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Biometrics 63, 558–567

June 2007DOI: 10.1111/j.1541-0420.2006.00725.x

Hierarchical Spatiotemporal Matrix Modelsfor Characterizing Invasions

Mevin B. Hooten,1,∗ Christopher K. Wikle,2 Robert M. Dorazio,3 and J. Andrew Royle4

1Department of Mathematics and Statistics, Utah State University, Logan, Utah 84322-3900, U.S.A.2Department of Statistics, University of Missouri, Columbia, Missouri 65211, U.S.A.

3USGS Florida Integrated Science Center and Department of Statistics, University of Florida,Gainesville, Florida 32611-0339, U.S.A.

4USGS Patuxent Wildlife Research Center, 12100 Beech Forest Road, Laurel, Maryland 20708, U.S.A.∗email: [email protected]

Summary. The growth and dispersal of biotic organisms is an important subject in ecology. Ecologistsare able to accurately describe survival and fecundity in plant and animal populations and have developedquantitative approaches to study the dynamics of dispersal and population size. Of particular interest arethe dynamics of invasive species. Such nonindigenous animals and plants can levy significant impacts onnative biotic communities. Effective models for relative abundance have been developed; however, a betterunderstanding of the dynamics of actual population size (as opposed to relative abundance) in an invasionwould be beneficial to all branches of ecology. In this article, we adopt a hierarchical Bayesian frameworkfor modeling the invasion of such species while addressing the discrete nature of the data and uncertaintyassociated with the probability of detection. The nonlinear dynamics between discrete time points areintuitively modeled through an embedded deterministic population model with density-dependent growthand dispersal components. Additionally, we illustrate the importance of accommodating spatially varyingdispersal rates. The method is applied to the specific case of the Eurasian Collared-Dove, an invasive speciesat mid-invasion in the United States at the time of this writing.

Key words: Animal dispersal; Detection probability; Hierarchical Bayesian models; Invasive species;Population models; Spatiotemporal dynamics.

1. Introduction1.1 Modeling Invasive SpeciesThe growth in population and dispersal of biotic organisms asa function of time has been recognized as an important sub-ject throughout the relatively short history of ecology as ascience (Bullock, Kenward, and Hails, 2002). Ecologists havelong been able to accurately describe survival and fecundityin plant and animal populations (Renshaw, 1991, Chapter 1;Bullock et al., 2002) and have also developed quantitativemodel-based approaches to study the dynamics of dispersal(Clark et al., 2003; Nathan et al., 2003). Of particular interestare the dynamics of invasive species (Kolar and Lodge, 2001).Such nonindigenous animals (and plants) can levy significantimpacts on native biotic communities (Elton, 1958; Shigesadaand Kawasaki, 1997, Chapter 1). Multiple approaches havebeen proposed to model invasions, many of which stem fromearly work by Fisher (1937) and Skellam (1951) in diffu-sion models and biological waves (Shigesada and Kawasaki,1997, Chapter 3). These led ultimately to more complexmetapopulation models, matrix models, and state-space mod-els (e.g., Hanski, 1999, Chapter 12; Caswell, 2001, Chap-ter 4; Borchers, Buckland, and Zucchini, 2002, Chapter 13).Often such models have been considered in the deterministic

sense or considered with known stochastic components forpurposes of simulation (Renshaw, 1991, Chapter 2; Kot,2001, Chapter A; Turchin, 2003, Chapter 5; Hastings et al.,2005). Statistical estimation of parameters in such models hasconventionally consisted of likelihood-based methods, linearregressions, time-series methods, and more recently, hierar-chical approaches (Shigesada and Kawasaki, 1997; Caswell,2001; Calder et al., 2003; Buckland et al., 2004; Thomaset al., 2005).

Recent advances in computational efficiency have allowedfor the implementation of sophisticated hierarchical Bayesianmodels (e.g., Calder et al., 2003; Clark, 2003; Wikle, 2003;Buckland et al., 2004; Su, Peterman, and Haeseker, 2004;Thomas et al., 2005; Wikle and Hooten, 2006; Hooten andWikle, 2007). It has been shown that such models have muchto offer, including more precise and less biased parameter es-timation (Calder et al., 2003), accounting for multiple sourcesof uncertainty (Wikle, 2003), and the ability to describe non-linear spatiotemporal dispersal and growth in relative abun-dance (Hooten and Wikle, 2007), as well as dynamics ofpopulation demographics (Buckland et al., 2004). Many pre-vious studies have been concerned with the rate of spreadonly (e.g., Arim et al., 2006). However, models specifically

558 C© 2006, The International Biometric Society

Hierarchical Spatiotemporal Matrix Models 559

formulated for describing the spread of invasive species ex-plicitly on spatio-temporal domains have relied on a state-space framework for implementing discretized partial differ-ential equations to model a log-transformed Poisson intensitythat represents “relative” abundance (Wikle, 2003; Wikle andHooten, 2006; Hooten and Wikle, 2007). Although proven tobe informative and useful, these formulations lack an intu-itive appeal, as parameter estimates arise from a latent, log-transformed process that may also induce a growth modelmisspecification (Hooten and Wikle, 2007). Conjugate full-conditional distributions highlight the primary mechanicaladvantage of these models in that large multivariate param-eter updates can easily be obtained from a Markov chainMonte Carlo (MCMC) algorithm. The combination of non-conjugacy, non-Gaussianity, and high-dimensional parameterspaces presents significant implementation challenges. At thesame time, such model specifications allow for a more intu-itive setting where identification and estimation of posteriorparameter distributions comes more naturally.

