction

opulation model

arameter estimation

atastrophes

a b s t r a c t

We consider a range of models that may be used to predict the future persistence of popula-

tions, particularly those based on discrete-state Markov processes. While the mathematical

theory of such processes is very well-developed, they may be difficult to work with when

attempting to estimate parameters or expected times to extinction. Hence, we focus on dif-

fusion and other approximations to these models, presenting new and recent developments

in parameter estimation for density dependent processes, and the calculation of extinction

times for processes subject to catastrophes. We illustrate these and other methods using

data from simulated and real time series. We give particular attention to a procedure, due

to Ross et al. [Ross, J.V., Taimre, T., Pollett, P.K. On parameter estimation in population mod-

els, Theor. Popul. Biol., in press], for estimating the parameters of the stochastic SIS logistic

model, and demonstrate ways in which these parameters may be used to estimate expected

extinction times. Although the stochastic SIS logistic model is strictly density dependent

and allows only for birth and death events, it nonetheless may be used to predict extinction

times with some accuracy even for populations that are only weakly density dependent, or

that are subject to catastrophes.

important to find some balance between accuracy of predic-

. Introduction

easures of the risk of extinction, such as the expected timeo extinction, are used in population viability analysis (PVA)nd in subsequent decision-making procedures, to gauge thempact of various management actions. Since PVA results areegularly used in the allocation of conservation funding, whichn turn has a major impact on the future persistence of pop-lations, the calculation of quantities measuring the risk ofxtinction is of great importance. These are typically calcu-ated by assuming a specific model for changes in the popula-ion over time. Diffusion models, for example, are often used

ecause they are simple to analyse and can give rise to explicitormulæ for most quantities of interest. However, these mod-ls frequently lead to inaccurate predictions of critical values

∗ Corresponding author.E-mail address: ben.cairns@bristol.ac.uk (B.J. Cairns).

304-3800/$ – see front matter © 2006 Elsevier B.V. All rights reserved.oi:10.1016/j.ecolmodel.2006.07.018

© 2006 Elsevier B.V. All rights reserved.

such as the expected time to extinction. Hence, managementdecisions based on these predictions may be similarly flawed.

A more appropriate model for describing the behaviour ofthe population in question may be a discrete-state Markov pro-cess describing the actual number of individuals in the popu-lation. The most commonly-used such models are birth–deathprocesses or extensions thereof which allow for catastrophicevents. Unfortunately, while these may be more appropriatefor modelling the dynamics of the population in question,they can be more difficult to work with, from both analyti-cal and computational points of view. For these reasons, it is

tions based on the models and tractability of the method ofprediction. Advances can be made by considering the limit-ing processes that correspond to these discrete-state models;

l i n g

20 e c o l o g i c a l m o d e l

in particular, Ornstein–Uhlenbeck processes and piecewise-deterministic processes with stochastic jumps. These modelsmay still provide inaccurate predictions of extinction times,but should show improvement over the simplest Brownianmotion approximation.

We consider populations that have density dependentdemographic rates (in a specially-defined sense), and whichmay also be subject to environmental catastrophes. In partic-ular, we assume that these populations may be modelled bycontinuous-time Markov chains – the stochastic SIS logisticmodel (Weiss and Dishon, 1971) with or without binomialcatastrophes occurring at a constant rate – and comparethe accuracy of several approximations to the expectedtime to extinction. We contrast the various advantagesof several methods for predicting extinction times for theabove-mentioned models, and compare these predictionsfor simulated data and a population of Bay checkerspotbutterflies using model parameters estimated from data. Wepay particular attention to the question of whether the extraanalytical and computational effort required for the morecomplex models is necessary to inform decision-making in aconservation context.

We find that a variety of different models may give compa-rable results for measures of the risk of extinction (such as theexpected time to extinction). This is true even in the situationswe examine where catastrophes are known to play a role inpopulation dynamics, but are not modelled when analysingthe data. One model we consider in detail, the stochastic SISlogistic model and its Ornstein–Uhlenbeck diffusion approx-imation, is particularly robust in allowing for either strongor weak limiting of populations by their carrying capacities,and in adjusting for catastrophic events. A simpler geometricBrownian motion approximation may also provide reasonableresults, but is less reliable due to shortcomings in its estima-tion of the population ceiling. Finally, we determine empiri-cally that heuristic approximation methods for the stochasticSIS logistic model subject to catastrophes can provide accu-rate values for the expected time to extinction when the trueparameters are known. The present lack of a suitable estima-tion procedure for these models would preclude their wideruse, but fortunately other models, such as the stochastic SISlogistic model and its diffusion approximation, can providereasonable estimates of the expected time to extinction.

