A Bayesian approach to the ecosystem inverse problem

Ecological Modelling 168 (2003) 39–55

A Bayesian approach to the ecosystem inverse problem

Michael Dowda,∗, Renate Meyerba Department of Mathematics and Statistics, Dalhousie University, Halifax, NS, Canada B3H 3J5

b Department of Statistics, University of Auckland, Auckland, New Zealand

Received 6 August 2002; received in revised form 1 April 2003; accepted 7 May 2003

Abstract

This study investigates a probabilistic approach for the inverse problem associated with blending time-dependent ecosystemmodels and observations. The goal is to combine prior information, in the form of ecological dynamics and substantive knowledgeabout uncertain parameters, with available measurements. Posterior estimates of both the time-varying ecological state variablesand the model parameters are obtained, along with their uncertainty. Ecological models of interacting populations are consideredin the context of a nonlinear, non-Gaussian state space model. This comprises a nonlinear stochastic difference equation for theecological dynamics, and an observation equation which relates the model state to the measurements. Complex error processesare readily incorporated. The posterior probability density function provides a complete solution to the inverse problem. Bayes’theorem allows one to obtain this posterior density through synthesis of the prior information and the observations. To illustratethis Bayesian inverse method, these ideas are applied to a simple ecosystem box model concerned with predicting the seasonalco-evolution of a population of grazing shellfish and its two food sources: plankton and detritus. Observations of shellfish biomassover time are available. Lognormal system noise was incorporated into the ecosystem equations at all time steps. Ingestion andrespiration parameters for shellfish growth are considered as uncertain quantities described by beta distributions. Stochasticsimulation was carried out and provided predictions of the model state with uncertainty estimates. The Bayesian inverse methodwas then used to assimilate the additional information contained in the observations. Posterior probability density functions forthe parameters and time-varying ecological state were computed using Markov Chain Monte Carlo methods. The ecologicaldynamics spread the measurement information to all state variables and parameters, even those not directly observed. Probabilisticstate estimates are refined in comparison to those from the stochastic simulation. It is concluded that this Bayesian approachappears promising as a framework for ecosystem inverse problems, but requires careful control of the dimensionality for practicalapplications.© 2003 Elsevier B.V. All rights reserved.

Keywords:Bayesian statistics; Inverse methods; Marine ecosystem model; Shellfish growth; Data assimilation

1. Introduction

Inverse problems are concerned with combiningdynamical models and measurements in order toproduce optimal estimates for the state of a system.

∗ Corresponding author. Tel.:+1-902-494-1048;fax: +1-902-494-5130.

E-mail address:[email protected] (M. Dowd).

They have a well established role in fields such asgeophysics(Tarantola, 1987)and physical oceanog-raphy (Bennett, 1992). Inverse methods are nowreceiving increased attention in ecological modelling.For instance, in marine ecology there is emerginginterest in the development of inverse techniques,also known as data assimilation, for ocean ecosystemmodels(Harmon and Challenor, 1997; Hofmann andLascara, 1998; Vallino, 2000; Natvik et al., 2001).

0304-3800/$ – see front matter © 2003 Elsevier B.V. All rights reserved.doi:10.1016/S0304-3800(03)00186-8

40 M. Dowd, R. Meyer / Ecological Modelling 168 (2003) 39–55

Such developments are being driven, in part, by therevolution in observing technologies. For example,the foundational elements of the aquatic food chain—the plankton—are now routinely surveyed by satelliteremote sensing(Kahru and Mitchell, 2001). Theyare also being measured using new in situ sensorsbased on acoustics(Batchelder et al., 1995), optics(Cullen et al., 1997), and image analysis(Sierackiet al., 1998). The purpose of this study is to ad-dress the ecosystem inverse problem, or how to bestuse information from (i) time-dependent ecosystemmodels, (ii) available observations, and (iii) prior in-formation on parameters, to produce estimates for thetime-varying ecological state and model parameters,along with their uncertainty.

Ecological systems are characterized by complexand nonlinear interactions between populations. Math-ematical models of such systems can exhibit complexdynamical behavior(May, 1972; Edwards, 2001).System identification is an important problem and un-certainty arises due to poorly understood model struc-ture (Reichert and Omlin, 1997; Omlin and Reichert,1999). Stochastic elements also play a role in terms ofboth parameter uncertainty(Annan, 2001)and envi-ronmental forcing(Marion et al., 2000). Such modelerrors and stochastic forcing are collectively knownas system noise. Measurements for ecological statevariables (populations numbers or biomass) are oftensparse, noisy, and have distributions which are far fromGaussian(Soudant et al., 1997; Dowd et al., 2003).

