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Ecological Modelling 220 (2009) 1681–1689

Contents lists available at ScienceDirect

Ecological Modelling

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

ierarchical demographic approaches for assessing invasion dynamics ofon-indigenous species: An example using northern snakehead (Channa argus)

an Jiaoa,∗, Nicolas W.R. Lapointea, Paul L. Angermeiera,b, Brian R. Murphya

Department of Fisheries and Wildlife Sciences, Virginia Polytechnic Institute & State University, Blacksburg, VA, 24061-0321, USAUnited States Geological Survey, Virginia Cooperative Fish and Wildlife Research Unit1, Virginia Polytechnic Institute & State University, Blacksburg, VA, 24061-0321, USA

r t i c l e i n f o

rticle history:eceived 27 July 2008eceived in revised form 31 March 2009ccepted 4 April 2009vailable online 9 May 2009

eywords:ierarchical demographic approachon-indigenous speciesisk assessmentorthern snakehead

a b s t r a c t

Models of species’ demographic features are commonly used to understand population dynamics andinform management tactics. Hierarchical demographic models are ideal for the assessment of non-indigenous species because our knowledge of non-indigenous populations is usually limited, data ondemographic traits often come from a species’ native range, these traits vary among populations, and traitsare likely to vary considerably over time as species adapt to new environments. Hierarchical models read-ily incorporate this spatiotemporal variation in species’ demographic traits by representing demographicparameters as multi-level hierarchies. As is done for traditional non-hierarchical matrix models, sensi-tivity and elasticity analyses are used to evaluate the contributions of different life stages and parametersto estimates of population growth rate. We applied a hierarchical model to northern snakehead (Channaargus), a fish currently invading the eastern United States. We used a Monte Carlo approach to simu-late uncertainties in the sensitivity and elasticity analyses and to project future population persistenceunder selected management tactics. We gathered key biological information on northern snakeheadnatural mortality, maturity and recruitment in its native Asian environment. We compared the modelperformance with and without hierarchy of parameters. Our results suggest that ignoring the hierar-chy of parameters in demographic models may result in poor estimates of population size and growthand may lead to erroneous management advice. In our case, the hierarchy used multi-level distributionsto simulate the heterogeneity of demographic parameters across different locations or situations. The

probability that the northern snakehead population will increase and harm the native fauna is consid-erable. Our elasticity and prognostic analyses showed that intensive control efforts immediately prior tospawning and/or juvenile-dispersal periods would be more effective (and probably require less effort)than year-round control efforts. Our study demonstrates the importance of considering the hierarchyof parameters in estimating population growth rate and evaluating different management strategies for

speci

non-indigenous invasive

. Introduction

Intensive field or laboratory research does not guarantee solu-ions to an introduced species problem, and waiting for enough

mpirical data on the effects of a new invader may delay manage-ent or allow irreversible ecological damage. Controlling invasions

s typically more difficult after invaders have spread (Simberloff,003), and early control is often the best solution. Initial con-

∗ Corresponding author at: Department of Fisheries and Wildlife Sciences, 100heatham Hall, Virginia Polytechnic Institute and State University, Blacksburg, VA,4061-0321. USA. Tel.: +1 540 231 5749; fax: +1 540 231 7580.

E-mail address: yjiao@vt.edu (Y. Jiao).1 The unit is jointly supported by U.S. Geological Survey, Virginia Polytechnic Insti-

ute and State University, Virginia Department of Game and Inland Fisheries, andildlife Management Institute.

304-3800/$ – see front matter © 2009 Elsevier B.V. All rights reserved.oi:10.1016/j.ecolmodel.2009.04.008

es.© 2009 Elsevier B.V. All rights reserved.

trol efforts can be guided by preliminary modelling analysesbased on a review of the biological and ecological character-istics of a non-indigenous species in its native environment,though biological characteristics may change as the species adaptsto new environments. Such analyses can be updated as moreempirical data become available or as management objectiveschange.

Matrix models have been used to study populations of manyaquatic species and stocks, including turtles, sharks, and bonyfishes, and have been used to evaluate management options forinvasive species (Crouse et al., 1987; Caswell, 2001; Frisk et al.,2002; Govindarajulu et al., 2005). Demographic data can be used

to predict population dynamics, especially when historical recordsof absolute abundance or relative abundance do not exist (Kieth andWindberg, 1978; Krebs et al., 2001). This is almost always the casefor recently introduced species. Examples of using matrix models topredict population trends of non-indigenous and invasive species

1 odellin

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682 Y. Jiao et al. / Ecological M

nclude Lampo and De Leo (1998), Parker (2000), and Govindarajulut al. (2005).

