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Ecological Modelling 250 (2013) 72– 80

Contents lists available at SciVerse ScienceDirect

Ecological Modelling

jo u r n al hom ep age : www.elsev ier .com/ locate /eco lmodel

n the evidence of an Allee effect in herring populations and consequences foropulation survival: A model-based study

api Sahaa, Amiya Ranjan Bhowmickb, Joydev Chattopadhyayb, Sabyasachi Bhattacharyab,∗

Government College of Engineering & Textile Technology, 1 Barrack Square, Berhampore, IndiaAgricultural and Ecological Research Unit, Indian Statistical Institute, 203, B. T. Road, Kolkata 700108, India.

r t i c l e i n f o

rticle history:eceived 14 June 2012eceived in revised form 18 October 2012ccepted 21 October 2012vailable online 5 December 2012

eywords:llee effectxpected time to extinctionrobability of extinction

a b s t r a c t

We propose a deterministic model for a population subject to a strong Allee effect and undertake a modelbased study on commercially valuable herring fish population. We analyze the time series of two herringpopulations from the Icelandic and Canadian regions from the Global Population Dynamics Databasewith GPDD Id 1765, 1759. The parameters for the proposed models are estimated using Nonlinear LeastSquares and Grid Search procedures. Confidence intervals for the parameters are computed using bothNonlinear Least Squares and regression bootstrap estimates. In the stochastic counterpart of the modelwe consider demographic noise to estimate different extinction measures viz. probability of extinctionand expected time to extinction. The data histogram of population size is well approximated throughthe quasi-stationary distributions of the proposed stochastic model. The hypothesis of the presence of a

heta-logistic modelonservationPDDtochastic differential equation

strong Allee effect is prominent in both of the herring populations. The presence of a strong Allee effect inthese two populations makes them more vulnerable to extinction. External perturbation or uncontrolledharvesting may drive the populations below the Allee threshold where the probability of extinction ishigh. We suggest that, our analysis can have a huge impact on understanding extinction patterns andenable us to identify demographic threats and guide decision making in conservation management. Inaddition, a similar analysis can be used in understanding the conservation status for other species.

. Introduction

Empirical evidence for severe worldwide depletion of commer-ially valuable fish stocks is plentiful. It is observed that many ofhe inshore sub-populations of Atlantic Herring (Clupea harengus),xample of an important marine species having both biologicalnd economic value, had already been driven to extinction or areuffering from unprecedented depletion in abundance (Smedbolnd Stephenson, 2001). There may be a tendency among conser-ationists to regard Allee effects as factors which will only kick inhen populations are already so small as to be doomed to extinc-

ion. Fisheries scientists have long been aware of the possibility ofllee effects (also termed as depensation) in marine populations

Myers et al., 1995; Liermann and Hilborn, 1997) and have calledor investigation of the “depensatory process”, because many stocksre exploited at low density.

The Atlantic herring is a small, pelagic plankton-feeder thatrows to a maximum of 17 in. and 1.5 pounds. Atlantic herringlupea harengus are found in the pelagic zone of marine waters,

∗ Corresponding author. Tel: +91 9433897120; fax: +91 33 2577 3049.E-mail addresses: bapi.math@gmail.com (B. Saha), sabyasachi@isical.ac.in

S. Bhattacharya).

304-3800/$ – see front matter © 2012 Published by Elsevier B.V.ttp://dx.doi.org/10.1016/j.ecolmodel.2012.10.021

© 2012 Published by Elsevier B.V.

as well as coastal zones throughout their geographic reach. Adultsmigrate across hundreds of miles of ocean during their life span(Wikipedia, 2012; GMRI, 2012). In the winter, schools of migrat-ing Atlantic herring can join forces, forming massive expanses offish as far as the eye can see. In the North Atlantic, people haveobserved herring schools measuring up to 4.5 billion cubic meters(over 4 cubic kilometers) in volume, with densities of up to 1 fishper cubic meter (Radakov, 1973). Studies have shown that herringtends to increase their swimming speed with increased light inten-sity (Batty et al., 1990). In experiments with simulated sounds oftoothed whales, a major impact on herring has been observed – theystop eating, swim downward and form schools actively (Wilson andDill, 2002). Herring larvae Clupea harengus L. are among the mostthoroughly studied marine organisms, and several comprehensivemodels of a wide array of the basic biology have been developedfor this species (Beyer, 1980; Beyer and Laurence, 1981; Kiørboeet al., 1987; Kiørboe and Saiz, 1995; Arrhenius and Hansson, 1993;Heath, 1993). Studies on the dispersal and population dynamics ofthe herring population based on structured metapopulation modelcan be found in Ware and Schweigert (2001), and Johannessen et al.