So-called matrix models (i.e., first-order Markov models)offer a very flexible framework for describing population dy-namics in time, space, and age classes (Caswell, 2001). Muchcan be learned about population dynamics through matrixmodel simulations. Neubert and Caswell (2000) provide anextensive discussion of demographic matrix models and theeffects of demographics on dispersal; however, they do not es-timate parameters on large spatiotemporal domains explicitlyfrom a model-based statistical perspective. Conventional pa-rameter estimation in such models requires extensive datasetsfrom well-designed experiments (Caswell, 2001, Chapter 6).Although large-scale spatiotemporal data are becoming morereadily available, they are often subject to multiple sources ofuncertainty (e.g., observer bias and irregularly sampled loca-tions and times) and not specifically collected to accommo-date complicated space–time demographic models (Hastingset al., 2005).

Although much more general as a class of models, ma-trix models can be specifically employed to model the dy-namics of invasive species in space and time and, therefore,such models are the focus of this paper. We approach theproblem hierarchically, with an intuitive data model, flexi-ble matrix process model comprised of growth and dispersalcomponents, and parameter model consisting mostly of non-conjugate priors. This approach has the advantage of appear-ing less complex than those proposed previously (e.g., Wikleand Hooten, 2006; Hooten and Wikle, 2007). The general formof the spatiotemporal matrix model, however, is suggested bydiscretized partial differential (Hooten and Wikle, 2007) andintegro-difference equations (Wikle, 2002), yet the direct ma-trix model approach proposed here allows for more generaldispersal.

In most ecological studies, the state variable of interest can-not be observed directly because of bias related to imperfectdetection. In fact, studies ignoring such sampling assumptionsgenerally focus on modeling “relative abundance” only (e.g.,Wikle, 2003; Hooten and Wikle, 2007). While models for rel-ative abundance are useful, it is often necessary to make di-rect inference about the size of a population (i.e., numbersof organisms) at a given set of locations. Monitoring effortsfor invasive species often result in count data, that is, the

number of individuals observed at each location (or area).In cases where less than the total number of individuals isobserved (i.e., imperfect detection), a binomial data model issuggested. In general, nonidentifiability issues arise when con-sidering such models (Olkin, Petkau, and Zidek, 1981; Carrolland Lombard, 1985; Raftery, 1988). For cases where prior in-formation about the probability of detection is available, wepropose a data model with identifiable population size param-eters that accounts for inherent uncertainty associated withnondetection bias.

1.2 The Eurasian Collared-Dove in the United StatesLarge scale spatiotemporal ecological datasets consisting ofquantitative population information are not common, espe-cially for invasive species (Caswell, 2001; Hastings et al.,2005). Fortunately, long-term monitoring efforts such asthe North American Breeding Bird Survey (BBS; Robbins,Bystrak, and Geissler, 1986) exist and can provide high-dimensional space–time invasive species data (North Amer-ican BBS data can be obtained at the following web site:http://www.pwrc.usgs.gov/bbs/).

Specifically, the Eurasian Collared-Dove (ECD, Streptopeliadecaocto) is a nonindigenous species in North America forwhich data have been collected since its introduction in theearly 1980s (Romagosa and Labisky, 2000). Although similarin appearance to the Ringed Turtle-Dove, ECD presence is notso benign as it rapidly spread through Europe in the 1900sand poses a possible threat to native ecosystems in NorthAmerica (Hengeveld, 1993).

During the peak of the avian breeding season each year formost of the United States and Canada, skilled observers col-lect bird population data along roadside survey routes. Eachsurvey route is 24.5 miles long with stops at 0.5-mile intervals.At each stop, a 3-minute point count is conducted, duringwhich every bird seen or heard within a 0.25-mile radius isrecorded. The spatial unit of observation for the routes alongwhich data are collected presents a significant challenge tospatial modeling, thus, certain simplifying assumptions aboutthe data must be considered in the construction of a spa-tiotemporal modeling framework. Herein we assume the dataoccur as counts located at the route centers; at a continen-tal scale, we assume the effect of aggregating over routes willbe minimal. Note that because of this aggregation assump-tion, any parameters corresponding to route locations shouldbe interpreted with the original spatial unit of observationin mind. Additionally, the BBS data are subject to multiplesources of uncertainty, especially within-site variability (Linket al., 1994; Sauer, Peterjohn, and Link, 1994). Knowledgeof such intricacies in the data provides a significant motiva-tion for the development of models that explicitly account forknown sources of uncertainty, particularly at the data level.

ECD currently occupies the Eastern United States al-though the invasion is ongoing (see Figure 1). Models forrelative abundance have suggested that the ECD displays typ-ical invasive behavior (Hooten and Wikle, 2007). That is, theunderlying dynamics of the invasion can be characterized bycommon density-dependent population growth and dispersalprocesses. Romagosa and Labisky (2000) speculated that theECD would likely colonize the entire United States in a few

560 Biometrics, June 2007

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Figure 1. BBS counts for ECD at BBS route centers for1986–2003. Points denote sample locations at which ECD wasobserved and circles illustrate magnitude of count data at eachroute location.

decades. The approach that follows allows for the rigorousprobabilistic evaluation of such claims.

2. MethodsWe adopt the general hierarchical model framework for spa-tiotemporal processes as in Wikle, Berliner, Cressie (1998),Calder et al. (2003), Wikle (2003), and Hooten and Wikle(2007), as proposed for time-series by Berliner (1996),whereby we construct the model in three stages: data, pro-cess, and parameters. The hierarchical approach allows usto construct and estimate a complex joint distribution viaa sequence of simpler, and more intuitive, conditional dis-tributions. This allows us to incorporate different forms ofuncertainty within each level of the hierarchy. The square

bracket and conditional notation (i.e., [a | b]) is used hereafterfor simplicity and denotes a conditional probability distribu-tion. That is, [a | b] implies the conditional probability distri-bution of a given b.