The matrix model is based on a matrix of life-history parame-er values (projection matrix) multiplied by a vector of populationize (age or size or stage structured). A projection matrix sum-arizes the life history of a population via estimates of fertility

or recruitment), survival, and probabilities of transition betweentages in a n × n matrix, and therefore determines the dynamicsf the population. Matrix models can help to analyze potentialhanges in a population due to exploitation or other factors explic-tly represented in the projection matrix; however, their utility as

management tool depends strongly on the reliability of modelnputs (Frisk et al., 2002). In many cases, the biological parameterssed in the projection matrix are estimated from a limited numberf studies and thus have high uncertainty. The resulting model out-ut may lead to faulty management advice because of inaccuracies

n model inputs. Thus, it is important to incorporate plausible vari-bility in parameter values of models, and to present model outputss probability distributions (Clark, 2003).

Intraspecific demographic variability is common in aquaticpecies, particularly among populations and subpopulations inifferent geographic locations (Tyler and Rose, 1994). This vari-bility has been ignored in many studies where demographic datarom the native range of a non-indigenous species were used. Theesulting outputs can lead to an incomplete understanding of pop-lation demography and faulty management advice. Hierarchicalemographic models can be used to simulate intraspecific vari-bility (Clark, 2003) and are a type of matrix model in which theriors of the parameters in the matrix are given a probabilisticpecification in terms of further parameters, known as hyperpa-ameters (Gelman et al., 1995; Clark, 2003). This type of model,hich includes consideration of variability among individuals and

mong temporal and spatial scales, is needed to predict popula-ion trends of non-indigenous species because our knowledge ofuch species is usually limited and their demography commonlyaries over time as they integrate into new ecosystems. Hierar-hical demographic models represent a middle ground betweenraditional models that ignore intraspecific variability and over-tted models that assign parameter estimates to each individual

n a population (Clark, 2003).Sensitivity and elasticity analyses are useful in evaluating

elative contributions of population parameters to estimates ofopulation growth rates, and these analyses can be conducted

n both non-hierarchical and hierarchical demographic modelsCaswell, 2001). The projection matrix and its variants are partic-larly well-suited to predicting population responses to plausiblecenarios. The effects of various control and management actions onpopulation can be evaluated through prognostic analyses by pro-

ecting population persistence subsequent to each action (Crouset al., 1987; Frisk et al., 2002).

Modelling analyses are useful in understanding different stagesf the invasion process, including introduction, establishment,pread, and impact (Lampo and De Leo, 1998; Parker, 2000;ovindarajulu et al., 2005). Modelling analysis in the early stages

ypically relies on information from an invading species’ nativeange because knowledge of its biology immediately after invasions extremely limited (Simberloff, 2003). However, information usedn later invasion stages, such as the spread stage, may come fromirect study of a newly invading population (Govindarajulu et al.,005).

In this study, we used northern snakehead (C. argus) to demon-

trate the use of hierarchical demographic models and their valuen characterizing population trends for non-indigenous species (Mat al., 1999; Wu et al., 2000b; Liu et al., 2002). Northern snake-ead, native to Asia, has established a reproducing population in theotomac River (Odenkirk and Owens, 2005); however, our knowl-

g 220 (2009) 1681–1689

edge of northern snakehead in North America is very limited. Thebroad environmental tolerances of northern snakehead in its nativeenvironment indicate a strong potential to establish across much ofNorth America (Herborg et al., 2007), and suggest that geographicvariability in its demographic traits is likely (Ma et al., 1999; Wu etal., 2000b; Liu et al., 2002).

Evaluating the potential for the northern snakehead invasion toproceed to the impact stage is a high priority for water-resourcemanagers. Federal and state agencies, in partnership with researchinstitutions, are studying the biology of this species in North Amer-ica, including its habitat preferences, spawning, feeding, homerange, movement, risk of population increase in the PotomacRiver, and the effectiveness of potential management tactics (U.S.Geological Survey, 2006). Northern snakehead appears to havespread to all inhabitable areas of the Potomac River that are acces-sible without human interference (i.e., intentional transplants)(Lapointe and Angermeier, 2009).

In this study, we demonstrated how hierarchical demographicmodels can be used to predict population trends of non-indigenousspecies and inform managers regarding the efficacy of plausiblecontrol tactics. Specifically, we: (1) built an age-structured, hierar-chical demographic model for northern snakehead; (2) comparedoutputs from hierarchical and non-hierarchical demographic mod-els based on this example; (3) conducted sensitivity and elasticityanalyses to show how measures of age-specific fertility and sur-vival, as well as the parameters used in modelling fertility andsurvival, influence population growth rates; and (4) evaluated theeffectiveness of potential management tactics with respect to theirinfluence on population persistence and their ability to controlnorthern snakehead.

2. Models and methods

2.1. A hierarchical age-structured demographic model

The Leslie matrix population projection technique (Leslie, 1945)is commonly used for demographic analysis:

ANt = Nt+1. (1)

An age-structured population projection matrix A then takes theform below:

A =

⎡⎢⎢⎢⎢⎢⎢⎢⎣

f1 f2 · · · fn fn+1

s1 0 · · · 0 0

0 s2 · · · 0 0

......

. . .. . .