2. The models

We will consider a number of models for a biological popula-tion with per-capita birth and death rates that are functionsof the population density, rather than of the population size.That is, we assume that the functional relationships betweenthe birth and death rates and the population size x have theform n f (x/n) where n might be equal to a carrying capacityK or population ceiling N. We will call population processesthat possess such rates density dependent. In its common usagein ecology, the term ‘density dependence’ refers to a strong

tendency of some populations to decline when above somecarrying capacity; the carrying capacity is simply the popula-tion size above which this tendency begins. The definition wegive here is a more literal ‘dependence on the population den-

2 0 1 ( 2 0 0 7 ) 19–26

sity’, of any strength, of the given form. Density dependentprocesses of this form are among the most important in mod-elling biological populations of all types. Notable examplesinclude most of the classical models for epidemics, the Levinsmodel for metapopulations (Levins, 1969), and early modelsfor human populations (e.g. Verhulst, 1838). In this paper,our goal is to compare estimates of the time to extinctionobtained from continuous-time Markov chain (CTMC) mod-els with those obtained from approximations to these modelsinvolving either central limit-type results or heuristic approx-imation schemes.

Each of the models detailed below is summarised inTable 1.

2.1. Continuous-time Markov chains

For our purposes in this paper, we use only the transition ratematrix Q of the processes discussed in this subsection, whichhas off-diagonal elements qij giving the rate of jumps fromstate i to state j, and diagonal elements qii = −qi, where qi rep-resents the rate of jumps from state i (qi =

∑j�=i

qij). For furtherdetails on the extensive theory of CTMCs see, for example,Norris (1997).

2.1.1. The stochastic SIS logistic modelWe will begin by assuming that the underlying populationschange over time according to a stochastic SIS logisticmodel (henceforth, simply the SIS model). Such models arenamed for their use in the study of epidemics in whichsusceptible individuals may become infective, then recover tobecome susceptible again (susceptible–infective–susceptible);however, they are broadly applicable as models for densitydependent populations. The SIS model has transition rates,for i and j in {0, 1, . . . , N},

b(i) = qi,i+1 = �i

(1 − i

N

), (1)

d(i) = qi,i−1 = �i, (2)

for births and deaths, respectively. The per-capita birth anddeath rate parameters are, respectively, � and �. N is themaximum size of the population or population ceiling. Forboth parameter estimation and calculation of expected timesto extinction, the SIS model may be approximated by adiffusion, as detailed in Section 2.2 and Appendix A (see alsoRoss et al., in press). Note that both b(i) and d(i) are of the formN f (i/N), so the SIS model corresponds to a density dependentprocess in the sense defined above.

2.1.2. Birth, death and catastrophe modelsIn some cases the main drivers of mortality in a populationmay be catastrophic events causing mass, rather than indi-vidual death. A number of authors have explicitly consideredMarkov chain models in which catastrophic mortality plays animportant role, including models for populations of flour bee-tles, Trilobium (Mangel and Tier, 1993, 1994), Crabeater seals,

Lobodon carcinophagus (Wilcox and Elderd, 2003), and of Cali-fornia spotted owls, Strix occidentalis occidentalis (Andersen andMahato, 1995). In this birth, death and catastrophe model, wereplace some or all individual death by death due to catas-

e c o l o g i c a l m o d e l l i n g 2 0 1 ( 2 0 0 7 ) 19–26 21

Table 1 – Summary of the models presented in Section 2 and their parameters

Model Parameters Interpretation

Stochastic SIS logistic process (SIS) � Birth rateOrnstein–Uhlenbeck diffusion (OU) � Death rate

N Population ceiling

Birth, death and catastrophe process (BDCP) � Birth ratePiecewise-deterministic Markov process (PDMP) � Death rate

� Catastrophe ratep Killing probabilityN Population ceiling

tatat

c

a

d

wf

2

A(ip{∞Ftd

otfitetNtaafdbsa�

p

Geometric Brownian motion (GBM)

rophic events, occurring at a constant rate �, which affect eachnd every individual in the population independently, killinghem with a certain probability p. In this situation, we have b(i)s in (1) above, with an additional rate of jumps down from io j (<i−1) as

(i, j) = qij = �

(i

j

)(1 − p)jpi−j, (3)

nd the rate of jumps from i to i − 1 as

(i) = qi,i−1 = � + �i(1 − p)i−1p (4)

hich is simply the sum of the rates for individual death andor catastrophic events that cause only a single death.