2.1 Data ModelIn general, recording counts of individuals (or groups) at var-ious locations over time is a common data collection schemefor monitoring plant and animal abundance. Because animalsare usually detected imperfectly in most animal surveys, suchdata often represents a subset of the true, unknown, numberof organisms at each time and location. If the assumption ismade that each organism at a given time and location is ob-served independently given that the organism is present, thena binomial data model would be reasonable. Let ni,t representthe observed number of organisms at location i and time t fora total of m spatial locations and T times, while the true,unknown number of organisms at that location and time isdenoted ni,t . In the case of the ECD, the ith location refers tothe location of the BBS route center, thus ni,t and N i,t referto the observed and true number of ECDs aggregated overthe ith BBS route, respectively. If the probability of observ-ing (i.e., detecting) each organism is θ, then we would have(temporarily) the following data model specification:

ni,t |Ni,t, θ ∼ Binom(Ni,t, θ) i = 1, . . . ,m; t = 1, . . . , T.

It would be preferable to use heterogeneous detection prob-abilities, yielding the data model: ni,t ∼ Binom (N i,t , θi,t).However, substantial data would be required to estimate suchhighly parameterized models in the absence of additional in-formation. One feasible approach is to absorb much of theuncertainty regarding the probability of detection with a dis-tribution on θ.

2.1.1 Estimation of detection probability. Although simplepoint counts are perhaps the most common sampling proto-col employed in avian surveys, they present a problem in theestimation of N i,t when the detection probability (θ) is un-known and must be estimated as well (Royle, 2004). The twoparameters are not identifiable in the absence of additionalinformation. A minor extension of the sampling protocol canprovide information on both θ and N i,t , though its imple-mentation specifically involving the ECD is not feasible usingBBS observations due to the lack of replicate data. Thus, inorder to estimate the true population size (N i,t) we need togain some a priori understanding of the probability of detec-tion. Consider separately a sampling protocol wherein eachlocal population (i.e., each sample unit) is visited on k sep-arate occasions, yielding a sequence of independent binomialcounts ni,t (for t = 1, . . . , k). Assuming the true populationsize, Ni , remains fixed during the period required to collect kobservations, the likelihood is the product binomial,

f(ni,1, . . . , ni,k |Ni, θ)

=∏t

Ni!

(Ni − ni,t)!ni,t!θni,t(1 − θ)Ni−ni,t .

(1)

The general instability of the maximum likelihood estimatefor Ni based on (1) is well known (Olkin et al., 1981; Car-roll and Lombard, 1985; Raftery, 1988), and this deficiencymay explain why the sampling protocol is not widely used in


practice. However, when this sampling protocol is applied atreplicate spatial locations and additional model structure isimposed on the location-specific abundance parameters, theestimation problem becomes more tractable (Royle, 2004).For example, if Ni ∼ Poisson (λ), then λ and θ are wellidentified.

This approach can be used in the presence of replicate datato obtain a posterior distribution for θ under a Bayesian im-plementation of equation (1) (Royle and Dorazio, 2006). Aspreviously mentioned, in the case of the ECD, the operationalBBS survey does not yield replicate counts at each BBS sam-ple location. However, a study conducted by Link et al. (1994)obtained replicate counts on a number of BBS routes withinthe ECD range. Specifically, we used replicate data from a dif-ferent species (Mourning Dove, Zenaida macroura) that sharessimilar detectability characteristics with ECD to inform θ.The posterior for the detection probability of the MourningDove is the standard form of a beta distribution with param-eters αθ and βθ. We do not expect the estimate of θ (in termsof αθ and βθ) to represent the detectability of the ECD ex-actly; however, based on similarities in dove behavior we canexpect it to be a reasonable estimate, especially consideringthat we are allowing for the uncertainty in θ through the betadistribution.

2.1.2 Incorporation of detection probability. The informa-tion about detection probability based on the Mourning Dovedata provides the best estimate of θ we could hope to attaingiven available data and methods. In fact, given the identi-fiability issues between N i,t and θ we cannot learn anythingfurther about the probability of detection in a hierarchicalspatiotemporal model without additional replicate data. How-ever, by allowing the Mourning Dove estimate to be our priorfor θ and then treating it as a nuisance parameter, we canintegrate it out of our data model while accounting for itsuncertainty (Berger, Liseo, and Wolpert, 1999). This suggeststhe following modified data model,

[ni,t |Ni,t]

=

∫ 1

0

[ni,t |Ni,t, θ][θ]dθ i = 1, . . . ,m; t = 1, . . . , T, (2)

which analytically yields a beta-binomial data model whenthe prior distribution for θ is indeed specified as a beta dis-tribution (i.e., θ ∼ Beta (αθ, βθ)):

ni,t |Ni,t, αθ, βθ ∼ Beta-Binom(Ni,t, αθ, βθ) i = 1, . . . ,m;

t = 1, . . . , T. (3)

Hence, this beta-binomial model is used hereafter as the datamodel component in the hierarchical matrix model.

2.2 Process ModelAlthough the estimation, spatial prediction, and temporalforecasting of population size are of primary concern in thissetting, the underlying ecological process is also of interest.That is, knowledge about the true process from which thedata arise is useful for characterizing and studying the in-vasion. Many previously proposed methods split the processmodel into two stages where the first stage is a nonlinear(and hence nonconjugate) transformation of the second stage(a Gaussian state equation). This provides a computationaladvantage yet detracts from the intuitive nature of the process

itself. We propose a process model with intuitive constructionwhile retaining scientifically meaningful dynamic behavior.

We adopt a natural and common model for the true numberof individuals at a given location and time. That is, the truepopulation (N i,t) is assumed to be conditionally Poisson withintensity (λi,t) depending hierarchically on population growthand dispersal parameters.