As a result of the uncertainty in ecological mod-els and measurements, parameter estimation has beenidentified as an important issue for ecological mod-elling (Jorgensen, 2001). A commonly used approachinvolves identification of the important subset of pa-rameters in a model application, whereupon their nu-merical values are determined based on available data(Harmon and Challenor, 1997; Omlin et al., 2001).The ecological state, with uncertainty estimates, canthen be reconstructed from model runs using these es-timated parameters. Ecological parameter estimationmay be formally carried out using optimization meth-ods(Lawson et al., 1995; Vallino, 2000; Lehman et al.,2001), or from a Bayesian perspective(Dilks et al.,1992; Harmon and Challenor, 1997; Annan, 2001;Borsuk et al., 2001). Here, we consider not only pa-rameter estimation, but also the simultaneous joint es-timation of the time-varying ecological state, taking

account of information and uncertainty in models, pa-rameters and observations.

The state space representation provides a usefulframework for the ecosystem inverse problem. It iscomprised of a time-dependent dynamic model and anobservation equation; it includes stochastic elementsfor both model and parameter uncertainty, as wellas observation errors(Kitagawa and Gersch, 1996).Nonlinear ecosystem models and associated measure-ments, characterized by complex non-Gaussian errorstructures, are readily expressed as generalized statespace models. Simplifications are possible which leadto useful, but suboptimal, inverse techniques. For ex-ample, if the model and measurement equations areassumed linear with additive Gaussian errors, stateestimation algorithms such as the Kalman filter, andits associated fixed interval smoother, can be used(Jazwinski, 1970). Approximate nonlinear extensionsof these filters have been used for ecological models(Ennola et al., 1998; Natvik et al., 2001). In contrast,ignoring model uncertainy allows for use of efficientparameter estimation techniques based on optimiza-tion methods, which may be viewed as nonlinearregression(Thompson et al., 2000). Unfortunately,ecological models often do not satisfy the criterianecessary for these simplications and the effective-ness of such state estimation techniques is unclear(Vallino, 2000).

It has been argued that estimation of the ecologicalstate using models and data should rely on Bayesianstatistics(Dilks et al., 1992; Aldenberg et al., 1995;Steinberg et al., 1997; Harmon and Challenor, 1997).These approaches are based on a probabilistic specifi-cation of prior information pertaining to the ecologicaldynamics and model parameters, as well as specifica-tion of distributional assumptions for the observations.Bayesian application synthesizes this informationleading to posterior probability density functions de-scribing the parameters and state of the ecological sys-tem.Omlin and Reichert (1999)andQian et al. (2003)provide recent overviews of Bayesian methods usedfor parameter estimation in ecological modelling. Weoffer an extension of these methods applicable to moregeneral nonlinear, non-Gaussian ecological systemsand which jointly estimates the state and parameters.

This paper is structured as follows.Section 2in-troduces the theory and background to the Bayesianapproach for the ecosystem inverse problem. The

M. Dowd, R. Meyer / Ecological Modelling 168 (2003) 39–55 41

nonlinear, non-Gaussian state space model is used toprovide a flexible framework for incorporating non-linear ecosystem models and measurement operatorswith complex error processes. InSection 3, a simple,but not unrealistic, shellfish ecosystem model is ap-plied to illustrate how posterior state and parameterestimates can be determined from prior informationand observations under a variety of distributional as-sumptions. Stochastic simulation carried out in theabsence of data, is contrasted with the inverse ap-proach which includes measurements. Summary andconclusions follow inSection 4.

2. Model

Models of interacting populations in ecology fre-quently take the form of a nonlinear system of coupledordinary differential equations, i.e.

x = f(x; θ). (1)

In marine ecology, these include, for example, theprototypical phytoplankton–zooplankton–nutrientmodel(Steele and Henderson, 1981; Edwards, 2001),as well as the shellfish ecosystem models consideredin Section 3of this study(Raillard and Ménesguen,1994; Dowd, 1997; Chapelle et al., 2000). In theabove,x is a vector that represents the state of the sys-tem andx = dx/dt, wheret is time. The componentsof x are each identified with the populations compris-ing the ecological system. The functionf(x; θ) de-scribes the interactions between populations in termsof mathematical descriptions of the flow of matter orenergy. These fluxes are dependent on the populationlevels,x, as well as an underlying set of parameters,θ. The parameters contain information on biologicalrates and environmental forcing. Given initial con-ditions, x(0), and the parameters,θ, integration of(1) provides the time sequencex(t). This procedureis usually carried out numerically as the complexityand nonlinearity of most ecological models generallyprecludes analytic approaches. Numerical integrationmethods are based on the discretization of the govern-ing equations, i.e. converting the differential equationsto difference equations. This operation means that thestatex is now defined at a finite number of instancesin time. This discrete time state is designatedxt fort = 0, . . . , T .