.2. Diffusion and deterministic approximations

number of approximation methods exist for the SIS modeland other density dependent CTMCs). One such methods the deterministic approximation for density dependentrocesses, which is obtained by scaling the process from0, 1, . . . , N} to [0, 1] by dividing through by N, and letting N →

to obtain a functional law of large numbers (Kurtz, 1970).or example, the SIS model has a deterministic approxima-ion given by the solution to the ordinary differential equationx/dt = �x(1 − x) − �x, for x(0) ∈ [0, 1].

Deterministic approximations are often not useful inbtaining measures of extinction risk, since they may predicthat the population reaches a steady state at some non-zeroxed point; though it may take a very long time indeed, even-ual extinction is to be expected in almost all population mod-ls. Pollett (2001) compares two methods for approximatinghe expected time to extinction when the population ceiling

is large: an asymptotic formula, and the numerical solutiono the hitting times of a diffusion approximation. Specifically,n Ornstein–Uhlenbeck (OU) diffusion approximation aroundstable fixed point in the deterministic dynamics may be

ound by subtracting the mean of the process, given by theeterministic approximation described above, and scalingy

√N rather than N (see Pollett, 2001). This OU process is

tationary, Gaussian, Markovian, and has parameters that

re again given explicitly in terms of the original parametersand �.

We use this OU approximation primarily as a method ofarameter estimation, and apply these estimates to obtain-

rd Growth rate meanvr Growth rate varianceN Population ceiling

ing expected times to extinction for the SIS model. In somecases we give expected times to extinction from both the SISmodel and its OU approximation; for further details of suchcomparisons see Pollett (2001).

2.3. Piecewise-deterministic Markov processes

The birth, death and catastrophe model of Section 2.1 issuch that the birth rate remains density dependent, and so,between catastrophes, this process should behave much likeany other density dependent process. Hence, if we conditionedon no catastrophes occurring, it would be possible to approx-imate the change in the population size over time by a deter-ministic process (an approximation based on the law of largenumbers limit, rather than the diffusion approximation cor-responding to the central limit). In order to obtain such anapproximation, we would scale the population size by divid-ing through by N and letting N → ∞.

When we consider catastrophes of the form given in (3) insuch a limit, we find that the discontinuous nature of catas-trophic events does not disappear into the average dynamicsas do birth and death events. Suppose that we condition on acatastrophe occurring when the population size is a particu-lar proportion of the population ceiling N, say xc/N (≤1). Then,letting N → ∞, the population size just after the catastropheconverges almost surely to (1 − p)xc, by the strong law of largenumbers.

The birth, death and catastrophe process is not densitydependent in the sense required for a diffusion or determin-istic approximation of the type discussed in Section 2.2. How-ever, it is density dependent ‘between catastrophes’, and thesize of a catastrophe – conditional on its occurrence and thepopulation size just before the catastrophe – converges to aconstant proportion of this population size as N → ∞. Thissuggests an approximation for the birth, death and catastro-phe process that is somewhat analogous to the deterministic,law of large numbers approximation for density dependentprocesses.

In this approximation, between catastrophes the pop-ulation dynamics are described by an ordinary differentialequation, given by the solution to dx/dt = �x(1 − x) − �x. Then,

catastrophes occur according to a Poisson process with rateparameter �, and when they occur they instantly reduce thesize of the population to a proportion (1 − p) of the populationsize just before the catastrophe. That is, if S is the time of a

l i n g 2 0 1 ( 2 0 0 7 ) 19–26

Fig. 1 – One hundred years of population sizes obtained bysimulating an SIS process (black, triangles) and abirth-and-catastrophe process (grey, circles) with the same

22 e c o l o g i c a l m o d e l

catastrophe, then x(S) = (1 − p)(limt↑S x(t)). Since this processis Markovian and exhibits deterministic behaviour betweenstochastic jumps, it is known as a piecewise-deterministicMarkov process (PDMP). The class of PDMPs includes almostevery Markovian process that is not a diffusion (Davis, 1984),including CTMCs, but is relatively little-studied in general.We propose a simple but robust procedure for numericalcalculation of the expected time to extinction of this PDMP(Appendix B). We will compare the exact expected extinctiontimes for birth, death and catastrophe processes with theapproximation given by a piecewise-deterministic Markovprocess (Section 2.3).