(2009)(see also, (Smedbol et al., 2002)).

By definition, a positive correlation between population sizeand its per capita growth rate at low population sizes is known asthe Allee effect (Allee et al., 1949). Marine populations have been

l Mod

coltcM2fibppa2fItmnhc

thfJmwetstmm

pisiaim

dt((aiPetmh

2

t(dasoCb

B. Saha et al. / Ecologica

onsidered as less susceptible to depensatory mortality becausef large effective population size with wide dispersal of planktonicarvae. But the over exploitation of many fisheries (Clark, 1990) andhe increasing empirical evidences of severe depletion of commer-ially valuable fish stocks worldwide (Maroto and Moran, 2008;cManus et al., 1997; Mous et al., 2000; Dronova and Spiridonov,

008), have made depensatory dynamics an important study insheries science and management. For example, Atlantic cod haseen reduced to a few percent of historical abundances: North Seaopulation has declined 90% since 1970 (Rowe et al., 2004), theopulation around Newfound land has decreased to 1% of 1960s,

loss of approximately 1.5 billion breeding individuals (COSEWIC,003). At least two billion reproductive individuals have been lostrom these areas alone (Rowe et al., 2004; Courchamp et al., 2008).n the Canadian Atlantic region, overfishing was so uncontrolledhat almost 77–99% of the spawning biomass of several cod (Gadusorhua) stocks were reduced (Myers et al., 1997). Beverton (1990)oted that the Icelandic spring-spawning herring (Clupea harengusarengus) has failed to appear even twenty years, since populationollapse.

The overall decrease in the herring population also caughthe attention of many fisheries scientists because, this speciesas significant socioeconomic value and biological importance

rom management perspectives (Roughgarden and Smith, 1996;akobsson, 1985). The worldwide collapse of fisheries stocks are

ainly due to over fishing, over exploitation, contamination ofater with poisonous elements (Lundstedt-Enkel et al., 2010), dis-

ase (Marty et al., 2003) etc. Bjørndal and Lindroos (2004) reviewedhe monitoring and management process of North-Sea herringhared by the EU and Norway and discussed the consequences ofhe management actions. Taking a discrete-time game-theoretic

odeling approach, they showed that the EU should be allocatedore than a half of the total allowable catch quotas.Exploratory data analysis based on population time series will

rovide more insights in understanding the species growth behav-or and its associated extinction pattern. Additionally, model basedtudies have potential impact on analyzing such growth dynam-cs. Although the deterministic models are simpler to handle andnalyze than their stochastic counterparts, stochasticity must bencorporated in order to handle the unpredictability of complex

arine systems.In this article, we propose a model for herring population

ynamics that has a strong Allee effect. For this study we usewo datasets from the Global Population Dynamics DatabaseGPDD) maintained by the National European Research CouncilNERC) with Id 1765 and 1759 (NERC, 2010). We also formulate

stochastic differential equation incorporating demographic noisen the proposed model and estimate different extinction measures.arameters of the proposed model are estimated by both Nonlin-ar Least Squares (NLS) and parametric bootstrap regression andheir significance is justified in the ecological context. Our analysis

ay have important applications in conservation management oferring populations.

. The data

In the present study we validate our analysis using popula-ion time series data of 24 years (1947–1970) for Icelandic herringGPDD Main Id. 1765) and of 15 years (1961–1976) for Cana-ian herring (GPDD Main Id. 1759) (see Fig. 1). These data setsre available in the GPDD which is a vast data base of various

pecies around the world containing records of population sizesver different years <http://www.sw.ic.ac.uk/cpb/cpb/gpdd.html>.anadian Tech. Report of Fisheries & Aquatic Sciences, 2024 haseen reported as the original data source of dataset 1765 (Myers

elling 250 (2013) 72– 80 73

et al., 1995). For this dataset (GPDD Id 1765) sampling was donein the Palaearctic Biogeographic Zone of Iceland, Europe. For theother dataset (GPDD Id 1759), sampling was done in the NearcticBiogeographic Zone of Canada, North America. Both of these her-ring populations are found in the pelagic region and live at depthsof 250 m. The sample units are given in tonnes.

3. The model

A deterministic single species model often takes the formdx(t)/dt = � (x (t)), where x(t) is the population size at time tand �(x) is a bounded, continuous function representing thedensity dependant growth process. Various forms of density depen-dant relationships are found in the literature. Different forms of�(x) demonstrate different types of growth mechanism, such asnegative density dependent (e.g. logistic), density independent(exponential), and positive density dependent (Allee effect). Wewill concentrate on the positive density growth mechanism (Alleeeffect).