...

0 0 0 sn 0

⎤⎥⎥⎥⎥⎥⎥⎥⎦

, (2)

where n is the observed maximum age of individuals or longevityof the species, and fi in the first row of matrix A is the age-specificfertility. The variable si is the probability of surviving from age i tothe next age (i + 1) in year t.

si = 1 − exp(−Mi) if mortality is caused by natural reasons only;

si = 1 − exp(−Mi − Fi) if both natural mortality and harvest

mortality exist,

(3)

where Mi is the natural mortality of age i individuals, and Fi is theharvest mortality of age i individuals. Values for si all lie between 0

and 1, while the values of fi are by definition positive.

Hierarchical and non-hierarchical models differ substantially inhow parameters are represented. In a non-hierarchical model, theparameters of fi and si are described as a distribution. For exam-ple, for si∼N(S, �2

s ) (follows a normal distribution with mean S

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Y. Jiao et al. / Ecological M

nd variance �2s ); both S and �2

s are deterministic values. How-ver, in a hierarchical model, both S and �2

s can be explainedr modeled by further parameters, known as hyperparametersGelman et al., 1995; Clark, 2003). That is, S may follow anotherrobability distribution (e.g., uniform distribution U(a, b)) and �2

say follow another probability distribution (e.g., normal distri-

ution N(c, d)). The parameters a, b, c, d are hyperparameters inur hierarchical demographic model; they represent the hierar-hy of variability across individuals, subpopulations, or temporalemographics. We used a hierarchical demographic model because

t seemed more reasonable for our case than a non-hierarchicalodel.

.2. Mathematical theory of population projection matrix

As in the non-hierarchical matrix models, each population pro-ection matrix A for a hierarchical matrix model has a dominantigenvalue � and a corresponding right eigenvector wi that repre-ents the stable age distribution of the population (Caswell, 2001).he dominant eigenvalue � of the matrix A is equal to er, where rs the intrinsic rate of increase of the population in the exponentialrowth population equation

t = N0ert . (4)

Both population growth rate and stable age-distribution areetermined by the population projection matrix A (Caswell, 2001).

n a non-hierarchical matrix model, A can be represented by param-ters in the matrix A itself. However, in a hierarchical matrix model,is determined by both parameters and hyperparameters, if any

xist. A sampling distribution for population growth rate �, decidedy the population projection matrix A.

.3. Sensitivity analysis

One benefit of constructing a population matrix is that we mayest how sensitive the population is to variations in fertility, matu-ity, natural mortality, and fishing mortality (if applicable). This isone by simulating changes in these parameters and then calculat-

ng � of the new matrix. Elasticity is the proportional sensitivityf model results to proportional changes of model parametersCaswell, 2001). By simulating the same proportional change forach age class successively, one can compare the relative effect ofhe different age classes on the population growth rate (Caswell,001).

The relative contributions of the matrix elements (survival andertility) to � can be compared. Survival and fertility are derivedn this model; they depend on other parameters: fishing mortalityF if it exists), natural mortality (M), recruitment per spawner, and

aturity at age. We also calculated the elasticity of � with respecto these parameters (de Kroon et al., 1986; Caswell, 2001). Becausef the difficulty in deducting all of the elasticity of � to x, we usedsimulation–approximation approach:

x

�

∂�

∂x= x

�

��

�x. (5)

.4. Prognostic analysis through population projection

An elasticity analysis of demographic models indicates whicharameters and age classes contribute more to population growthate, but it alone does not allow assessment of the best manage-

ent tactics (de Kroon et al., 2000; Caswell, 2001). The ability of

lternative management tactics to alter the values of parameterslso needs to be considered and may be of great value in settingolicies (de Kroon et al., 2000; Caswell, 2001). Thus, in addition tohe elasticity analysis, we conducted a prognostic analysis to evalu-

g 220 (2009) 1681–1689 1683

ate different population control tactics by projecting the populationpersistence expected under each tactic.

2.5. Environmental uncertainty and risk of population increase

A Monte Carlo simulation study was used to investigate theuncertainty in estimates of northern snakehead population growthrate, sensitivity analyses, and population projections. Risk assess-ments for invasive species have been widely promoted to informcontrol of invasive species (Bax et al., 2001; Andersen et al., 2004a,b;Staples et al., 2005). Risk of population increase is assessed as theprobability that the population will increase over time (Bartell andNair, 2003). Specifically, such assessments determine the likelihoodthat population growth rate � is larger than one (P1(� > 1)), or thelikelihood that population size after x years is larger than the cur-rent population size (P2(Nt > N0), where N0 represents the currentpopulation size).

When we used Monte Carlo simulation, we used Latin hypercubesampling to avoid extreme values and guarantee that the sample israndom but relatively uniformly distributed over each dimension.We also used the bounds of 0.2 and 1 as the minimum/maximumvalues to make the sample biologically reasonable.