2.4. Geometric Brownian motion

Foley (1994) considers a simple procedure for estimatingextinction risk to populations using geometric Brownianmotion (GBM). Under this model, the population changesaccording to a stochastic differential equation, such that, ifthe natural log of the population size is Y(t) = log X(t) at timet, then Y(t) is assumed to satisfy the stochastic differentialequation dY(t) = rd dt + √

vr dW, where W is the white noiseprocess. The parameters are the mean and variance of the nat-ural log of the growth rate of the population, rd and vr, respec-tively. The advantage of this model is that expected times toextinction can be found analytically in all cases through a sim-ple procedure for parameter estimation. However, while it isdensity dependent in the literal sense, it explicitly sidestepsthe issue of declining growth or increasing mortality at largepopulation sizes. Also, it is not proposed as an approximationto any particular discrete-state population model (however,it may be constructed as the limit of a random walk usingthe theory of density dependent processes described abovein Section 2.2). Foley uses simulation methods to explore thepossibility of density dependence in a range of ecological timeseries, but reports very little difference in the results.

3. Results

3.1. Simulated data

We simulated two time-series of data to examine the predic-tion of extinction risk in populations that may be subject tocatastrophes. In each case, the maximum size of the popu-lation was N = 400, the birth rate was � = 0.25 and the meandeath rate, either by individual death or catastrophic events,was 1/6. The first series of data was simulated from an SISmodel (Section 2.1) with death rate � = 1/6. The second serieswas simulated using a birth-and-catastrophe model (Section2.1), obtained by setting � = 0 so that there are no individualdeath events, but allowing for catastrophes at rate � = 2/3 withdeath probability p = 0.25. Each of these models represents apopulation that tends to increase when the population size issmall and decline when the population size is large. A hundredyears of each of these simulated time-series is given in Fig. 1.

We computed estimates for �, � and N using the OU approx-imation, and for rd and vr for the GBM model, assuming eitherthat N was 400, as used in the actual simulation, or that N wasunknown; these estimates were computed using the methods

population ceiling, birth rate and mean death rate.

in Appendix A, and are given in Table 2, Sim. 1 and Sim. 2, (a–c). The level of quasi-extinction is the population size belowwhich our hypothetical population is considered functionallyextinct or at least severely threatened. From these estimates,expected times to quasi-extinction below 40 individuals werecomputed for the various models (Figs. 2 and 3; see AppendixB for methods).

In the case of the data simulated from the stochastic SISlogistic model, the estimates obtained (Table 2, Sim. 1) pro-duced widely varying expected times to extinction. Only whenthe underlying process and the population ceiling N are knowndo the estimated results compare well with the true expectedtimes to extinction, although in this case the match is veryclose (Fig. 2). Not plotted in Fig. 2 are the results for theexpected time to extinction of the OU approximations; thesewere consistently 3–4 times larger than the results for thecorresponding SIS models. The greatest differences from thetrue values were obtained when the population ceiling N wasestimated alongside the demographic parameters in the SISmodel, and in the case of the GBM model (Fig. 2) which wasexpected to perform poorly in this case as it matches the trueprocess least well. (Note that the plotted domains of the esti-mated expected times provided by the various models differdue to differences in the estimates of the population ceiling N.)