Ni,t |λi,t ∼ Pois(λi,t) i = 1, . . . ,m; t = 1, . . . , T.

Assuming the Poisson intensity (λt ≡ [λ1,t, . . . , λm,t]′)

evolves according to first-order Markovian dynamics, we canutilize a matrix model framework (Caswell, 2001, Chapter 4).In the specific setting where we lack information about theage structure of a species, we have λt, the state vector, beingpropagated by a transition matrix H in the following evolu-tion equation.

λt = Hλt−1

= MGλt−1

= M(τ )G(K, r,λt−1)λt−1 t = 1, . . . , T. (4)

Notice in equation (4) that we have decomposed the transi-tion matrix H into a product of two distinct m × m matricescorresponding to growth (G) and movement (M) of the pop-ulation. This formulation allows the process, which is randomthrough its parameters (i.e., K, r, τ , and λ1 are all proba-bilistically specified in the next section) and quite flexible, toevolve in a conventional dynamical system. Additionally, itprovides a convenient and ecologically meaningful specifica-tion of the growth and dispersal parameters. Similar decompo-sitions of the transition matrix H in settings where populationdemographics is also being studied have been discussed (e.g.,Neubert and Caswell, 2000; Buckland et al., 2004), though thespecification presented here differs from those used previouslyin order to accommodate the specific ECD dataset as well asprediction of a high-dimensional spatially explicit process.

The matrix controlling population growth (G) can be pa-rameterized many ways. We propose a diagonal form wherepopulation growth at each location is described by a commondensity-dependent formulation given by the Ricker growthequation (Turchin, 2003, Chapter 3). Although other growthequations exist and are commonly used (e.g., Beverton-Holt,Malthusian, Gompertz), the Ricker form of growth is flexi-ble and allows for potentially chaotic behavior and oscillatorycharacteristics in population stability (Kot, 2001, Chapter A).Additionally, investigators wishing to perform inference onthe form of growth model specifically could implement vari-ous process models separately and compare their effect usingany number of model selection criteria (e.g., Deviance Infor-mation Criterion [DIC] or Bayes factors). Another approachwould be to consider multiple growth models simultaneouslyusing a mixture-based specification for the process model andreversible jump MCMC. In simulation, we have found theRicker growth model, with random parameters, to be veryflexible and capable of exhibiting behavior similar to that ofother forms of theoretical population growth. Thus, here wefocus on Ricker growth and let the ith diagonal element of Gbe defined as,

G(r,K, λi,t) = exp{r(1 − λi,t/K)}, t = 1, . . . , T, (5)


where K and r correspond to carrying capacity and growthrate, respectively. In cases where there are significant differ-ences in the carrying capacity (K) and growth rate (r) acrossspatial locations, one could consider parameterizations whereK and r are allowed to vary in space. Effective estimation ofspatially varying growth rates may be difficult in models al-ready heavily parameterized without informative prior knowl-edge. In the presence of data describing a nontransient process(i.e., a process where population size has reached carrying ca-pacity at all locations) it may be possible to estimate spatiallyvarying carrying capacities (K); in cases where a populationis still growing, however, these parameters will be difficult toestimate.

The dispersal of organisms in space is described in equa-tion (4) by the matrix M. Dispersal in this case is assumed tobe characterized by a spatially varying Gaussian kernel. Thatis, the population in a given location at the current time is aspatially weighted average of the population at surroundinglocations depending on the distances (di,j ) between locationsand dispersal parameters, τ . We let M be defined as:

M(τ ) = [(Mi,j)]m×m,

Mi,j ∝ exp

{−d2i,j

τj

},

(6)

where each row of M is constrained to sum to one with eachvalue between 0 and 1, as motivated by discretized diffusionequations (e.g., Wikle, 2002; Hooten and Wikle, 2007).

Other parameterizations of this dispersal matrix could beconsidered. For example, an alternative specification of thedispersal matrix, say M(alt), is one where each column (ratherthan row) of M(alt) is constrained to sum to one with 0 ≤M

(alt)i,j ≤ 1 for all i and j. This dispersal matrix is especially

suited to the “multisite” situation where the M(alt)i,j refer to

transition probabilities. In such a situation, where individualanimals are moving from one distinct “population” to anotherwith a certain probability, it is sensible to think of the tran-sition probabilities as: M

(alt)i,j = P(animal moves to location

i | animal is currently in location j). In a spatial setting, withdata pertaining to irregularly observed locations on a land-scape where there are open boundaries (as is typically thecase in invasion problems), a dispersal model needs to havethe flexibility that the quantity being dispersed can move outof the domain through the boundary. To help visualize thedifference between M and M(alt), consider the simplest case,where G = I (i.e., no growth in population); here M(alt) pre-serves the number of total organisms over time and only allowsthem to occur in distinct areas, whereas M does not accountfor every organism at each time and thus the total numberof organisms may fluctuate over time even though there is nogrowth component in the model. This effect of parameteriza-tion implies that when G = I, the parameters in M and Gmay not be completely identifiable because they could both becontributing to the overall growth in population. However, inthe situation considered here, with the ECD and BBS countdata, the preservation of population size resulting from the“probability parameterization” of M(alt) is not as importantas the dispersal behavior, thus we retain the parameterizationof M as in equation (6). Though the chosen specification for

M implies that the growth (i.e., K and r) and dispersal (i.e.,τ ) parameters may not be separable, simulations (detailed inHooten, 2006) verify that for the spatial and temporal do-mains considered in this application, the contribution fromM to overall growth is minimal. Still, rigorous interpretationof growth and dispersal parameters, separately, could be mis-leading, and thus it is best to interpret the effect of K, r, andτ simultaneously. Here, the emphasis is on the prediction oftrue population size (N i,t) in time and space and thus there isno need to interpret the parameters separately; however, weprovide the posterior results of these parameters for the sakeof completeness.