The general �-logistic model takes the form (Gilpin and Ayala,1973),

dx

dt= rx

(1 −

(x

K

)�)

, (1)

where r is the intrinsic growth rate, K is the carrying capacity ofthe environment, and � is the parameter describing the curvatureof the relationship and measures the density regulation aroundthe carrying capacity. Convex relationships (� > 1) imply that percapita growth rate varies little until population size is near carry-ing capacity, then drops rapidly. Concavity (� < 1) means that percapita growth rate is initially relatively high, so, small populationsgrow quickly, but then declines rapidly as population size increases.Large values of � correspond to a strong density dependence aboveK, where as small values of � allow larger fluctuations above K.

Thus in the �-logistic model the per capita growth rate is a mono-tonically decreasing function of population size. The basic idea isthat, as the population size increases, intra-specific competitionincreases, per capita growth rate decreases and vanishes at carry-ing capacity. In the case of low population size however, the Alleeeffect may be prevalent and there may exist a critical populationsize below which the per capita growth rate becomes negative. Thiscritical depensation level makes the population more vulnerableto extinction. Schooling fishes, in particular are more vulnerableto critical depensation levels due to overfishing (Courchamp et al.,2008, p. 177). Over harvesting in dense schools causes the popu-lation to decline so rapidly that it may not recover. The depletedschool is also more vulnerable to predators, that may lead tothe ecological and economic collapse of the fishery (Hilborn andWalters, 1992). In this case, the strength of density dependencemay play an important role and can have potential implications forthe preservation of animal populations (explained later, Section. 6).

In addition to the Allee threshold the strength of density reg-ulation also plays an important role. In order to include the Alleeeffect, we consider the following model,

dx

dt= rx (x − a)

(1 −

(x

K

)�)

, (2)

where a is the Allee threshold and the other parameters have thesame meaning as explained in Eq. (1). If a ≤ 0 the positive densitydependence still persists but there is no critical depensation. Thissituation which is called a weak Allee effect. When there is a crit-

ical depensation in the population (a > 0), the situation is called astrong Allee effect. The parameter � describes the strength of den-sity regulation at carrying capacity K. A simple stability analysissuggests that for � > 0, the equilibrium points 0 and K are stable and

74 B. Saha et al. / Ecological Modelling 250 (2013) 72– 80

1950 195 5 196 0 196 5 1970

020

040

060

080

0

Iceland Herring, ID 1765

N

1965 197 0 1975

020

040

060

080

010

00

Canadian Herring, ID 1759

N

ple u

tllrm(�

ha

Hsm

Fa1xta

Year

Fig. 1. Time series plot of Clupea harengus from Iceland and Canada region. The sam

he Allee threshold a is unstable; if the population falls below theevel of a, then the population will be attracted to the stable equi-ibrium point 0, ensuring extinction. Otherwise, the population willeach the carrying capacity K (see Fig. 2). It is to be noted that theodel is similar in form to the model considered by Gruntfest et al.

1997) in a metapopulation context with a strong Allee effect and = 1.

We also consider a limiting form of the model (2) as lim�→0

. Ignoring

igher order terms in � and taking the limit as � → 0 the model (2)pproaches

1x

dx

dt= r� ln(K)(x − a)

[1 − ln(x)

ln(K)

]ence, for a well defined model in the limit, r must tend to infinity

o that r� ln(K) approaches a finite constant, say r0, in the limitingodel (see Appendix A (2)), giving

1x

dx

dt= r0(x − a)

[1 − ln(x)

ln(K)

](3)

0 20 40 60 80 100 1200

10

20

30

40

50

60

70

80

time

popu

latio

n si

ze

K

Extinction

ig. 2. Solid lines: Numerical solutions of the logistic model modified to incorporaten Allee effect (2), for different initial population sizes. Parameters are r = 0.008, a =5, K = 58, � = 0.21. Dashed lines: local unstable and stable equilibria (2) at x = a and

= K. Any initial population size x0 starting above the critical threshold a convergeso carrying capacity, stable equilibria, where as when x0 is below critical threshold, then it converges to zero.

Year

nits are measured with tonnes. In the time series units are shown as tonnes×103.

4. Stochastic model

The variability between individual survival and reproductioncauses demographic stochasticity. Although the population pro-cess may be perturbed by random environmental changes, at lowpopulation density demographic stochasticity is more prevailing(Lande, 1998). Thus the extinction process of a given populationcan not be adequately analyzed without inclusion of demographicdisturbances (Dennis, 2002; Courchamp et al., 1999).