In the presence of catastrophes, the successes of the var-ious estimates (Table 2, Sim. 2) in predicting the expectedtimes to quasi-extinction are reversed (Fig. 3). In this case,the underlying process has catastrophes replacing individualdeath, which increases the apparent variability of the process.Consequently, the SIS model and OU approximation from theSim. 2 (b) estimates produced large values, up to 5.40 × 106 and1.90 × 107 years, respectively, and are omitted from Fig. 3. Wecompared the expected times to quasi-extinction from each of

the fitted models to the values computed from the underlyingbirth-and-catastrophe process and its PDMP approximationusing the true parameter values; the width of the plotted ‘rib-bon’ gives the precision of the numerical procedure for the

e c o l o g i c a l m o d e l l i n g 2 0 1 ( 2 0 0 7 ) 19–26 23

Fig. 2 – Expected times to quasi-extinction below 40individuals for the simulated time series withoutcatastrophes: the exact (black, solid) and estimated (black,dashed) times for SIS models with N known, the estimatedta

Pisevi

afhr

Fi(usaa

Table 2 – Estimates obtained from the JRH butterfly dataand the simulated stochastic SIS logistic (Sim. 1) andbirth-and-catastrophe (Sim. 2) data, using Appendix A:(a) for GBM, and for the SIS model: (b) assuming N isknown, and (c) assuming N is unknown

Estimate JRH Sim. 1 Sim. 2

(a)N 1998∗ 164 299rd −0.052∗ −0.0013 −0.0011vr 0.840∗ 0.0032 0.0571

(b)N 1998∗ 400 400� 4.8585 0.3233 1.6343� 3.9506 0.2144 1.0832

(c)N 280156 309 3035� 245.645 0.3849 3.4982

imes for an SIS model where N is unknown (grey, solid)nd the GBM approximation (black, dash-dotted).

DMP, which was computed in this case using a total of 1537ntervals (see Appendix B). The SIS and OU results were rea-onably accurate despite overestimating the expected time toxtinction. The GBM approximation also produced reasonablealues for the expected time to quasi-extinction, although its made less reliable by considerably underestimating N.

We also computed an additional piecewise-deterministic

pproximation by taking the estimates of �, � and N obtainedrom the Sim. 2 (c) estimates, and assuming that catastrophesad the true killing probability p = 0.25 so that the catastropheate was � = �/p. The mean times to quasi-extinction found in

ig. 3 – Expected times to quasi-extinction below 40ndividuals for: the SIS (black, solid) and OU approximationblack, dashed) using estimates with N unknown; thenderlying birth-and-catastrophe process used in theimulation (grey, solid), the piecewise-deterministicpproximation to this (grey, ‘ribbon’), assuming N = 400,nd the GBM approximation (black, dash-dotted).

� 245.215 0.2180 3.3269

∗ These values are due to Foley (1994).

this way were an order of magnitude smaller than the plottedvalues. This indicates that, although such a procedure mightbe a natural first guess for scientists attempting to extractinformation about catastrophes from estimates for individualmortality, it is unlikely to produce accurate results.

3.2. Time-series of real populations

We considered the time series of the Jasper Ridge ‘JRH’ popula-tion of Bay checkerspot butterflies, Euphydryas editha bayensis,sampled yearly between 1960 and 1986 (Harrison et al., 1991).As we discuss in the next section, this population is nowextinct. Although data exists up to the time of extinction, itis of interest here to compare approaches for prediction ofextinction risk using data and analyses from a time whenthe population was still extant. The data from this popula-tion was analysed by Foley (1994) in the context of the GBMmodel described in Section 2.4. Harrison et al. (1991) foundthat the observed data are consistent with the assumption ofdensity independence or only weak density dependence, inthe sense of a tendency to decline when above the carryingcapacity. We obtained estimates of birth and death rates (�and �, respectively) and of the population ceiling N from theOU approximation to the SIS model, using the same numeri-cal maximum likelihood procedure as in Section 3.1. We usedFoley’s estimates of parameters rd and vr for the GBM approxi-mation or found our own (see Appendix A). The best estimatesobtained using these procedures are given in Table 2.