As with the growth parameters, one could specify thisGaussian kernel to be spatially homogeneous in cases wherelimited data exist. For the application considered here, a mul-tivariate parameterization has been shown to accommodateregional differences in dispersal (Hooten and Wikle, 2007)when adequately estimated. The dispersal mechanism em-ployed here is more flexible than those previously proposed;thus we would expect patterns of dispersal to be different. Infact, because our emphasis is on prediction and estimation ofthe true population size (N i,t), by specifying a simpler modelwith the constraint that all τ i are equal, we can illustrate therisk of assuming homogeneous dispersal. Additionally, we cancompare model complexity and effectiveness using DIC (seeGelman et al., 2004, Chapter 6). Though no methods of modelcomparison and selection are without criticism, DIC focusedon the most important level in complex hierarchical models(such as the data model in this case) can provide a meansof comparing the overall effect of different parameterizationswhile taking into account model complexity.

2.3 Parameter ModelThe underlying process (equation 4) in this hierarchical modelis a dynamical system with random components. Conven-tionally, latent processes are formulated with additive errorto account for misspecification of the model. Such specifi-cations are very general and often result in nonidentifiablevariance components (e.g., Hooten and Wikle, 2007) and pro-hibit long-range prediction due to additive error propagatingthrough the predictive forward model. Accounting for the pro-cess model error through probabilistic growth and dispersalparameter specifications alleviates such complications whileoffering sensible flexibility in the latent process.

With the nuisance parameter, θ, integrated out, we pro-pose a very simple specification for prior distributions of theremaining model parameters. Assuming that we have inde-pendence between priors (i.e., [K, r, τ ,λ1] ∝ [K][r][τ ][λ1]), welet:

K ∼ Gamma(αK = 4, βK = 50),

r ∼ N(μr = 0.5, σ2r = 0.25),

log

(τ

τ̃

)∼ N

((μτ

μ̃τ

), Στ =

(Σ1,1

τ Σ1,2τ

Σ2,1τ Σ2,2

τ

)), (7)

log

(λ1

λ̃1

)∼ N

((μλ

μ̃λ

), Σλ =

(Σ1,1

λ Σ1,2λ

Σ2,1λ Σ2,2

λ

)), (8)


where τ̃ and λ̃1 are the parameter values at a set of loca-tions where we are interested in making predictions. Also,the Gamma distribution on K has a mean 200 and vari-ance of 10,000 and is parameterized such that βK is thescale parameter. Στ and Σλ are characterized by expo-nential spatial hyperpriors on the covariance structure ofthe dispersal and intensity process at the initial time (i.e.,t = 1), respectively. Specifically, [μτ ]i = [μ̃τ ]j = log(2), ∀i, jand [Στ ]i,j = log(1.1)[exp(−2 ||di,j ||)] represent the hyperpri-ors for τ while μλ = μ̃λ = log(15) for locations in extremesouthern Florida (i.e., the location where ECD was introducedinto the United States) with zero elsewhere and [Σλ]i,j =log(1.1)[exp(−2 ||di,j ||)] represent the hyperpriors for λ1. TheEuclidean distance between locations i and j is represented by||di,j ||. The spatial specification of τ and λ1 allows for pos-terior predictive estimation at locations and times withoutdata as well as likely autocorrelation in the parameters (seeWeb Appendix A). Hyperpriors are specified based on theirability to exhibit reasonable growth and dispersal behavior insimulations. For example, we know that the population sizeis generally growing over time and thus μr was specified to bepositive, although r could still be negative depending on σ2

r

and the influence of the data. Recall that prior specificationfor θ (i.e., θ ∼ Beta (1.7, 7.7), implying a mean of 0.18 andstandard deviation of 0.12) was based on posterior estimatesof detection probability from a separate model and datasetwith repeated measurements and then integrated out, as dis-cussed in the data model section of this article.

3. ResultsMethods for estimating the posterior distribution for themodels considered here include MCMC and various formsof importance sampling (Gelman et al., 2004, Chapter 11).Importance sampling is especially desirable in settings withnonconjugate full-conditional distributions and nonlinear re-lationships between parameters; however, it suffers fromdegeneracy problems, especially in situations with high-dimensional parameter spaces, as is the case here. Metropolis–Hastings updating within a Gibbs Sampler (one form ofMCMC) is an alternative approach for sampling, but requirescareful selection of proposal distributions that can sufficientlyexplore the parameter space. We concentrate on the MCMCapproach for estimation although other sampling algorithmsmay also provide adequate results.

Figure 2. Maps representing the posterior mean (E(τ |ni,t), ∀i, t) and standard deviation ((V (τ |ni,t))1/2, ∀i, t) of dispersal.

Several multivariate Metropolis–Hastings parameter up-dates slow the mixing of MCMC samples, thus the Gibbssampler was allowed to run for 200,000 iterations to ensurethat the posterior parameter space was explored sufficiently.Resampling was used to remove correlation between realiza-tions. Convergence, which was assessed by visually inspect-ing parameter chains, occurred before the burn-in period of20,000 iterations, which were used for calculation of posteriorsummary statistics. Metropolis–Hastings acceptance rates forthe parameter updates varied between 20% and 40%. For ad-ditional details about the sampling algorithm and predictions,see Web Appendix A.