Inverse problems are concerned with estimating thestate of a system by combining mathematical mod-els with measurements. They may be viewed as fit-ting models to data(Thacker and Long, 1988), oras data interpretation in the context of a dynamicalmodel. In the absence of inverse methods, a typicalmodelling exercise for estimating the state of the eco-logical system takes the following form. The forwardmodel(1) is integrated to produce predicted values forthe state. These predictions are retrospectively com-pared to available observations. Parameter values, orthe state variables themselves, are then adjusted so thatthe discrepancy between observations and predictionsis minimized. (If this ‘tuning’ exercise proves unsuc-cessful, the model structure itself may be modified.)One view of inverse methods is as a systematic meansof carrying out the above procedure. Such parameterestimation relies on optimization methods and has pro-vided the foundation for many current applications inecology(Lawson et al., 1995; Vallino, 2000; Lehmanet al., 2001). An alternative view of the inverse prob-lem is based on the Bayesian perspective. That is, weseek a (posterior) description of the ecological statebased on the synthesis of all available prior knowl-edge of the system, and information from the mea-surements. The perspective is one of a conjunction ofdifferent sources of information, rather than as an op-timization problem. Bayesian methods in ecologicalmodelling have to date considered mainly parameterestimation using data(Dilks et al., 1992; Aldenberget al., 1995; Harmon and Challenor, 1997; Borsuket al., 2001; Qian et al., 2003). Here, we investigate aBayesian approach to the ecosystem inverse problemwherein the joint posterior probability density func-tion of the time-varying ecological state and modelparameters are simultaneously determined under verygeneral assumptions about the underlying form of theerror processes.

Prior information on the ecological system takes theform of model dynamics and substantive knowledgeabout parameters. Each of these are subject to somelevel of uncertainty and therefore naturally describedin a probabilistic manner using the joint probabilitydensity function (pdf) of the state and parameters. Asobservations are subject to measurement error, the in-formation stemming from the data is given in the formof a sampling distribution, or the so-calledlikelihoodfunction. The overall goal is to update theprior pdf,


in light of new information obtained from the obser-vations, to obtain a jointposteriorpdf of the state andparameters. The only mathematically rigorous way toperform this update is through the Bayesian paradigm.It provides a way of formalizing the process of learn-ing from data so as to update prior beliefs in accor-dance with recent notions of knowledge synthesis. Theprimary advantage of this approach is a complete char-acterization of the ecological state as a random vari-able, i.e. through obtaining its full probability densityfunction. State estimates and measures of uncertaintycan then be derived from these quantities. These donot depend on the assumptions of asymptotic normal-ity underlying classical estimation methods. The fulldistributional profile of the state is provided, and anydeviations from normality can be easily detected. Themain difficulty is that for nonlinear and non-Gaussiansystems, a characteristic of most ecological systems,computationally intensive Monte Carlo based methodsmust be used. Adopting a sampling-based approachto posterior computation, however, has various advan-tages. In particular, it is easy to derive anymarginalposterior pdfs, or posterior pdfs of any transformedparameter of interest.

The basic framework for our probabilistic approachto inverse modelling is the nonlinear state spacerepresentation(Kitagawa and Gersch, 1996). Definezt = (x′t ,0′)′ as the augmented state vector(Kitagawa,1998), where prime (′) denotes matrix transpose. Thisincludes both the ecological state variables as well asthe model parameters, some of which may be timedependent. Available observations are given asyt . Anonlinear state space representation of this systemtakes the form

zt|zt−1 = g(zt−1, nt) (2)

yt|zt = h(zt, vt) (3)

for t = 1, . . . , T . Here, the notationA|B denotes theconditional distribution ofA givenB. The functionsg(zt−1, nt) andh(zt, vt) are possibly nonlinear func-tions of the state and the noise inputs. The first equa-tion represents the ecological dynamics. The functiong includes the discretized dynamics of(1) modifiedto account for the augmented state. It is assumed thata Markovian form is sufficient, wherein the state de-pends only on values at the previous time step. Thesystem noise term,nt , represents model errors and pa-

rameter uncertainty. The observation equation is givenby (3) and relates the observations to the state throughthe nonlinear operatorh. This serves the dual pur-pose of interpolating the measurements to the com-putational grid, and converting measured quantitiesto modelled variables. Observation noisevt has alsobeen included. The statezt and the observationsyt arerandom vectors and the representation(2) and (3)de-scribes a nonlinear and non-Gaussian system. It is anextension of the well-known linear state space model

xt|xt−1 = Gxt−1 + nt (4)

yt|xt = Hxt−1 + vt (5)

whereG andH are matrices, and additive Gaussianwhite noise sequencesnt ∼ Normal(0, Rt) andvt ∼Normal(0, St) are used.

For notational convenience, define the followingquantities:Y = (y′

1, . . . , y′T )

′ andZ = (z′1, . . . , z′T )

′.These contain the full state and complete observationssets for the period of interest. The quantityP(Z|Y)represents the posterior joint probability density func-tion of the stateZ conditioned on the observation setY , i.e. after taking the data into account. Knowledgeof P(Z|Y) provides a complete solution to the inverseproblem. It embodies all statistical information aboutthe state coming from the two sources: the observa-tions, and the prior knowledge of the ecological dy-namics and parameters.

Further insight into the inverse problem can begained by expandingP(Z|Y) using Bayes’ theorem,i.e.