In our analysis of the JRH data, we calculated the riskof extinction proper: the event that the population droppedbelow 1 individual. Expected times to extinction were calcu-lated from the GBM model, the SIS model and its OU approxi-mation, using the estimates from Table 2 JRH (a and c). Theseresults are given in Fig. 4. The JRH (b) estimates, for whichN = 1998 was considered known, produced expected times to

extinction of more than 1013 years, considerably longer thanmodern estimates for the current age of the universe. Theexpected times to extinction for the OU approximation usingthe JRH (b) estimates are assumed to be similarly large, but

24 e c o l o g i c a l m o d e l l i n g

Fig. 4 – Estimated mean time to extinction for the JRHpopulation of Bay checkerspot butterflies. Foley’s (1994)

approximation (dash-dotted), an SIS model (solid) and itsOU approximation (dashed) assuming N is unknown.

consequently we were unable to compute them accurately.While such large values are not a priori unreasonable for popu-lations in general, they may safely be assumed to be extremeoverestimates in the present case and we omit these valuesfrom Fig. 4 in order to better see the relationship betweenthe other results. The SIS and OU models produced similarexpected times to extinction from the JRH (c) data, which dif-fered by approximately an order of magnitude from the resultsfor the GBM (obtained by Foley, 1994).

4. Discussion

Our analysis of the JRH data provides some interesting insightsinto the relationship between the various models presentedhere. First, the estimates for � and � differ markedly depend-ing on whether or not N is assumed to be known (Table 2). TheN-unknown estimates support the conclusion of Harrison etal. (1991): since the estimates for � and � are so close, the popu-lation tends to be small, relative to N, and hence b(i) ≈ �i over awide range of population sizes. Consequently, the populationdoes not exhibit a strong downward trend in size followingdeviations from its long-term mean size. This result indicatesa degree of flexibility in the SIS model and the estimation pro-cedure that we propose when N is assumed to be unknown,because, while the model allows for a tendency to decline atlarge population sizes, it does not require this tendency to bestrong. As a wide variety of density dependent processes maybe analysed using this approach, it appears to provide a robustmethod for assessing whether populations are strongly lim-ited by their carrying capacity.

The SIS, OU and GBM models for the butterfly populationshowed somewhat different predictions for the expected timeto extinction, although the SIS model and its OU approxima-

tion gave very similar values (Fig. 4). The GBM model pro-duced predictions that were similar to those obtained fromthe SIS and OU models assuming an unknown N, unlike thosefound when assuming N = 1998. This suggests that both the

2 0 1 ( 2 0 0 7 ) 19–26

SIS/OU and GBM approximations can give comparable esti-mates despite the difference in their structure and estimatesfor N. However, because the estimate of N for the GBM model isderived from only the maximum observed value, the degree ofcompatibility between these results may vary. A comparisonof the results of these models may serve as a useful test of thereliability of predictions of the time to extinction, especiallywhen considering density dependence.

The Bay checkerspot butterfly is now extinct on JasperRidge, with the JRH population thought to have gone extinctin 1998 due to a combination of extensive habitat loss anda period of increased climatic variability beginning in 1972(McLaughlin et al., 2002). It is interesting to note that eachof the expected times to extinction plotted in Fig. 4 are quiteclose in relative terms to the observed time of extinction—12years after the last measurements used in the present study.In absolute terms, the SIS and OU times suggest a higher levelof risk, as might be presumed, with the benefit of hindsight,to have been the case.

The simulated SIS data demonstrates that care must betaken with estimates derived from the OU approximation.In this instance only the expected times to quasi-extinctionobtained for the known population ceiling N produced accu-rate results, and the OU models were less accurate than theirSIS counterparts. While the OU approximation can producesuitable estimates for use with the SIS model in producingmeasures of (quasi-) extinction risk, it is not always so reli-able when used directly in assessing such risk. Hence, werecommend that even when the OU approximation is usedto estimate parameters, the SIS model should be used in cal-culating measures of extinction risk.

Because there is at present no suitable method for esti-mating parameters for birth, death and catastrophe modelsfrom real-world data, we constructed data (Fig. 1) simulatedfrom a CTMC birth-and-catastrophe model (without a sepa-rate individual death term) in order to assess the performanceof the OU and GBM approximations in dealing with the pres-ence of catastrophic mortality. Again, both these and the SISmodel derived from the OU estimates of �, � and N, gave resultsthat are roughly similar to the exact results for the underly-ing birth-and-catastrophe process (Fig. 3). This suggests thatthese models are able to adjust for the effects of catastrophes.

Finally, the PDMP model appears to give very good approx-imations to the mean times to (quasi-) extinction for birth-and-catastrophe processes (Fig. 3); these were consistentlyoverestimates, but followed the pattern of the true valuesquite closely. However, the mathematical links between thebirth, death and catastrophe process and the PDMP approxi-mation remain to be properly described, and in the absenceof a procedure for estimating birth and death rates togetherwith catastrophe rates and killing probabilities, the utility ofthis model in dealing with real-world data will remain limited.Fortunately, with careful use, models such as the OU approxi-mation are adequate in at least some of these circumstances.