2.6. An application

2.6.1. Northern snakehead dataWe searched ASFA (Aquatic Science and Fisheries Abstracts), VIP

Information (a Chinese database) and WANFANG DATA (a Chinesedatabase) to collect information on northern snakehead biologyand ecology. Data on maturity at age, recruitment, and survival ofnorthern snakehead were collected from a latitudinal range com-parable to the Potomac River in both the native and introduced(Eurasian) ranges of northern snakehead (Table 1). Because of thelimited number of publications on its population dynamics in itsnative environment, we assigned uncertainty (variability) levelsarbitrarily as CV = 20–40%, which have been used as a reasonableuncertainty level for fisheries data (Walters, 1998). Northern snake-head demographic characteristics vary among locations in its nativerange, as shown by differences in mean mortality rates and fecun-dity, but we found no quantitative measures of this variability (Maet al., 1999; Ma and Xie, 2000; Wu et al., 2000). The hyperparam-eters, parameters in multi-level distributions (Gelman et al., 1995)that we used in the hierarchical demographic model are listed inTable 1. Details of these parameters are described in the followingsection.

2.6.2. Life-history parametersKey life-history parameters include longevity, proportion of

maturity at age (Maturityi), age-specific survival, and recruitment.The observed maximum age of northern snakehead individuals inits native environment is 15 years, which was used as its potentiallongevity in this study. The product of age-specific recruitment-per-spawner and the % of matured fishes is fi. Recruitment-per-spawner(recruitment of age-1 fish per spawner), is based on the survival rateof fish of age-2 and older.

R1/spawner = N1/15∑i=2

Ni Maturityi

Ni+1 = Nisi

(6)

This is not a perfect way to estimate recruitment-per-spawner,

and it assumes a population with stable age distribution. However,because no records/studies are available on survival from egg toage-1 fish, this becomes a reasonable choice to estimate northernsnakehead recruitment of age-1 fish per spawner. Parameters a, b,c, d are hyperparameters in the hierarchical demographic model

1684 Y. Jiao et al. / Ecological Modelling 220 (2009) 1681–1689

Table 1Parameter values and their uncertainties in the snakehead hierarchical demographic model and scenarios of partially or non-hierarchical demographic model parametersdesigned for comparison with the hierarchical model. CV is coefficient of variation. See the text for explanations of other parameters and distributions. Sources of parametersare Ma et al. (1999), Ma and Xie (2000), and Wu et al. (2000).

Scenarios Parameters Mean values and their distributions Elasticity

S1: Base model: hierarchical demographic model Maturity at age Age 1: 0; age 2: 0.84; ages 3–15: 1.CV = 20–40% for age 2 fish; 0 for theother age groups

0.18 (for maturity of age-2fish); 0.16 (for a); 0.17 (for b);−0.32 (for c); −0.31 (for d);−0.09 (for CVs in both RPS andM)

Recruitment (fish of age 1)per spawner (fish of age2–15) (RPS)

RPS∼N(u, �2u ); u∼U(a, b);

CV = 20–40%; �u = (u × CV)

Natural mortality for fishes(age 1 to 14)

M∼N(M, �2M

); M∼U(c, d);CV = 20–40%; �M = (M × CV)

S2: Partial-hierarchical demographic model 1 Maturity at age Age 2: 0.84; Ages 3–15: 1.CV = 20–40% for age 2 fish; 0 for theother age groups

0.17 (for maturity of age-2 fish);0.16 (for a); 0.17 (for b); −0.89(for M); −0.007 (for CV in RPS)

RPS RPS∼N(u, �2u ); u∼U(a, b);

CV = 20–40%; �u = (u × CV)Natural mortality M ∼ U(c, d)

S3: Partial-hierarchical demographic model 2 Maturity at age Age 2: 0.84; ages 3–15: 1. CV = 20–40%for age 2 fish; 0 for the other agegroups

0.17 (for maturity of age-2 fish);0.46 (for RPS); −0.30 (for c);−0.31 (for d); −0.10 (for CVs in M)

RPS RPS ∼ U(a, b);Natural mortality M∼N(M, �2

M); M∼U(c, d);

CV = 20–40%; �M = (M × CV)S4: Non-hierarchical demographic model 1 Maturity at age Age 2: 0.84; ages 3–15: 1. CV = 20–40%

for age 2 fish; 0 for the other agegroups

0.17 (for maturity of age-2 fish);0.46 (for RPS); −0.88 (M)

RPS RPS ∼ U(a, b);Natural mortality M ∼ U(c, d)

S5: Non-hierarchical demographic model 2 Maturity at age Age 2: 0.84; ages 3–15: 1. CV = 20–40%for age 2 fish; 0 for the other age

0.09 (for maturity of age-2 fish);0.45 (for RPS); −0.89 (M)

(v

iodppdriatfip

hemmCcChR�tM

�id(s

RPSNatural mortality

Table 1), and represent the hierarchy of individual demographicariability.