Estimation of the posterior distributions of model parame-ters provides useful information about the underlying dynam-ics of the ECD invasion. Although the focus here is on theestimation and prediction of the population size at variouslocations and times, knowledge about the process driving thepopulation size can also be informative. In this setting wecan evaluate population dispersal as well as growth paramet-rically, though, inference on dispersal and growth parameters,separately, should only serve as a general guideline becausethe degree of separation is unknown. However, the combinedeffect of growth and dispersal is valid and can be assessed byinterpreting the process itself (i.e., λt). Simulated processeson similar spatial and temporal domains suggested that theshared contribution from K, r, and τ to overall populationsize is negligible (as evidenced by only a slight bias in theposterior estimates of parameters based on simulated data);however, we provide the following posterior summaries witha cautionary warning.

Posterior distributions for the univariate parameters K(carrying capacity) and r (growth rate) yielded 95% credi-ble intervals of [103, 201] and [0.26, 0.32], respectively. Pos-

terior standard deviations for K and r (i.e., (V (K |n))1/2 =

24.3, (V (r |n))1/2 = 0.02) were very precise compared to their

prior standard deviations (i.e., (V (K))1/2 = 100, (V (r))1/2 =0.5), thus implying that there was sufficient data to informthe growth of the process.

Statistics from the posterior distribution of multivariate pa-rameters can be expressed as spatial maps. Prediction andestimation at a set of regularly spaced locations provides aconvenient way to view the posterior parameter spaces. Notethat all figures containing maps herein are displayed as imageswith grid cell intensity representing parameter magnitude at


the location of the grid cell center. The image format is forvisualization only (as an alternative to variable sized points)and should not be misinterpreted as “areal” support. That is,each pixel in the images represents a point at the pixel center,rather than the area of the pixel itself.

Consider the posterior mean and standard deviation mapsof dispersal τ (Figure 2). As with the growth parameters, theposterior standard deviations for dispersal were much smallerthan the prior standard deviation (i.e., max(V (τ i |n)) <<V (τ i)). A primary concern of ecologists and resource man-agers related to invasive species is the extent and magnitudeof the invasion at a given time (including future times). Inthe case of the ECD, the mean posterior population size pre-sented in Figure 3a illustrates the estimated magnitude andextent of the invasion at a set of times (including predictions)given the sample data. By the year 2016, the ECD populationis predicted to be at or near carrying capacity for the portionof the United States shown in the figures. Posterior credibleinterval growth curves for individual locations can reveal atwhat time the population is projected to reach its carryingcapacity (i.e., the total number of ECDs a sampled locationcan support).

By implementing a simpler form of the model with ho-mogeneous dispersal rates (i.e., τ i = τ , ∀i) we can compareand assess what, if any, advantage spatially varying disper-sal contributes to estimation and prediction. Using DIC (e.g.,Gelman et al., 2004, Chapter 6) focused on the data model, wecan compare the two models in terms of their predictive capac-ity while penalizing for model complexity. Denote model 1 asthe proposed model allowing for heterogeneous dispersal andmodel 2 as the simplified model with homogeneous dispersal.Then, the resulting number of effective parameters for eachmodel is: pD1 = 133 and pD2 = 130. The DIC for each modelis: DIC 1 = 13,052 and DIC 2 = 13,107.

4. DiscussionConsider, first, the posterior 95% credible interval for theRicker growth rate (r). In situations where there is no changein population size due to dispersal, a growth rate of r < 0 im-plies a decrease in population while r > 0 implies an increase.Here r does not overlap zero, implying that there is indeedgrowth in the population, in addition to dispersal. In thiscase, where dispersal contributes to population size, and be-cause an invasion generally constitutes a growing population,we can expect r to be positive but smaller than it would be ina population without dispersal. Also, recall from equation (5)that when r → 1 it implies that G(K, r, λi,t) → exp {1 −λi,t/K}; a much simpler growth model. Here, the posteriorcredible interval of r suggests that it is significantly differ-ent from 1, thus the assumption of a two parameter growthequation is reasonable.

Recall that the carrying capacity is modeled with an overallterm (K); thus, the estimate for E(K |n) could be thought ofas representing the overall mean while V (K |n) represents theuncertainty associated with carrying capacity. The possiblespatial heterogeneity in carrying capacity is not accounted forexplicitly here for the sake of parsimony; situations may exist,however, where such a multivariate carrying capacity param-eter is indeed identifiable. The posterior credible interval for

Figure 3. (a) Posterior (and posterior predicted) meanof ECD population size throughout the United States from1986 to 2016. (b) Also, the absolute difference between poste-

rior means for N(2)t and N(1)

t (i.e., the difference in estimatesof population size for the model with homogeneous dispersalversus the model with heterogeneous dispersal). Overall shadecorresponds to absolute difference and solid white circles rep-resent large positive differences while empty white circles rep-resent large negative differences.


1985 1990 1995 2000 2005 2010 2015 2020

050

10

01

50

20

0

95% credible intervals for N

year

co

un

t

Figure 4. Credible intervals for posterior population growth of ECD (Ni,t) from 1986 to 2020 for locations in south Florida(solid lines) and northwest Wyoming (dashed lines) simultaneously.

the carrying capacity used here (K) suggests that the numberof individuals a given location can support is quite variable.This is not surprising given the lack of prior information aboutthis parameter and the fact that the invasion is still ongoingand has not reached saturation at all locations. For situationswhere the population size at more locations had reached car-rying capacity, the variability in K could likely be reduced byconditioning on spatial covariates.

From the parameter estimates in Figure 2 we can see thatthere is a clear and significant area of low dispersal in the re-gion of northern Florida (in the southeast portion of the mapin Figure 2a). The effect of this will slow the spread of ECDwestward out of Florida while northward, along the east coast,ECD dispersal remains uninhibited. Note that the values forthe mean parameter estimates of τ are directly related tothe scale of the map distance units, and therefore the generalpattern rather than the magnitude should be interpreted here.That is, the use of a different coordinate system would resultin different parameter values, but retain the same pattern.