Although this is a only a preliminary study, the evidencepresented herein suggests that the SIS model, with parameter

estimates provided by its OU approximation, can providereasonable approximations to measures of extinction risk,such as the expected time to extinction. The GBM model mayalso be capable of providing reasonable predictions, but it

n g

mteaes

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MATLAB on the appropriate boundary value problem. We

e c o l o g i c a l m o d e l l i

ay be prone to over- or under-estimate the expected timeo extinction, depending on the underlying process. This mayven be true when catastrophic declines in population arefeature of the process, however care should be taken to

xamine the reliability of these estimates, for example, byimulation methods such as we have used here.

cknowledgements

his work was supported by PhD scholarships to each of theuthors from the Australian Research Council Centre of Excel-ence for Mathematics and Statistics of Complex Systems. Theork of B.C. was also funded by the BBSRC. The authors would

ike to thank Phil Pollett for providing code to compute theean time to extinction for the OU approximation, and the

eferees for their helpful suggestions.

ppendix A. Parameter estimation

.1. The SIS model

ere we summarise our parameter estimation method for theIS model. For a general presentation of the method please seeoss et al. (in press). The parameter estimation method useshe OU approximation as follows. Firstly, the OU process istrongly stationary if we start it in equilibrium: Z(0) ∼ N(0, �2),here �2 = �/� = � (<1) (see Pollett, 2001; Ross et al., in press).herefore, assuming that the population is initially at quasi-quilibrium around the carrying capacity (true equilibriumeing the state of extinction) we have for large N that

N(0) ∼ N

(x∗,

�2

N

),

here x∗ = 1 − � is the stable fixed point of the density depen-ent deterministic approximation, and N is the populationeiling (see Pollett, 2001; Ross et al., in press). We may thenpproximate Cov(XN(s), XN(s + t)) by

(t) := 1N

Cov(Z(s), Z(s + t)) = c(0) exp(B|t|), (A.1)

here c(0) = �2/N and B = −(� − �) (see Pollett, 2001; Ross et al.,n press). This explicitly gives the correlation structure of theaussian vector (XN(t1), XN(t2), . . . , XN(tn)), where n is the num-er of observations, and thus we know its likelihood function:

(x) = 1√(2�)n|V|

exp[−1

2(x − x̄)V−1(x − x̄)′

], (A.2)

here x̄ = (x̄1, x̄2, . . . , x̄n), x̄i = x∗, for all i = 1, 2, . . . , n, and V ishe covariance matrix with elements vi,j,

=

⎛⎜⎜

v1 v1,2 v1,3 · · · v1,n

v1,2 v2 v2,3 · · · v2,n

⎞⎟⎟ ,

⎝ ...

......

......

v1,n · · · · · · · · · vn

⎠

here vi,i = vi = �2/N and vi,i+s = (�2/N) exp(B|ti − ti+s|).

2 0 1 ( 2 0 0 7 ) 19–26 25

We may therefore evaluate the (joint) maximum likelihoodestimators, which are the values of � and � that maximise(A.2). Explicit calculation of these is not feasible if the sam-ple size is large. Hence, we use the Cross-Entropy Method ofnumerical optimisation (Rubinstein and Kroese, 2004) to findthe parameters which maximise the likelihood function, butother numerical optimisation procedures should be similarlyeffective. When the population ceiling N is unknown, we esti-mate it by using the approximation NXN(0) ∼ N(x∗N, �2N) andthen calculate the joint maximum likelihood estimators withthe addition of N as an unknown parameter.

Note that because the OU approximation is achieved byletting the maximum population size tend to infinity, this pro-cedure is best for large population sizes. Further, unequallyspaced sampling of the process does not present difficulties,as can be seen from the covariance structure (A.1).

A.2. Geometric Brownian motion

Foley (1994) proposes a range of tools for estimating theparameters rd, vr and N using the GBM approximation.For comparison, we employ the simplest such approach. Ifr = Y(t) − Y(t − 1), where Y(t) is the natural logarithm of thepopulation size at time t, then rd is just the arithmetic meanof r, and vr is just the variance of r, as computed using thestandard unbiased estimators. The population ceiling N ischosen in this instance as one more than the largest observedpopulation size.