Many stochastic models of population growth can be approxi-mated by a diffusion process (including a discrete time birth–deathprocess). This generally takes the form

dx(t) = � (x(t)) dt +√

v (x(t))dW(t), (4)

where dx(t) is the approximate population size change in timeinterval dt, and dW(t) has a normal distribution with zero mean andvariance dt. The infinitesimal mean � (x(t)) specifies the underlyingdeterministic trend, while the infinitesimal variance v (x(t)) corre-sponds to stochastic fluctuations. Engen et al. (1998) showed thatv(x) = �2

dx corresponds to demographic stochasticity and v(x) =

�2e x2 corresponds to environmental stochasticity. These equations

have become vital tools in modeling populations in stochastic envi-ronments. Incorporating demographic stochasticity in model (2),we have,

dx

dt= rx(x − a)

(1 −

(x

K

)�)

+ �d

√xdWt. (5)

The computation of the stationary distribution of the above equa-tion is provided in the appendix.

4.1. Expected time to extinction

The diffusion approximation is often used to describe popula-tion fluctuations in discrete time models. For the present analysis

the infinitesimal mean is � (x) = rx (x − a)(

1 −(

xK

)�)

, for the

limiting model � (x) = rx (x − a)(

1 − ln(x)ln(K)

)and the infinitesimal

variance is v(x) = �d2x. We define extinction of the population to

occur when x =1 (Sæther et al., 1998). The mean time to extinctioncan be derived from the distribution of the sojourn time – that is,the expected time spent at each population size before extinction,

given by the following Green’s function (Karlin and Taylor, 1981),

G (x; x0) = 2m (x) S (x) x < x0

= 2m (x) S (x0) x ≥ x0,(6)

l Modelling 250 (2013) 72– 80 75

w

I1∫

4

pbc2gab

It�ieftf

5

am2upmh(o

E

Tdtn

0 20 0 40 0 60 0 800

−1.

0−

0.5

0.0

0.5

1.0

Population size

pgr

B. Saha et al. / Ecologica

here x0 is the initial population size and

S (x) =∫ x

1

s (u) du,

s (u) = exp

[−2

∫ u

1

� (z)v (z)

dz

]= exp

[−2

∫ u

1

� (z)�2

dz + �2

e z2dz

],

m(x) = 1v(x)s(x)

.

f we consider the extinction of the population at x = 1 (Engen et al.,998), then the required expected time to extinction is given by,

∞

1

G(x; x0)dx =∫ x0

1

G(x; x0) dx +∫ ∞

x0

G(x; x0) dx

.2. Probability of extinction

When it comes to a stochastic population, the first passagerobability (i.e. the probability of attaining a large population sizeefore attaining a small one); and the mean time to extinction areommonly used to evaluate population viability (Drake and Lodge,006). We are interested in evaluating the probability of extinctioniven different initial population sizes. Let � (n; a, b) be the prob-bility that the population reaches a before reaching an upper size, starting at n, where 0 < a ≤ n ≤ b. A standard formula gives,

� (n; a, b) =

∫ b

n

exp [−ϕ (x)] dx∫ b

a

exp [−ϕ (x)] dx

ϕ (u) = 2

∫m (u)v (u)

du

(7)

n the above expression the (exponentiated) constant of integra-ion cancels in the numerator and denominator. As a function of n,(n; a, b) is equal to 1 when n = a, is strictly monotone decreasing

n the interval (a, b), and is equal to 0 when n = b. The probability ofxtinction starting from population size n under this model is foundrom (7) by letting a → 0 and b→ ∞. An explicit analytical form forhe probability of extinction is not available due to the complicatedorm of the chosen model (2).

. Statistical analysis

Statistical modeling and inference play a central role inddressing how density dependence acts on by testing theoreticalodels against time series data (Polansky et al., 2008; Nedorezov,

011). We follow the usual convention in denoting the vector val-ed parameter =

(r, a, �, K

)in the space � of all admissible

arameter values. Let {xt}nt=1 be the given time series data and

odel the fluctuations in the data on a log transformed scale. Weave estimated per capita growth rate by ln

( xt+1xt

)in the interval

t, t + 1) and consider it as a response variable, say yt. The regressionf y on x is represented by

(Y |x) = �(x, ˇ) = r(x − a)

(1 −

(x

K

)�)

.