The accuracy of the distributions of parameters is very importantn yielding reliable results (Frisk et al., 2002). In our case, becausef limited references, it is impossible to develop a well-informedistribution for parameters of natural mortality and recruitment-er-spawner. However, we did find a range of the mean of thearameter values in the literature, which enabled us to use uniformistributions for the means of parameters. Our literature reviewevealed that mean values of natural mortality varied among stud-es (Ma et al., 1999; Wu et al., 2000; Liu et al., 2002). Fecundity variedmong studies as well (Ma and Xie, 2000; Wu et al., 2000). Uncer-ainty and probability density functions (pdf) for natural mortality,shing mortality, maturity at age, and recruitment-per-spawner areresented in Table 1.

To examine the differences between hierarchical and non-ierarchical models, we compared population growth ratestimates under scenarios where part or all of the hierarchical infor-ation was ignored. Scenario 1 is the hierarchical demographicodel, in which RPS∼N(u, �2

u ); u∼U(a, b); a = 0.4929; b = 1.5537;V1 ∼ U(20%, 40%), �u = (u × CV1); M∼N(M, �2

M); M∼U(c, d),= 0.3049; d = 0.6113. CV2 ∼ U(20%, 40%), �M = (M × CV2), i.e.,V = 20–40%, which corresponds to each mean sample of theyperparameters. Scenario 2 ignores the hierarchy of M, in which,PS∼N(u, �2

u ); u∼U(a, b); a = 0.4929; b = 1.5537; CV1 ∼ U(20%, 40%),u = (u × CV); M ∼ U(c, d); c = 0.3049; d = 0.6113. Scenario 3 ignoreshe hierarchy of RPS, in which, RPS ∼ U(a, b); a = 0.4929; b = 1.5537;∼N(M, �2

M); M∼U(c, d), c = 0.3049; d = 0.6113; CV2 ∼ U(20%, 40%),

M = (M × CV). Scenario 4 ignores the hierarchy of both RPS and M,n which RPS ∼ U(a, b); a = 0.4929; b = 1.5537; M ∼ U(c, d); c = 0.3049;= 0.6113. These are the ranges of RPS and M found in the literature

Table 1). Some may suggest that a wider range of mean valueshould be used for non-hierarchical models to incorporate uncer-

groupsRPS ∼ U(a, b); a = 0.1; b = 2M ∼ U(c, d); c = 0.1; d = 1.0.

tainty due to individual variability. Thus, in a final scenario thatignores hierarchy of RPS and M, we extended the range of meanvalues for both RPS and M (Scenario 5, Table 1). The distributionsof RPS and M are RPS ∼ U(a, b); a = 0.1; b = 2 and M ∼ U(c, d); c = 0.1;d = 1.0, which are wider uniform distributions than those used inScenario 4. The values of a, b, c, and d used in this scenario arearbitrary to accommodate the idea that wider distributions maycapture uncertainties reflected in the hierarchical models throughmulti-level distributions.

2.6.3. Evaluation of possible population control tacticsWe evaluated selected population-control tactics (Table 2) as

a prognostic analysis by projecting population persistence undereach tactic. Northern snakeheads nest during summer and larvalfish do not disperse for weeks (Lapointe and Angermeier, 2009).Based on this phenomenon, three management tactics were evalu-ated: (1) to harvest all age classes year-round; i.e., survival for all ageclasses is si = exp(−Mi − Fi); (2) to harvest adult just before spawn-ing occurs, i.e., RPS and spawner size are the values in the basemodel multiplied by exp(−F); and, (3) to harvest larval fish beforethey disperse; i.e., fi is exp(−Fi − Mi). For each tactic, fishing mortal-ity was assumed to be 0.2 year−1 for the harvested age classes. Wecompared the resulting population growth rate, ratio of populationsize after harvest to population size with no control, and persis-tence risk (the probability that population size will remain stableor increase). For each management tactic, we further evaluated theeffect of increasing fishing mortality from 0.2 year−1 to 0.4 year−1,and estimated fishing mortality needed to keep the population sta-

ble corresponding to each management tactics. A five-year timeperiod was used, which is moderate relative to the longevity ofnorthern snakehead. Since our interest here is not the absolutechange in population size, but the relative change in populationsize, 5 years is long enough to evaluate population changes.

Y. Jiao et al. / Ecological Modelling 220 (2009) 1681–1689 1685

Table 2Alternative management tactics and their corresponding influences as assessed from projection and sensitivity analyses. Initial population age structure is assumed to bethe stable age structure. F is fishing mortality, and NSi

is the population size in year 5 when management scenario i (S1:S3) is used; NS0 is the population size in year 5 whenmanagement Scenario 0 (S0) is used. See the text for explanations of other parameters and distributions.