Consider 95% credible intervals for a location in southFlorida and a location in northwest Wyoming (i.e., North-west United States) simultaneously (Figure 4). At the time ofthis writing, very few, if any, ECDs have likely dispersed as faras Wyoming, whereas the population sizes in most of Floridaare nearly at carrying capacity. Although, by the year 2020,Wyoming will be near carrying capacity with high probabil-ity as well, as speculated by Romagosa and Labisky (2000).Such extrapolations (in time and space) are generally not ad-visable; however, the predictions in this case are a result ofthe estimated dynamics of the invasion and serve as the onlyinsight into the probable population sizes at future times andlocations based on the available data.

The DIC analysis for comparing the two models suggeststhat after correcting for the number of unconstrained parame-ters, model 1 still provides better predictions due to the lowerDIC value. Additionally, Figure 3b illustrates the relative dif-ferences between models and shows how assuming homoge-nous dispersal can lead to inflated and deflated populationsize estimates in both time and space. In Figure 3b the ar-eas with solid white circles imply that model 2 (homogeneousdispersal) is overestimating the population size relative to thepredictions resulting from model 1, while the areas with emptywhite circles indicate that model 2 is underestimating popu-lation size relative to the predictions resulting from model 1.These areas of misestimation clearly evolve over time and canbe quite misleading. Overall, the DIC analysis and differencemaps suggest that a spatially varying dispersal parameter isindeed important in the model.

Overall, we have found that by using a priori knowledgegained from models proposed by Royle (2004) and Royle andDorazio (2006), it is possible to estimate parameters thatcharacterize a nonlinear and non-Gaussian dynamical sys-tem underlying the dispersal of invasive species. This use ofsubjective probability illustrates a strength of Bayesian mod-els in complicated hierarchical settings where conventionallikelihood-based parameter estimation would not be feasible.From a scientific perspective, it is important to note thatprevious dynamical spatiotemporal population models havelargely ignored the detectability of the species, whereas herewe are directly accounting for detection as well as uncer-tainty associated with detection. In the specific case of theECD, this represents the only way to gain an understandingof the spatiotemporal dynamics in true population size (i.e.,N i,t).


The model proposed in this article adopts the intuitive andsimple yet powerful form of a matrix model (e.g., Caswell,2001). Although information about the age distribution ofECD over time and space is not available, this model frame-work could easily accommodate such data. In fact, severalapproaches have been taken to implement the demographicversion of matrix models in a hierarchical framework (e.g.,Buckland et al., 2004; Thomas et al., 2005).

It is important to note that this form of the proposed modelrelies heavily on the estimated dynamics of the invasion inorder to predict population size at future times and unsam-pled locations. For cases where there exists more completedata for less of a generalist species, the model could easilybe extended to accommodate covariate effects on model pa-rameters, such as growth rate (r), carrying capacity (K), anddispersal (τ ). In cases where such information is available,predictions could also serve as a way to evaluate scenarios onhow to control the spread of invasions. Many other extensionsto the model are possible concerning detectability. For exam-ple, if detection was known (or estimated a priori) to varyby location or time (possibly through association with covari-ates), then it could be included in the exact same frameworkonly with θi,t in the data model rather than θ. Furthermore,in cases where repeated counts are observed for each loca-tion and time, a modeling framework similar to that of Royleand Dorazio (2006) could be implemented to estimate θi,t , inaddition to the process and dynamics.

The main advantage of this model over other similar in-vasive species models is that it is intuitive and simple yetflexible enough to accommodate various specifications andsources of uncertainty (e.g., such as the uncertainty in de-tection). It provides tangible graphical output (in addition tonumerical output) and is based on rigorous data-driven sta-tistical methodology, the basic principles of which ecologistsand resource managers can understand and utilize for scien-tific endeavor and decision making.

Supplementary MaterialsWeb Appendix A referenced in Sections 2.3 and 3 is availableunder the Paper Information link at the Biometrics web sitehttp://www.tibs.org/biometrics.

Acknowledgements

CKW and MBH’s work was supported by National Sci-ence Foundation grant DMS-0139903 and USGS Coopera-tive Agreement 1434-HQ-RU-1556. RMD and JAR’s workwas supported by grants from the USGS National Park Mon-itoring Project. The authors thank the reviewers of thismanuscript for suggesting relevant literature and many help-ful comments and also John R. Sauer for providing the datafrom Link et al. (1994) and general advice on modeling datafrom the North American BBS.

References

Arim, M., Abades, S. R., Neill, P. E., Lima, M., and Marquet,P. A. (2006). Spread dynamics of invasive species. Pro-ceedings of the National Academy of Sciences, USA 103,374–378.

Berger, J. O., Liseo, B., and Wolpert, R. (1999). Integratedlikelihood methods for eliminating nuisance parame-ters. Statistical Science 14, 1–28.

Berliner, L. M. (1996). Hierarchical Bayesian time-seriesmodels, In Maximum Entropy and Bayesian Methods,15–22. Dordrecht, The Netherlands: Kluwer AcademicPublishers.

Borchers, D. L., Buckland, S. T., and Zucchini, W. (2002).Estimating Animal Abundance: Closed Populations. Lon-don: Springer-Verlag.

Buckland, S. T., Newman, K. B., Thomas, L., and Koesters,N. B. (2004). State-space models for the dynamics of wildanimal populations. Ecological Modelling 171, 157–175.

Bullock, J. M., Kenward, R. E., and Hails, R. S. (2002). Dis-persal Ecology. Malden, Massachusettes: Blackwell Sci-ence Ltd.