he distribution of yt conditional on xt is assumed to be normallyistributed with mean �(xt, ˇ) and variance �2

xt. Together with

he assumption that observations are independent, this defines aonlinear regression model. We use NLS to estimate the fixed but

Fig. 3. Assuming the limiting model, the model predicted values are plotted in thescatter plot of per capita growth rate and population size, GPDD Id 1765. r = 0.001,a = 1022, K = 1610, R2 = 0.7591, Adj . R2 = 0.7109.

unknown parameter ˇ by minimizing the residual sum of squaresfunction

RSS(ˇ) =n∑

t=1

xt

(yt − �

(xt, ˇ

))2,

with weights wt = xt .We use the Gauss-Newton algorithm implemented in the R-

package for statistical computing to minimize the residual sum ofsquares. Convergence of this iterative algorithm depends heavilyon the initial choices of the model parameters. Convergence of theleast-squares procedure is sometimes uncertain while fitting realdata to highly nonlinear models. The Grid Search Method is an alter-native option for parameter estimation, as well as for choosing theinitial guess for NLS.

Given the initial range of parameter values, each range is dividedinto an equal number n of partitions. With (n + 1) points in the parti-tion of the space of each of the four parameters, (n + 1)4 quadruplets(r, a, �, K) of parameter combinations are available. For best fitting,we compute error sum of squares at each of these combinations andchoose the quadruplet having the minimum error sum of squares.The grid points may be selected in such a way that the parameterestimates can be found with desired accuracy level.

We can use the standard asymptotic properties of NLS estimatesfor statistical testing when we have reasonable size of data. Forsmall samples, one can estimate the uncertainty in parameters bysimulating repeated datasets (say, B) from the given data matrixusing bootstrap. For these B bootstrap samples, we have B bootstrapestimates of the model parameter vector using NLS. A nonparamet-ric CI can be constructed accordingly from the percentiles of thebootstrap sampling distribution (see Section 6).

6. Results and discussion

We observed that the plot of realized per capita growth rateagainst population size for both datasets of herring populations

exhibits a strong Allee profile (see Fig. 3). We used NLS to esti-mate the parameters of the Icelandic herring dataset (GPDD Id1765) with reasonable length (24) of time series (using model(2)). Confidence intervals for the parameters are computed using

76 B. Saha et al. / Ecological Mod

Table 1NLS and bootstrap estimates for Atlantic herring, GPDD Id 1765.

Parameters NLS estimate (± SE) Bootstrap estimate Bootstrap CI

r 0.11 (± 0.03) 0.011 (0.005, 0.017)

breuGtsdaHfirGteoedp

a 214.4 (± 0.39) 226.1 (150.25, 414.69)

K 723.3 (± 0.37) 719.9 (617.20, 781.28)

oth NLS and regression bootstrap estimates. For Canadian her-ing dataset (GPDD Id 1759) the length of time series is not longnough to guarantee the convergence of least square procedure. Wesed Grid Search method to compute the parameter estimates. Therid Search procedure lacks the asymptotic properties of parame-

er estimates from inferential view point. In general, when sampleize is very short, noisy, contains missing values, standard proce-ures like NLS or Maximum likelihood estimation may fail to fit

suitable model. In those cases Grid Search may perform well.owever, using this procedure non-parametric type bootstrap con-dence interval can be constructed for the parameter estimates byesampling the data matrix a large number of times, and runningrid Search procedure for each of these bootstrap sample. But it is

o be noted that it requires a significant amount of computationalffort which is beyond the present endeavor. The performance

f the parameter estimates can be cross-checked by plugging thestimates in density function to approximate the quasi-stationaryistribution of population sizes. Estimated parameter values arerovided in Table 1 for Icelandic herring.