Management alternatives Corresponding parameter changes � (mean) (NSi/NS0 ) *1 *2 *3

S0: base mode: no changes 1.13 1 0.78S1: F = 0.2 and 0.4 for all age groups si = exp(−Mi − Fi) 0.99, 0.86 0.58, 0.35 0.17 0.45, 0.16

0.47, 0.18S2: F = 0.2 and 0.4 for spawners just before spawning RPS and spawner size are the values in the base

model ×exp(−F)1.07, 1.01 0.29, 0.08 0.45 0.65, 0.50

0.16, 0.01S3: F = 0.2 and 0.4 for larvae before they disperse fi is exp(−Fi − Mi) 1.01, 0.91 0.60, 0.39 0.23 0.51, 0.21

* rom s(

3

3m

ceea2aem1madapMew

Fginetdi

the mean population growth rates (Fig. 1), confidence intervals ofthe elasticity and reproductive age contributions were wider forthe hierarchical demographic model than for the non-hierarchicalmodel (figures not shown).

1: population size changes in 5 years (mean). *2: F that can keep the population fsecond row).

. Results

.1. Comparing hierarchical and non-hierarchical demographicodels

Both the mean and variance of population growth rate estimatesan differ with and without considering the hierarchy of the param-ters in the projection matrix. Mean estimates of natural mortalitystimates based on life history or meta-analysis (Pauly, 1980; Quinnnd Deriso, 1999) varied widely (Table 1, Ma et al., 1999; Wu et al.,000). In many cases, the range of these estimates has been useds the distribution of the natural mortality, and variability of eachstimate was ignored. In this study, the hierarchical demographicodel resulted in a mean population growth rate � = 1.13, but � was

.16 without considering the hierarchy of natural mortality esti-ates. The corresponding 95% confidence intervals were 0.83–1.48

nd 0.96–1.40, respectively, with the interval from the hierarchicalemographic model being much wider (Scenarios 1 and 3, Fig. 1and c). Similar patterns were observed when the hierarchy of the

arameters RPS and when the hierarchy of the parameters RPS and

were not considered (Scenarios 2 and 4, Fig. 1b and d). In thisxample, extending the mean values of RPS and M resulted in muchider confidence intervals: 0.61–1.61 (Scenario 5, Fig. 1e).

ig. 1. Probability distribution of growth rate �. (a) From the hierarchical demo-raphic model; (b) hierarchy of natural mortality is ignored; (c) hierarchy of RPSs ignored; (d) hierarchies of both natural mortality and RPS are ignored; and (e)on-hierarchical demographic model but with wider parameter ranges. Histograms:mpirical distribution of the population growth rates; red curves: fit of the his-ograms under the assumption of normal distributions. y axis is the probabilityensity function of growth rate �. (For interpretation of the references to colour

n this figure legend, the reader is referred to the web version of the article.)

0.49, 0.20

preading. *3: Risk of population increase P1 = P(� > 1) (first row) and P2 = P(N5 > N0)

3.2. Assessment of northern snakehead

The predicted population growth rate of northern snakehead islarger than 1.0 and the variance of this estimate is very high (Fig. 1a).Population growth rate is most sensitive to the fertility of age-2fish and survival of age-1 fish (Fig. 2a and b). For all age groups,survival affects growth rate estimates more than fertility. The meanreproductive contribution of age-1 fish is around 0.04, but increasesto 0.075 for age-2 to age-10 fish and then gradually decreases to0.04. Variances of these estimates are also very large (Fig. 2c). For

Fig. 2. Results from northern snakehead projection matrix. (a) Elasticity of popula-tion growth rate to age-specific fertility; (b) elasticity of population growth rate toage-specific survival; and (c) reproductive contribution. Continuous line representsmean estimates, and dotted lines represent 95% probability intervals.

1686 Y. Jiao et al. / Ecological Modelling 220 (2009) 1681–1689

F rojectC nd do

dpaioHtreegotH

cmttruparrtat

ig. 3. Sensitivity analysis of population growth rate � to model parameters in pontinuous lines represent mean estimates of � under different parameter values, a

Population growth rate was sensitive to all parameters but toifferent degrees. The analysis of � sensitivity to different modelarameters showed that increasing the proportion of age-2 fish thatre mature increased the mean and variance of � (Fig. 3). Increas-ng the hyperparameters a and b when modelling the distributionf recruitment-per-spawner also increased the mean values of �.owever, increasing the hyperparameters c and d when modelling

he distribution of natural mortality decreased population growthates (Fig. 3). These influences were clearer in an analysis of thelasticity of population growth rate relative to different param-ters (Table 1). The influences of hyperparameters c and d werereater than others. Population growth rate decreased when CVsf both natural mortality and recruitment-per-spawner increased,hough influence of these CVs was much smaller (Table 1 and Fig. 3).owever, the uncertainty of � increased as CVs increased (Fig. 3).