P(Z|Y) = P(Y |Z)P(Z)P(Y)

(6)

P(Z|Y) ∝ P(Y |Z)P(Z). (7)

Now, consider each of the terms on the right-hand sideof (6):

1. P(Z) represents the prior pdf of the ecological state.It is based on prior knowledge of the model dy-namics and parameters. The Markov property ofthe dynamics(2) allows us to write

P(Z) = P(z0)

T∏t=1

P(zt|zt−1). (8)

This implies thatP(Z) is comprised of a product ofthe pdf of the initial conditions,P(z0), and all the


transition pdfs,P(zt|zt−1). These transition densi-ties are identified with the ecological dynamics(2).At time t, P(zt|zt−1) acts to transform the pdf ofthe ecological state at timet − 1, P(zt−1), to itsnew pdf at timet, P(zt).

2. P(Y |Z) is the joint pdf of the observations,Y ,conditioned on knowledge of the stateZ, or thelikelihood function. If theyt in (3) are condition-ally independent given the statezt , the observationEq. (3) implies

P(Y |Z) =T∏t=1

P(yt|zt). (9)

This term, therefore, incorporates the observationalerrors.

3. P(Y) is the marginal pdf of the observations. Itacts as a normalization factor for the numerator in(6). Expanding using conditional probability andmarginalization yields

P(Y) =∫ ∞

−∞P(Y |Z)P(Z)dZ. (10)

Evaluation of this term involves high dimensionalintegration (equal to the dimension of the stateZ). In the past, this was the main difficulty thathindered the application of Bayesian inference.For modern posterior computation using simula-tion techniques, this normalization constant is notrequired.

The above decomposition(7) of the quantities inBayes’ formula (ignoring the normalization constant)shows how the prior pdf of the ecological state is up-dated using the likelihood function to yield the pos-terior pdf of the state. The posterior pdf can thus beregarded as a synthesis of different sources of infor-mation.

If we make the assumption of a linear form for themodel and observations equations and assume addi-tive and independent Gaussian errors (i.e. as in(4)and (5)), the problem of evaluatingP(Z|Y) reducesto the time-dependent estimation problem of optimalsmoothing(Bryson and Ho, 1969; Jazwinski, 1970).For this case, the pdf of the state is Gaussian and com-pletely characterized by its mean and covariance. Thesolution is provided by application of the Kalman fil-ter and associated fixed interval smoother. Unfortu-nately, most ecological models and measurements fall

into the category of nonlinear and non-Gaussian sys-tems, with conditional distributions that are no longernormally distributed. Various approximations to, or as-sumptions on, the pdfs are used to obtain recursiveformulae for state estimation. Some of the examplesare the extended Kalman filter(Jazwinski, 1970), theGaussian-sum filter(Alspach and Sorenson, 1972), thedynamic generalized linear model(West et al., 1985),and the non-Gaussian filter and smoother(Kitagawa,1987; Hodges and Hale, 1993). However, these ex-tensions are not always effective. Optimization meth-ods are similarly restricted, estimating only the meanof the state using, say, a variational framework whichminimizes the weighted sum of squares of the errors(Bryson and Ho, 1969; Bennett, 1992). The generalsolution forP(Z|Y) for nonlinear and non-Gaussiansystems relies on Monte Carlo methods(Kitagawa,1987).

Markov Chain Monte Carlo (MCMC) methods havebecome well established for practical applications ofBayesian methods(Gelman et al., 1995). MCMC tech-niques are general Monte Carlo methods that approxi-mate the generation of samples from the posterior pdfP(Z|Y) when this distribution is so high-dimensionaland complex that it cannot be sampled directly. Thebasic idea of MCMC is to generate a Markov chainwith a stationary (or ‘limiting’ or ‘long run’) distribu-tion P(Z|Y). This idea of using the limiting behaviorof a Markov chain came almost as early as the orig-inal Monte Carlo technique, at least in the particlephysics literature(Metropolis et al., 1953), but it re-quired computational power that was not available inthose early days. Two major techniques devised to cre-ate Markov chains with the desired stationary distri-bution are Gibbs sampling(Geman and Geman, 1984;Gelfand and Smith, 1990)and the Metropolis–Hastingalgorithm (Metropolis et al., 1953; Hastings, 1970).The interested reader is referred toGilks et al. (1996),Gamerman (1997), andRobert and Casella (1999)formore information on the theory and application ofMCMC methods. In the ecological modelling litera-ture,Harmon and Challenor (1997)provide an appli-cation to ecological parameter estimation, andQianet al. (2003)further discusses implemetation issues.For the user, important practical concerns include: en-suring convergence of the Markov chain to stationar-ity, and making sure that the sampled portion of theMarkov chain has traversed the entire parameter space


(or achieved ‘mixing’). Generally, a ‘burn-in’ period isallowed for the system to adjust prior to drawing sam-ples, followed by careful assessment of convergencediagnostics as suggested inCowles and Carlin (1996).This ensures that samples drawn from the chain aretruly representative ofP(Z|Y). Nonlinear state spacemodels were first examined byCarlin et al. (1992)andFruehwirth-Schnatter (1994)using the Gibbs sampler.An overview of more advanced MCMC techniques forsequential state space modelling is given inDoucetet al. (2001).