Calder, C., Lavine, M., Muller, P., and Clark, J. S. (2003).Incorporating multiple sources of stochasticity into dy-namic population models. Ecology 84, 1395–1402.

Carroll, R. J. and Lombard, F. (1985). A note on N estimatorsfor the binomial distribution. Journal of the AmericanStatistical Association 80, 423–426.

Caswell, H. (2001). Matrix Population Models. Sunderland,Massachusettes: Sinauer Associates, Inc.

Clark, J. S. (2003). Uncertainty and variability in demographyand population growth: A hierarchical approach. Ecology84, 1370–1381.

Clark, J. S., Lewis, M., McLachlan, J. S., and HilleRisLam-bers, J. (2003). Estimating population spread: What wecan forecast and how well? Ecology 84, 1979–1988.

Elton, C. A. (1958). The Ecology of Invasions by Animals andPlants. London: Methuen.

Fisher, R. A. (1937). The wave of advance of advantageousgenes. Annals of Eugenics 7, 355–369.

Gelman, A., Carlin, J. B., Stern, H. S., and Rubin, D. B.(2004). Bayesian Data Analysis, 2nd edition. New York:Chapman & Hall.

Hanski, I. (1999). Metapopulation Ecology. New York: OxfordUniversity Press.

Hastings, A., Cuddington, K., Davies, K., et al. (2005). Thespatial spread of invasions: New developments in theoryand practice. Ecology Letters 8, 91–101.

Hengeveld, R. (1993). What to do about the North Americaninvasion by the Collard Dove? Journal of Field Ornithol-ogy 64, 477–489.

Hooten, M. B. (2006). Hierarchical SpatioTemporal Models forEcological Processes. Ph.D. Dissertation, University ofMissouri. Columbia, Missouri.

Hooten, M. B. and Wikle, C. K. (2007). A hierarchicalBayesian nonlinear spatiotemporal model for the spreadof invasive species with application to the EurasianCollared-Dove. Environmental and Ecological Statistics,in press.

Kolar, C. S. and Lodge, D. M. (2001). Progress in invasionbiology: Predicting invaders. Trends in Ecology and Evo-lution 16, 199–204.

Kot, M. (2001). Elements of Mathematical Biology. Cambridge:Cambridge University Press.

Link, W. A., Barker, R. J., Sauer, J. R., and Droege, S. (1994).Within-site variability in surveys of wildlife populations.Ecology 74, 1097–1108.


Nathan, R., Perry, G., Cronin, J. T., Strand, A. E., and Cain,M. L. (2003). Methods for estimating long-distance dis-persal. Oikos 103, 261–273.

Neubert, M. G. and Caswell, H. (2000). Demography anddispersal: Calculation and sensitivity analysis of inva-sion speed for structured populations. Ecology 81, 1613–1628.

Olkin, I., Petkau, A. J., and Zidek, J. V. (1981). A comparisonof N estimators for the binomial distribution. Journal ofthe American Statistical Association 76, 637–642.

Raftery, A. E. (1988). Inference for the binomial N parameter:A hierarchical Bayes approach. Biometrika 75, 223–228.

Renshaw, E. (1991). Modelling Biological Populations in Spaceand Time. Cambridge: Cambridge University Press.

Robbins, C. S., Bysrak, D. A., and Geissler, P. H. (1986). TheBreeding Bird Survey: Its first fifteen years, 1965–1979.Fish and Wildlife Service Resource Publication 157, US-DOI, Washington D.C.

Romagosa, C. M., and Labisky, R. F. (2000). Establishmentand dispersal of the Eurasian Collared-Dove in Florida.Journal of Field Ornithology 71, 159–166.

Royle, J. A. (2004). N-mixture models for estimating popula-tion size from spatially replicated counts. Biometrics 60,108–115.

Royle, J. A. and Dorazio, R. M. (2006). Hierarchical modelsof animal abundance and occurrence. Journal of Agri-cultural, Biological, and Environmental Statistics 11(3),249–263.

Sauer, J. R., Peterjohn, B. G., and Link, W. A. (1994). Ob-server differences in the North American Breeding BirdSurvey. Auk 111, 50–62.

Shigesada, N. and Kawasaki, K. (1997). Biological Invasions:Theory and Practice. New York: Oxford University Press.

Skellam, J. G. (1951). Random dispersal in theoretical popu-lations. Biometrika 38, 196–218.

Su, Z., Peterman, R. M., and Haeseker, S. L. (2004). Spatialhierarchical Bayesian models for stock-recruitment anal-ysis of pink salmon (Oncorhynchus gorbuscha). CanadianJournal of Fisheries and Aquatic Sciences 61, 2471–2486.

Thomas, L., Buckland, S. T., Newman, K. B., and Harwood,J. (2005). A unified framework for modelling wildlifepopulation dynamics. Australian and New Zealand Jour-nal of Statistics 47, 19–34.

Turchin, P. (2003). Complex Population Dynamics. Princeton,New Jersey: Princeton University Press.

Wikle, C. K. (2002). A kernel-based spectral model for nonGaussian spatiotemporal processes. Statistical Modelling:An International Journal 2, 299–314.

Wikle, C. K. (2003). Hierarchical Bayesian methods for pre-dicting the spread of ecological processes. Ecology 84,1382–1394.

Wikle, C. K. and Hooten, M. B. (2006). Hierarchical Bayesianspatiotemporal models for population spread. In Appli-cations of Computational Statistics in the EnvironmentalSciences: Hierarchical Bayes and MCMC methods. NewYork: Oxford University Press.

Wikle, C. K., Berliner, L. M., and Cressie, N. (1998). Hier-archical Bayesian space-time models. Environmental andEcological Statistics 5, 117–154.

Received December 2005. Revised August 2006.Accepted September 2006.

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