The harvest tactics examined varied in their effectiveness forontrolling northern snakehead. Without instituting any controleasures, the risk of population increase is 78% (Table 2). Among

he management tactics, S1a and S1b (F = 0.2 and 0.4 year−1, respec-ively, for all age classes) resulted in 17% and 24% decreases,espectively, in population growth rate and, after 5 years, a pop-lation that was 59% and 35%, respectively, of the population sizeredicted without harvest (Table 2 and Fig. 4). The tactics S2and S2b (F = 0.2 and 0.4 year−1, respectively, just before spawning)

esulted in 5% and 11% decreases, respectively, in population growthate, and after 5 years, a population that was 29% and 8%, respec-ively, of the population size predicted without harvest (Table 2nd Fig. 4). The tactics S3a and S3b (F = 0.2 and 0.4 year−1, respec-ively, for larval fishes before they disperse) resulted in 11% and 19%

ion matrix A shown as � distribution changes under different parameters values.tted lines represent 95% probability intervals of � under different parameter values.

decreases, respectively, in population growth rate and, after 5 years,a population that was 60% and 39%, respectively, of the populationsize predicted without harvest (Table 2 and Fig. 4).

The risks of population increase when using P1 = P(� > 1) werevery similar to those obtained when using P2 = P(N5 > N0) in S1 andS3. The small differences reflected the limited number of simulationruns and differences in the initial population structure. However,in S2 the difference was larger; P1 = P(� > 1) is much larger thanP2 = P(N5 > N0) because of the changes in spawner size when thistactic is employed. Spawner size is not a parameter in our esti-mates of population growth rate, which is determined only by theprojection matrix, but is important in estimating population sizebecause it is a product of the projection matrix and populationsize of the previous year. After comparing all changes in populationgrowth rate, population size, and risk of population increase, weconclude that management Scenario 2 is most effective at reducingthe probability of population increase (Table 2).

4. Discussion

4.1. Need for hierarchical approaches and costs of ignoringintraspecific variability

Hierarchical approaches in population modelling are especially

important for non-indigenous species impact assessments, whichusually borrow information from species’ native ranges. Whenusing such information, populations from different geographiclocations may exhibit different life-history traits. Additionally,non-indigenous species may express different life-history traits as

Y. Jiao et al. / Ecological Modelling 220 (2009) 1681–1689 1687

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ig. 4. Changes in the population growth rates � and relative population size in yeaiddle: F is for spawners just before spawning. Bottom: F is for larvae before they di

nd dotted lines represent 95% probability intervals of � under different parameter

hey adapt to new environments. Even within native populations,any species show individual variability related to environmen-

al change, harvest, and changes in population composition (Clark,003; Jiao et al., 2009). This suggests a need for using hierarchi-al demographic models instead of non-hierarchical demographicodels, which ignore such information, when conducting risk

ssessments for invasive species.Hierarchical demographic models enable scientist and man-

gers to consider the variability among individuals in responseo spatial heterogeneity and other factors, such as different fish-ng pressures or large environmental changes. Northern snakeheadhowed variation in mean estimates of its survival rate, fecundity,nd recruitment pattern in different populations across its nativeange. Our hierarchical models produced wider confidence inter-als than non-hierarchical models, which incorporated only theange of mean parameters values, as shown in Figs. 1b, c, and 2dnd Table 1. When variability of mean values was incorporated intour non-hierarchical models by using a wide range of mean param-ter values to incorporate the variability of each observed meanalues, much wider confidence intervals resulted. This pattern mayot be realistic given that useful information was likely missing, ashown in Fig. 1e and Table 1. In this application, the heterogeneity ofurvival rates (recruitment-per-spawner) was efficiently modeledhrough hyperparameters, which were also used in the hierarchicalemographic model. The inference of population growth rate and

he evaluation of selected management tactics were more accu-ate than those based on non-hierarchical models, which ignoredome useful information. Because hierarchical demographic mod-ls incorporate uncertainty explicitly, management decisions basedn inferences from them are more reliable (Clark, 2003).

response to different fishing mortality. Top: fishing mortality F is for all age classes.. Continuous lines represent mean estimates of � under different parameter values,

s.

4.2. Utility of demographic models for predicting trends ofnon-indigenous populations

The design of effective tactics for controlling invasive species isoften delayed by a lack of basic demographic information in thenew environment (Simberloff, 2003), as is the case for northernsnakehead in the Potomac River. Continued research on its biologyis warranted but waiting for data on the potential effects of a newinvader may delay management actions and make control more dif-ficult after invaders establish and spread (Simberloff, 2003). Matrixmodels can be parameterized even if available data are minimal andcan then inform management strategies in the early stages of theinvasion (Heppell et al., 2000; Govindarajulu et al., 2005). In oursnakehead example, because studies of its invasion are limited, weused biological information on northern snakehead in several lakesand rivers in China, summarized this information in appropriatedistributions, and constructed the projection matrix based on thesesummaries. We then evaluated the efficacy of different manage-ment tactics through elasticity and population-projection analyses.Modelling analyses that use information from species’ native envi-ronments can facilitate immediate design of control programs.Elasticity and prognostic analyses are effective tools for evaluatingdifferent management tactics (Heppell et al., 2000; Caswell, 2001).Moreover, this approach is applicable to a wide range of potentiallyinvasive species.