Appendix B. Mean times to extinction

B.1. Continuous-time Markov chains

The expected time to extinction for a CTMC is relatively easyto calculate in many cases, using the transition rate matrixQ (e.g. constructed from (1) to (3)). First, we restrict Q to thenon-extinct population sizes; we remove all those rows andcolumns with entries corresponding to (quasi-) extinct pop-ulation sizes, and call this restricted matrix M. Then, theexpected time to extinction is the minimal, non-negative solu-tion � to M� = −1. In most cases, this system of linear equa-tions can easily be solved using virtually any numerical com-puting package. Difficulties may arise if M is particularly largeor complex in structure, in which case the expected times toextinction for a corresponding OU process or PDMP may beuseful, if such approximations are valid. Where there are nosuch approximating processes, approximate expected timesto extinction may still be calculated by truncating M (Mangeland Tier, 1993; Cairns and Pollett, 2005).

B.2. Ornstein–Uhlenbeck processes

We follow Pollett (2001) and find the expected times toextinction for the OU model using the bvp4c procedure in

choose slightly different bounds between which to evaluatethe solution, to allow for quasi-extinction and to make theupper bound equivalent to N. See Pollett (2001) for furtherdetails.

l i n g

stochastic and deterministic models of an epidemic. Math.

26 e c o l o g i c a l m o d e l

B.3. Geometric Brownian motion

Foley (1994) gives the derivation of the expected time to extinc-tion for a population under the assumption that the popu-lation size changes approximately according to a GBM. Theexpected time to extinction Te(x0), when the log of the initialpopulation size is x0, is given by

Te(x0) = 2x0

vr

(n − x0

2

), rd = 0,

where rd is the mean of the natural log of the populationgrowth rate, vr the variance in the log growth rate, and n =log N is the log of the population ceiling, or,

Te(x0) = 12srd

(e2sn(1 − e−2sx0 ) − 2sx0),

if rd �= 0, where s = rd/vr. Because of the symmetry of the pro-cess in the log of the population size, we may adjust for quasi-extinction by subtracting the log of the quasi-extinction levelfrom both x0 and n (in the above, we have assumed the extinc-tion level is equal to 1).

B.4. Piecewise-deterministic Markov processes

Expected times to extinction for PDMPs have been consideredanalytically by Hanson and Tuckwell (1981, 1997). We employ arather different, unsophisticated, yet very robust approach toobtain numerical solutions. The expected times to extinction�(u), for u ∈ [0, 1], satisfy a delay differential equation that maybe solved implicitly, by use of an integrating factor and care-ful selection of boundary values, to obtain a (delay) integralequation: �(u) then also solves

�(u(t; ue)) = 1�

+ e�t

∫ ∞

t

� e−�x�((1 − p)u(t; ue)) dx. (B.3)

Here, as earlier, � is a growth rate constant, � the rate of occur-rence of catastrophes, and p is the killing probability. Theconstant ue ∈ (0, 1) is the quasi-extinction level, and u(t; ue)is the solution to du/dt = �u(1 − u), with u(0) = ue. In order toproceed, we note that (B.3) has the form � = K�, where K is thefunctional given by the right-hand side of the equation, andhence the expected time to extinction is the minimal, non-negative fixed point of K.

We exploit the form of (B.3) by approximating the integralby its lower and upper Riemann sums, respectively, K and K̄,evaluated over a mesh chosen so that if x is a mesh point thenso is (1 − p)x. Then, it is easy to show that, because K h ≥ K g

and K̄ h ≥ K̄ g if h(u) ≥ g(u) for all u (the functionals are mono-tone increasing), the minimal non-negative fixed points of K

and K̄ bound � from below and above, respectively, (see Lemma

2.11.1 of Gihman and Skorohod, 1972). These fixed points canbe found using numerical linear algebra. Then, by refining themesh on which the Riemann sums are calculated, the boundson � can be tightened, and the use of interval analysis (see, e.g.

2 0 1 ( 2 0 0 7 ) 19–26

Hansen and Walster, 2004) can ensure that numerical errorsin the calculation of these bounds are also accounted for.

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