r

200 40 0 6

200

400

600

800

a

0.005 0.010 0.015 0.020

Fig. 4. Scatterplot matrix of bootstrap estimates of the

elling 250 (2013) 72– 80

We observed that NLS estimate of � (0.003) for Icelandicherring is not statistically significant at 5% level and obviouslypretty close to zero. So for this specific case the limiting form(3) of this model (2) might be a suitable choice. We fit thedata to model (3) and observed that, all the model parame-ters namely growth rate (r = 0.11 ± 0.03 [SE]), Allee threshold(a = 214.4 ± 0.39), carrying capacity (K = 723.3 ± 0.37) are signifi-cant (t = 3.514, t = 5.414, t = 19.188, df = 17, for r, a, K respectivelyand P < .001). The bootstrap confidence intervals of parameters are(rb = 0.011171, (0.0047, 0.017); ab = 226.1(150.25, 414.69);Kb = 719.9, (617.1963, 781.28)). The estimated demographicvariance is 3.413. The fitted curve is shown in Fig. 3 and cor-responding bootstrap histogram and confidence intervals aredepicted in Fig. 4. For Canadian herring the Grid Search estimatesare obtained as r = 0.52, a = 800, � = 0.0006, K = 1210. The estimateof � is close to zero as in case of Icelandic region. So we consider thelimiting model (3) and obtained the revised parameter estimatesas r1 = 0.001, a1 = 1022, K1 = 1610 (see Fig. 5). The estimate of r andr1 are very different, because in the limiting model the relationr1 ≈ r� ln K holds as � → 0 (see Appendix A(2)). From Fig. 5, wenote that the population size data beyond the Allee threshold arenot available, which leads to a relatively less reliable estimate ofcarrying capacity K.

To compute the stationary density, we used the Runge-Kutta

method of order 4 to solve the Kolmogorov forward differentialequation numerically. We used the ode45 routine implementedin MATLAB for numerical solution of differential equations. ForIcelandic and Canadian herring, plots of the stationary density

00 800

0.00

50.

010

0.01

50.

020

400 80 0 120 0 1600

400

800

1200

1600K

Atlantic herring data in Canada. GPDD ID 1765.

B. Saha et al. / Ecological Mod

0 200 400 600 800 1000 1200 1400 1600 1800−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

population size

PG

R

Fs

Fw

ig. 5. Assuming the limiting model, the model predicted values are plotted in thecatter plot of per capita growth rate and population size, GPDD Id 1759.

0 100 200 300 400 500 600 700 800 9000

0.5

1

1.5

2

2.5x 10−3

Population size

Pro

babi

lity

Fig. 6. Histogram plot for Icelandic herring (Panel (a)) and Canad

0 200 400 600 800 1000 1200 14000

0.2

0.4

0.6

0.8

1

Initial population size

Pro

b of

ext

inct

ion

ig. 7. �(n ; a, b) the probability of reaching population size a before size b, graphed as aith critical depensation with demographic variance �2

d= 5.78 (Panel (a)) and �2

d= 8.30 (

elling 250 (2013) 72– 80 77

and probability of extinction are depicted in Figs. 6 and 7 respec-tively. The expected time to extinction is calculated as 206 yearsfor Icelandic herring and 133 years for Canadian herring. If we haverecent time series data, the estimated expected time to extinctionmay be changed.

It is to be noted that change in extinction probability with thechange in initial population size (from (7)) is significant aroundthe unstable equilibrium point x = a. So the species is at the great-est risk of extinction when the population size lies below thecritical level a. In contrast, at the stable equilibrium x = K, thechange in the extinction probability with the change in populationsize is not so severe indicating that a small decrease in popula-tion size does not lead to extinction and above x = K probabilityof extinction decreases rapidly. The extinction probability for theIcelandic herring with the given initial population size (126 metrictonnes as given in GPDD) was found to be 0.8 and for the Cana-dian herring the extinction probability was found to be 0.93 withthe given initial population size (333.5 metric tonnes as given inGPDD).

Demographic variance together with the Allee effect has abig impact on the extinction vulnerability of species. In thequasi-stationary state the herring population is more likely tostay close to zero due to Allee effect. As demographic variance

35 240 445 650 855 10600

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8x 10−3

Population size

Pro

babi

lity

ian herring (Panel (b)) using estimated parameter values.

0 500 1000 1500 2000 25000

0.2

0.4

0.6

0.8

1

Initial population size

Pro

b of

ext

inct

ion

function of initial population size n, for (lower line) population growth model (2)Panel (b)) respectively.

78 B. Saha et al. / Ecological Mod

0 100 200 300 400 500 600 700 800 9000

0.5

1

1.5

2

2.5

3

3.5x 10−3

population size

prob

abili

ty

σd=4

σd=5

σd=6

σd=8

Histogram ofpopulation size

Fig. 8. Plot of stationary distribution with varying demographic variance �2d

. As �2d

increases the mode at carrying capacity becomes low and the risk of extinctionincreases.

0 100 200 300 400 500 600 700 800 9000

0.5

1

1.5

2

2.5

3x 10

population size

prob

abili

ty

θ = 1.3

θ = 0.9

θ = 0.5

θ = 0.1

Histogram ofpopulation size

Fig. 9. This figure describes the effect of the strength of density regulation at carry-ing capacity. If strength of density regulation is high at carrying capacity, i.e. large�, then the population is more likely to stay at carrying capacity. As � increases, themode increases at carrying capacity.