Demographic modelling analyses are useful in forecasting suc-cess of non-indigenous species at all stages of the invasionprocess (Lampo and De Leo, 1998; Heppell et al., 2000; Parker,2000; Govindarajulu et al., 2005). They can predict probability ofestablishment, spread, and population growth. By evaluating the

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robability of population growth over multiple temporal scales,hese models can provide useful predictions of ecological impact,iven that any non-indigenous species that becomes abundant orominant in an ecosystem will probably have major impacts onative biota (Simberloff, 2005). In our snakehead example, the pop-lation has a 78% probability of increasing if no control measures arepplied. Other available biological knowledge, such as the species’road environmental tolerances and diet (Wu et al., 2000a), alsouggest that the probability of population spread and risk of inva-ion are high. As the population increases over time, we expectorthern snakeheads to significantly alter the biotic community ofhe Potomac River.

An elasticity analysis alone does not predict the best manage-ent tactics (de Kroon et al., 2000; Caswell, 2001). Alternativeanagement tactics alter the values of parameters in the projectionatrix. We observed this in our study, such as when tactic S2 was

sed both RPS and spawner size changed. Thus, we advocate usingoth elasticity and prognostic analyses to evaluate the efficacy ofifferent tactics for population control.

.3. Predicting and controlling the northern snakehead invasion

The high risk of invasion for northern snakehead suggests aotential for major impacts on native biota, and thus a need for

mmediate control (Tan, 1996; Wilcove et al., 2000; Wu et al., 2000;irginia Invasive Species Council, 2005). Although specific impactsf northern snakehead have not been observed, it seems reasonableo expect that controlling its population size will limit potentialmpacts (Simberloff, 1998, 2005; Parker et al., 1999). Assessing therobability of population increase and evaluating different man-gement tactics for controlling population size are relevant tovaluating and planning for potential impacts. Our elasticity andrognostic analyses suggested that limiting the number of spawn-rs is the best control tactic.

Cost is an important consideration in non-indigenous speciesontrol (Bax et al., 2001; Maguire, 2004). Northern snakeheadarvae school for weeks before they disperse (Lapointe andngermeier, 2009), and the spawning season is mainly fromay to September (Ma and Xie, 2000; Lapointe and Angermeier,

009). Because tactics 2 and 3 can be applied in the same seasonnd location, simultaneously applying both tactics may enhancehe cost-effectiveness of controlling this non-indigenous species.herefore, we suggest that intensive fishing effort in summer,ocused on spawning and larval habitat, offers the greatest like-ihood of controlling northern snakeheads via harvest actions.ompared to fishing year-round, this tactic would require less timend effort and is likely more effective. Other control tactics per-inent to the Potomac River could also be considered, such asocusing harvest along snakehead movement routes, which couldncrease the cost-effectiveness of harvest. Such tactics could eas-ly be evaluated with our model and projection analysis, prior tomplementation.

We recommend further study of northern snakehead to advanceur understanding of its recruitment, survival, movement, andther life-history aspects in its new environment. Previous studiesave reported higher survival rates of non-indigenous species dur-

ng establishment due to low intraspecific density and a paucity ofredators and parasites (Fagan, 2002; Govindarajulu et al., 2005).thers have suggested possible Allee effects (Fowler and Ruxton,002; Drake, 2004). Such patterns are unexplored for northernnakehead. Our model parameters can be updated over time to track

opulation dynamics and better inform control efforts. Our prelim-

nary field data show that northern snakehead can move more than5 km during spring dispersal (Lapointe and Angermeier, 2009).hus, risk assessments may be further improved by incorporat-ng the geomorphology of the Potomac River and data on dispersal

g 220 (2009) 1681–1689

habits of northern snakehead into assessment models (Andersen etal., 2004a,b; Neubert and Parker, 2004; With, 2004).

As demonstrated here, hierarchical demographic models arepowerful tools for assessing the potential invasion of non-indigenous species, especially when life-history parameters arepoorly known. These models explicitly incorporate common varia-tion in life history among individuals responding to spatiotemporalheterogeneity, fishing effort, or other sources of environmentalvariation. Efforts to predict or control impacts of invading speciesoften must proceed without precise knowledge of life-historyparameters or the geographic source of the invader. Hierarchicaldemographic models are more flexible than non-hierarchical mod-els in allowing managers to incorporate such uncertainties intoeffective control strategies. Thus, hierarchical demographic mod-els provide more reliable inferences about an invading population’sgrowth rate and responses to different management tactics.

Acknowledgments

This project was supported in part by the USDA Cooperative StateResearch, Education and Extension Service, Hatch project #0210510to Y.J. and by a grant from the U.S. Geological Survey Invasive SpeciesProgram to P.L.A. et al. Any use of trade, product, or firm names doesnot imply endorsement by the U.S. Government.

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