In this study, we make use of MCMC methods asimplemented in the Bayesian Inference Using GibbsSampling (BUGS) software(Spiegelhalter et al.,1995). BUGS is available free of charge from

www.mrc-bsu.cam.ac.uk/bugs

for the operating systems UNIX, LINUX, and WIN-DOWS, among others. It comes with complete doc-umentation and two example volumes. BUGS hasbeen used for a Bayesian approach to nonlinearnon-Gaussian state space models in various differentdisciplines, such as fisheries stock assessment(Meyerand Millar, 1999; Millar and Meyer, 2000), econo-metrics(Meyer and Yu, 2000), and chaotic dynamicalsystems in physics(Meyer and Christensen, 2000).Our goal here is to assess the utility of a Bayesianapproach to inverse modelling in the context of asimple, but not unrealistic, coastal marine ecosystemmodel focused on shellfish growth.

3. Application

3.1. Shellfish ecosystem model

The Bayesian inverse framework was applied toa shellfish aquaculture ecosystem model. This is asimplified version of the ecosystem model inDowd(1997) which described a coastal marine ecosystemin a tidal inlet near Lunenburg, Nova Scotia, Canada.The focus is on the prediction of mussel growth andcarrying capacity on seasonal time scales in the con-text of its supporting ecosystem. The physical situa-tion is one of a semi-enclosed tidal inlet exchangingwater with the adjacent open ocean. A conceptualdiagram of the box model is shown inFig. 1. Thegoverning equations for the ecosystem box model are

M

P D

P∞ D∞

Fig. 1. Conceptual diagram for the shellfish ecosystem box model.Within the box, mussels,M, feed on plankton,P , and detritus,D. The freely floatingP andD are exchanged with their far-fieldP∞ andD∞ via water motion and mixing.

given by the following system of ordinary differentialequations:

M = (εfII − fRR)M (11)

P = −αPfI IMN +K(P∞ − P) (12)

D = −αDfI IMN +K(D∞ −D). (13)

Here, the biomass of an individual mussel is givenbyM, with N representing the number, or density, ofmussels. The concentration of phytoplankton and de-tritus is given byP andD, respectively. Their corre-sponding far-field values areP∞ andD∞. The modelcurrency is dry weight carbon (grams carbon or gC).Table 1summarizes the variables and parameters ofthe model. Scaling analysis of the more comprehensivemodel inDowd (1997)identifies the models(11)–(13)as containing the major ecological interactions im-portant at higher mussel densities, a conclusion alsosupported by the mussel growth models ofRoss andNisbet (1990). In general terms, this nonlinear modeldescribes predator (M) and prey (P andD) interac-tions in an open system(Nisbet et al., 1997). Morespecifically, it is an ecophysiological model of shell-fish culture of the type used to assess ecosystem ef-fects, bivalve growth, and carrying capacity(Raillardand Ménesguen, 1994; Dowd, 1997; Chapelle et al.,2000).

The time rate of change in the weight ofM isgiven by (11). This is measured as the net of gainsthrough ingestion and losses due to respiration, and the

http://www.mrc-bsu.cam.ac.uk/bugs/welcome.shtml


Table 1Defintion of quantities used in the shellfish ecosystem model,grouped according to variable type. For each quantity, the followinginformation is given: units, its numerical value (or source), and abrief definition. See text for more details

Quantity Units Value Definition

StateM gC Eq. (11) Individual mussel weightP gC m−3 Eq. (12) Phytoplankton biomassD gC m−3 Eq. (13) Detritus biomass

ParametersN m−3 1 Number (density) of musselsK day−1 0.1 Exchange/flushing coefficientkM gC m−3 1 Half-saturation constant for

ingestionI day−1 0.1 Ingestion rateR day−1 0.01 Respiration rateQ10

◦C−1 0.05 Physiological temperatureeffect

εP – 0.9 Assimilated fraction foringestedP

εD – 0.2 Assimilated fraction foringestedD

External inputsP∞ gC m−3 Fig. 2a Far-fieldP biomassD∞ gC m−3 Fig. 2b Far-fieldD biomassTemp ◦C Fig. 2c Temperature

Modulating functionsfI – Eq. (14) Modulation function forM

ingestionfR – Eq. (14) Modulation function forM

respiration

Derived parametersε(t) – Eq. (15) Assimilated fraction of

ingested rationαP(t) – Eq. (16) Fraction of plankton in sestonαD(t) – Eq. (16) Fraction of detritus in seston

quantity in brackets represents the scope for growth.The reference rates for ingestion and respiration aregiven byI andR, respectively. These rates are mod-ulated by the dimensionless functions

fI = exp{Q10 × Temp}(

P +DkM + P +D

),

fR = exp{Q10 × Temp}. (14)

BothfI andfR include a temperature effect controlledby the physiological rate constantQ10. The modula-tion function for ingestionfI also includes a satura-

tion response to increasing food levels (a Holling typeII functional response) and is controlled by the rateconstantkM . The ingested ration in(11) is scaled bya time-varying assimilation efficiencyε which mea-sures the fraction of food ingested which is availablefor growth. It is given by