0 200 400 600 800 1000 1200 1400

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

Allee threshold

mea

n tim

e to

ext

inct

ion

σd=4

σd=6

σd=8

σd=10

Fig. 10. Plot of mean time to extinction against the Allee threshold. We see that asAllee threshold decreases mean time to extinction increases and population staysat carrying capacity. It is important to note that, at low population the mean time toextinction is low with high variance (�d = 10). But as the population size increases thetrend seems to change to opposite direction. The decreasing rate of mean extinctiontime is low for higher variance (�d=10) than the lower one (�d = 4).

elling 250 (2013) 72– 80

decreases, there is an increase in the probability that the herringpopulation will stay around the stable equilibrium (carrying capac-ity K)(see Fig. 8). In Fig. 9, we have found that, when the populationis subject to stochastic fluctuations, the population with higherstrength at carrying capacity may survive for long time. The plot ofquasi-stationary density suggests that, with high �, there is a highprobability that the population will stay at carrying capacity ratherthan below the critical threshold level (see Fig. 9). When the Alleethreshold is small, then high demographic variance substantiallyreduces the mean time to extinction, where as with low variancethe population persists for long time. If the Allee threshold is high,then the mean time to extinction decreases and hence there is anincreased chance of extinction (see Fig. 10).

7. Conclusion

The existence of depensatory processes in demographic his-tory data of populations has implications in many different areas,such as pest control, optimal management of endangered species(Sinclair et al., 1998), optimal harvest policy (Quinn et al., 1993;Peterman, 1977) and population viability analysis (Dennis et al.,1991). Reduced survival at low population size or density increasesthe probability of extinction, which is studied for many speciesand populations including Salmonids (Montgomery et al., 1999;Swart et al., 1993), birds and many other populations (Fowler andBaker, 1991). Liermann and Hilborn (2001) suggested performingthe population viability analysis at a quasi-extinction level as mostof the time a very little information is known about the popula-tion dynamics at low density. In fisheries, uncontrolled harvestinghas the potential to reduce the populations to the point where thedepensatory process dominates.

We have analyzed the population time series data of two herringpopulations from the Icelandic and Canadian regions. The evi-dence of a strong Allee effect is prominent in both populations. Theparameters for the proposed models are estimated using NLS andGrid Search procedures. Many commercially important fish species,including herring, possess the characteristics of group defense andschooling (Radakov,1973; Courchamp et al., 2008). Species hav-ing such characteristics are at an increased risk of extinction dueto the Allee effect when severe exploitation occurs. Harvestedfish populations are highly subjected to the existence of criti-cal threshold. Moreover, many of the large fish populations havemetapopulation structure where subpopulation are connected bylarval transport (Crowder et al., 2000). In such case, there mayexist a critical number of subpopulations below which the wholepopulation becomes vulnerable to extinction. When the popula-tion size is large, schooling fish are more likely to be susceptibleto strong Allee effect because, over harvesting in densely popu-lated areas may lead per capita growth rate to become negative(Courchamp et al., 2008). Thus the schooling, generally consideredas defence against predators (Sæther et al., 1996), have become ahighly plausible explanation for collapse, as an isolated school maybecome extinct with uncontrolled harvesting (Courchamp et al.,2008, 1999; Hilborn and Walters, 1992).

Note that at low population density demographic stochasticityis more prevalent. We have considered the stochastic counterpartof the model by incorporating demographic noise to estimate dif-ferent extinction measures viz. expected time to extinction andprobability of extinction. our analysis can have a huge impacton understanding extinction patterns and enable us to identifydemographic threats and guide decision making in conservation

management. The estimated time to extinction for both of the her-ring populations suggests that these species might be facing somedisturbances in their respective geographic region and requireinvestigation.

l Mod

e1qpimpce

A

fltbR

A

od

wps

f

w

e

0

I

w

s

w

A

f

T∫duc

B. Saha et al. / Ecologica

There have been debates and cross debates regarding the exist-nce of the Allee effect in marine systems (Liermann and Hilborn,997; Shelton and Healey, 1999). Considering its alarming conse-uences and issues related to species extinction, it should be givenriority in conservation, fisheries management and stock rehabil-

tation (Frank and Brickman, 2000). In this paper we developedodel with a strong Allee effect, particularly focusing on herring

opulations. However we argue that the model has general impli-ations and can be applied to model populations subject to the Alleeffect across a wide range of taxa.

cknowledgements

We are grateful to the editor and the two anonymous reviewersor their valuable comments and providing corrections on the ear-ier version of the manuscript. The comments immensely improvedhe standard of the paper. Amiya Ranjan Bhowmick is supportedy Council of Scientific and Industrial Research (CSIR), Humanesource Development Group, New Delhi.

ppendix A.