ε = εPαP + εDαD (15)

whereεP andεD are the assimilation efficiencies formussels grazing onP andD, respectively. The fractionof P andS in the water at any time are approximatedby

αP = P∞P∞ +D∞

, αD = D∞P∞ +D∞

. (16)

The time rate of change of the freely floatingPandD components are described by(12) and (13),respectively. Exchange of these components with theadjacent open ocean is proportional to the concentra-tion difference in the state variables between the box(P,D) and the far-field (P∞,D∞), scaled by an ex-change coefficientK. This gradient–flux relation actsto coupleP andD to their far-field values in this opensystem. BothP andD populations are reduced by thegrazing activity of the mussel population (the prod-uct ofM andN) in direct proportion to their relativeabundance in the environment.

A baseline simulation for the model was con-structed as follows. The governingEqs. (11)–(13)were discretized using a Euler method (e.g.Boyceand DiPrima, 1986, Section 8.2). This results in aMarkovian form for the resulting nonlinear differenceequations. The time-dependent inputs (P∞, D∞, andtemperature) are based on idealized annual cycles andare shown inFig. 2. The remaining parameters re-quired for the simulation, and their numerical values,are given in the second grouping inTable 1. The modelwas constructed to have a time step of one week, witha view to expediting the Monte Carlo computationsat the heart of the probabilistic methods. Simulationruns covered one calendar year, i.e. from January 1to December 31. Initial conditions forP andD were0.03 and 1.2 gC m−3, respectively. These match theircorresponding far-field concentrations,P∞ andD∞,at that time. Initial mussel weight takes a value of0.1 gC (25 mm shell length). Further rationalizationfor these values can be found inDowd (1997).


0 10 20 30 40 500

0.1

0.2

time (weeks)

P∞

(gC

m−3

)

0 10 20 30 40 500.5

1

1.5

time (weeks)

D∞

(gC

m−3

)

0 10 20 30 40 500

5

10

15

20

time (weeks)

Tem

p (o C

)

(a)

(b)

(c)

Fig. 2. Inputs for the shellfish ecosystem model in the form of ide-alized annual cycles for: (a) phytoplankton far-field concentration,P∞, (b) detritus far-field concentration,D∞, and (c) temperature.The time axis covers the calendar year.

Fig. 3shows results from the baseline deterministicsimulation. The state variablesP andD are tightlycoupled to their far-field values due to the exchangeof water between the bay and open ocean. As the yearprogresses, they become increasingly depressed be-low these far-field values as individual mussel weightincreases, with a corresponding increase in the totalmussel biomass, resulting in enhanced grazing ofP

and D. Note that at the end of the yearP nearlyequilibrates withP∞ through exchange processes; itbecomes a small fraction of the total seston, and tem-perature effects slow mussel ingestion. The predicted

0 10 20 30 40 500

0.2

0.4

time (weeks)

M (

gC)

MM

obs

0 10 20 30 40 500

0.1

0.2

time (weeks)P

(gC

m−3

)

PP∞

0 10 20 30 40 500

0.5

1

1.5

time (weeks)

D (

gC m

−3)

DD∞

(a)

(b)

(c)

Fig. 3. Baseline results from the deterministic simulation of theecosystem model. (a) Predicted (solid line) and observed (trian-gle) individual mussel weightM. (b) SimulatedP cycle (solidline), with the far-fieldP∞ shown for reference (dashed line). (c)SimulatedD cycle (solid line), with the far-fieldD∞ shown forreference (dashed line). The time axis covers the calendar year.

annual trajectory of individual mussel growth is givenin Fig. 3a. Superimposed on this curve are observa-tions of mussel biomass based on field sampling of anensemble of individuals(Grant et al., 1993). After oneyear, mussel weight increases from its starting valueto about 0.4 gC (60 mm shell length). The simulatedmussel growth trajectory is generally consistent withthe observations, but is smoother and under-predictsthe observed changes occuring near the end of the year.

3.2. Stochastic simulation

Stochastic simulation predicts the evolution of theprobability density of the state variables. It takes into


account prior knowledge of the model and parame-ters, without reference to the measurements. That is,the state space model (2) is used to determine the jointpdf of the state,P(Z). According to(8), P(Z) is madeup of the pdf of the initial state transformed throughmultiplication by the time sequence of the varioustransition probabilities associated with the ecologicaldynamics. Monte Carlo simulation is used to approx-imateP(Z). This involves generating a large numberof realizations of the time trajectories of the state vari-ables, after which the results are ensemble averaged.A useful analogy is the generation of particle pathsaccording to a random walk, which is also a simpleMarkov process. In the continuum limit, the governingLangevin equation gives rise to an advection-diffusion,or Fokker–Planck, equation describing the time evo-lution of the pdf of particle position (e.g.Gardiner,1997).

The discrete version of our ecosystem model(11)–(13) is written as a nonlinear stochastic differ-ence equation, i.e.

xt = g∗(xt−1, θ)enxt (17)

wherext = (Mt, Pt,Dt)′. The vector functiong∗ con-

tains the ecosystem dynamics and is functionally de-pendent on the state at the previous time,xt−1, and theparameters,θ. A multiplicative error term exp{nxt } hasalso been included on the right-hand side of(17)to ac-count for discretization errors, as well as system noise.It is assumed that thenxt are zero-mean, normally dis-tributed random vectors whose components are mutu-ally uncorrelated and independent in time. A constantvarianceσ2

x is also assumed. This lognormal form forthe multiplicative system noise ensures that the statevariables remain non-negative. It also means that thevariance ofMt , Pt , andDt scales with their respectivetime-varying mean population level. This feature hasbeen observed for both phytoplankton(Dowd et al.,2003)and individual shellfish weight(Gangnery et al.,2001). Here, we assume thatσ2

x = 0.05 gC2 m−6.A subset of the model parameters,θ, in (17) are

also considered to be random variables. These havean associated noise processnθ, considered to beindependent ofnxt . For the purposes of this study, pa-rameter uncertainty is assumed to reside only in themussel ingestion and respiration rates,I andR, so thatθ = (I, R)′. Together, these quantities set the musselenergy balance, or scope for growth. The remaining

parameters were treated as deterministic and take onconstant values. The parameters are defined on a finiteinterval [θmin, θmax], such thatθ = θmin + nθ wherenθ = (θmax − θmin)β(a, b). Here,β(a, b) is the stan-dard beta distribution with parametersa andb. In thisapplicationnθ = (nI, nR)′, and are thus defined withrespect to the parametersI andR. For I, the range is[0, 0.15] and the distributionβ(5,2) is used. ForR,the range is [0, 0.035] and the distributionβ(2,5) isused. The range for possible parameter values is ratio-nalized based on literature values. The prior pdfs forI andR are skewed to the left and right, respectively.This assumes that ingestion is generally near its ex-pected value of 0.1 day−1, but may take on low valuesif grazing activity is reduced or suspended(Cranfordand Hill, 1999). The expected value of respiration is0.01 day−1 but in situations of physiological stressit may take on much higher values(Griffiths andGriffiths, 1987).

As in Section 2, the state vector can be redefinedsuch thatzt = (x′t ,0′)′, thereby mapping the abovemodel system into standard state space form(2). Thenoise term in(2) is modified such thatnt = (nx, nθ)′t .To implement this nonlinear state space model, ini-tial conditions forM, P , andD, and the parametersI andR are set following the baseline deterministicsimulation. These represent expected values for theirassociated random variables. The required prior in-formation on the ecological dynamics and parametershave been completely specified. Stochastic simulationis used next to determine the prior pdf,P(Z), of thestate.

Probability distributions and ensemble statisticswere computed for the state variablesM, P andDusing the shellfish ecosystem model. It was found thatapproximately 104 independent realizations of thetime series of the ecological state variables providedsufficiently stable statistics to compute the desiredestimates. Each of the independent model runs re-quired drawing a sample value for the system noiseand parameters at each time step during model inte-gration. The stochastic simulation was implementedusing the MATLAB software. (BUGS software wasalso tested to ensure consistency in the model im-plementation. However, many more model runs areneeded to establish viable statistics with BUGS sinceconvergence and mixing of the Markov chains wasvery slow for this measurement free case. This is


0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

time (weeks)

M (

gC)

0 10 20 30 40 500

0.05

0.1

0.15

0.2

0.25

time (weeks)

P (

gC m

−3)

0 10 20 30 40 500

0.5

1

1.5

time (weeks)

D (

gC m

−3)

0 6M (x10−1)

0

30prob dens

0

52

time

0 3P (x10−1)0

100

prob dens

0

52

time

0 15

D (x10−1)

0

10prob dens

0

52

time

(a) (b)

(d)(c)

(e) (f)

Fig. 4. Results from the stochastic simulation of the ecosystem model for the state variablesM, P , andD. The panels on the left (a, c,and e) show the time sequence of the median (open circles) and the 95% confidence interval. The panels on the right (b, d, and f) showthe time evolution of the probability density functions.


due to the high temporal correlations amongst suc-cessive components ofZ which are imposed by theMarkovian ecosystem model and which make Gibbssampling less efficient.) Results from this stochasticsimulation are given inFig. 4. These are reported as(i) time series of the median of the state along withits 95% confidence intervals (Fig. 4a, c, and e), and(ii) the time evolution of the pdf for each of the statevariables (Fig. 4b, d, and f).