Let {X(t), t ≥ 0} be a regular time homogeneous diffusion processn [0, ∞) with transition density p(t, x, y) for t > 0. The transitionensity satisfies Kolmogorov’s forward equation,

∂p(t, x, y)∂t

= 12

∂2

∂x2[�2(x)p(t, x, y)] − ∂

∂x[�(x)p(t, x, y)],

here �(x) and �2(x) are the infinitesimal mean and variance of therocess. If there exists a stationary density f(.), then it necessarilyatisfies

(x) =∫

f (y)p(t, x, y)dy forall t > 0,

here limt→∞

p(t, x, y) = f (x) and f satisfies the Kolmogorov’s forward

quation. Putting ∂f∂t

= 0, we have the following equation:

= 12

∂2

∂x2[�2(x)f (x)] − ∂

∂x[�(x)f (x)].

ntegrating, we have,

d

dx

[�2(x)

2f (x)

]− �(x)f (x) = 1

2C1 (8)

here C1 is a constant. Multiplying with the integrating factor

(x) = exp

{−∫ x [

2�(�)�2(�)

]d�

},

e can write (8) in the compact form

d

dxs(x)�2(x)f (x) = C1x(x).

nother integration with S(x) =∫

xs(y) dy gives

(x) = C1S(x)

s(x)�2(x)+ C2

1s(x)�2(x)

= m(x)[C1S(x) + C2].

he constants C1 and C2 are determined such that f(x) ≥ 0 and∞

0f (x) dx = 1. In our case, �(x) is a non-linear function and it is

ifficult to get a compact expression for extinction time. So wese a numerical method to solve the differential equation and toompute different extinction times.

elling 250 (2013) 72– 80 79

Appendix B.

Derivation of the form of the model (2) when � −→ 0Expected change in the per capita growth rate of a population

can be written as,

E

(X/X

)= r(X − a)(1 − (X/K)�)

We define

r0 = E(X/X|X = 1) = r(1 − a)(1 − K−�)

Therefore

r = r0

(1 − a)(1 − K−�)

and hence

E

(X/X

)=

(r0

(1 − a)(1 − K−�)

)(x − a)(1 − (X/K)�)

=(

r0

1 − a

)(X − a)

(1 − (X/K)�

1 − K−�

)=

(r0

1 − a

)(X − a)

(K� − X�

K� − 1

) (9)

First, we determine the value of r0 as � −→ 0. We have

r0 = r(1 − a)(1 − K−�)

Now

1 − K−� = 1 −(

1 − K − 1K

)�

= 1 −

[1 − �

(K − 1

K

)+

�(

� − 1)

2!

(K − 1

K

)2

− �(� − 1)(� − 2)3!

(K − 1

K

)3

+ . . .

]= �

(K − 1

K

)− �(� − 1)

2!

(K − 1

K

)2

+ �(� − 1)(� − 2)3!

(K − 1

K

)3

− . . .

= �

[(K − 1

K

)− (� − 1)

2!

(K − 1

K

)2

+ (� − 1)(� − 2)3!

(K − 1

K

)3

− . . .

]Now taking � −→ 0 in the above relation we get

= �

⎡⎢⎣(K − 1

K

)+

(K − 1

K

)2

2+

(K − 1

K

)3

3+ . . .

⎤⎥⎦= −�

⎡⎢⎣−(

K − 1K

)−

(K − 1

K

)2

2−

(K − 1

K

)3

3− . . .

⎤⎥⎦= −� ln

(1 − K − 1

K

)= −� ln

1K

= � ln K as � −→ 0.

(10)

Therefore,

r0 = r(1 − a)� ln K as � −→ 0.

Now taking the limit as � −→ 0 on both sides of Eq. (9), we have

lim�−→0

E

(X/X

)= lim

�−→0

(r0

1 − a

)(X − a)

(K� − X�

K� − 1

)

8 l Mod

To

R

A

A

B

B

B

B

B

C

C

C

C

C

DD

D

D

E

F

F

G

G

G

H

H

J

J

K

0 B. Saha et al. / Ecologica

his is an indeterminate form. By applying the L’Hospital’s rule, webtain,

=(

X − a

1 − a

)lim

�−→0r0

(K� ln K − X� ln X

K� ln K

)=

(X − a

1 − a

)r(1 − a)� ln K

(1 − ln X

ln K

)= r� ln K(X − a)

(1 − ln X

ln K

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