The time series of median mussel weight,M, hasslightly greater values than the corresponding resultsfrom the baseline deterministic simulation. This re-flects the use of distinct beta distributions as priorsfor the parametersI andR. Ingestion, therefore, tendsto be inflated relative to respiration, leading to in-creasedM growth. The 95% confidence interval gen-erally widens with increasing time. This results fromthe accumulation of uncertainty inM due to parameteruncertainty (the scope for growth computation entersthe state multiplicatively at every time step). A lessercontribution is also due to the system noise variancefor mussel growth which increases in magnitude asM increases. The time evolution of the pdf ofM alsoclearly shows rapid spreading of the variance. As thesimulation progresses, the higher order moments of thepdf of M, skewness and kurtosis, increase with time.The pdfs are not Gaussian, which is not surprisinggiven the nonlinear model and the use of beta distribu-tions for the parameters and multiplicative lognormalsystem noise. It is also interesting to note that theseerror bars resemble those from mussel growth trajec-tories determined by a population dynamics model ofinter-individual variability(Gangnery et al., 2001).

For both phytoplankton,P , and detritus,D, thetime evolution of the median follows closely the base-line deterministic simulation. The variance generallyscales with the level ofP , reflecting the properties ofthe system noise variance. However, the variance ofD, as well as its skewness and kurtosis, increases withtime. This feature results from mussel grazing activ-ity. That is, during weeks 0 to 20,P andD levels aredetermined mainly by exchange and their distributionis set by the lognormal system noise. After this period,M grazing begins to affect these variables; the increas-ing variance inM acts to increase the uncertainty inDas these populations co-evolve. This effect is also seenin P during weeks 20 to 40. The differential responseof the uncertainty inP andD to mussel grazing is

due to the strength of the coupling between these foodsources and grazing activity. Clearly, the ecologicallinkages in the dynamics provide a means to couplethe co-evolution of the pdfs of the state variables.

3.3. The inverse model

Inverse methods blend the information contained inthe model dynamics and parameters with available ob-servations in order to refine estimates for the state ofthe ecological system. This idea was illustrated froma probabilistic perspective using Bayes’ theorem(6).In Section 2, it was demonstrated that the goal for theinverse problem is to evaluateP(Z|Y), the joint pos-terior density of the state and parameters after takingthe observations into account. Inverse methods updatethe priorP(Z), as specified by (2) and sampled viastochastic simulation in the previous section, to yieldthe posteriorP(Z|Y) by using the observation in (3).Here, we make use of observations of mussel weight,

0 0.05 0.1 0.15 0.20

5

10

15

20

25

I (day−1)

prob

abili

ty d

ensi

ty

PriorPosterior

0 0.01 0.02 0.03 0.040

50

100

150

200

250

R (day−1)

prob

abili

ty d

ensi

ty

PriorPosterior

(a)

(b)

Fig. 5. Prior (dashed line) and posterior (solid line) probabilitydensity functions for the parametersI (a) andR (b).


0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

time (weeks)

M (

gC)

0 10 20 30 40 500

0.05

0.1

0.15

0.2

0.25

time (weeks)

P (

gC m

−3)

0 10 20 30 40 500

0.5

1

1.5

time (weeks)

D (

gC m

−3)

0 6M (x10−1)

0

30prob dens

0

52

time

0 3P (x10−1)0

70

prob dens

0

52

time

0 15

D (x10−1)

0

7prob dens

0

52

time

(a)

(c)

(e) (f)

(b)

(d)

Fig. 6. Results from application of the Bayesian inverse method for the state variablesM, P , andD. The panels on the left (a, c, and e)show the time sequence of the posterior median (open circles) and the 95% confidence interval. The panels on the right (b, d, and f) showthe time evolution of the posterior probability density functions.


M, as given inFig. 3a. Note that these are direct mea-surements of a single state variable. Hence, the ob-servation operator in(3) is defined quite simply: it isnon-null only when associated with the state variableM, and at times when observations are available. Theobservation errorvt in (3) is taken to be additive. Itis assumed to be an independent and identically dis-tributed normal random variable with zero mean andvarianceσ2

v = 10−4 gC2. This implies that the musselmeasurements are relatively precise.

MCMC methods were used to evaluate the poste-rior densityP(Z|Y). Implementation used the BUGSsoftware(Spiegelhalter et al., 1995). Approximately5×104 quasi-independent random samples ofP(Z|Y)were drawn from the simulated Markov Chain andused to construct the estimates for the posterior pdfsof the state and parameters. Prior to drawing thesesamples, a burn-in period of 5× 103 iterations (ormodel runs) were performed to ensure convergenceof the chain to stationarity, or a statistical steadystate. After this, 5× 105 iterations were computedwith samples drawn every 100 iterations, which wasthe approximate decorrelation time of the Markovchain. (These calculations took approximately 10 hon a 900 MHz Intel Pentium III PC.) To ensure ro-bustness and consistency in the results of the MonteCarlo integration, multiple realizations of the chainwere generated using different starting values for therandom variables, as well as different random numberseeds. The reader is referred toGilks et al. (1996)for more information on the setup and diagnosis ofMCMC simulations. Computational requirements arediscussed further inSection 4.

Fig. 5 shows estimates for the posterior pdfs of theparametersI andR, along with their associated priors.Note that the solution has constrained the posteriorpdfs to lie in the same interval as their respective pri-ors. The posterior pdf for the ingestion rate,I, closelyresembles the prior pdf, with only a slight shift in themode of the distribution. Assuming the posterior fol-lows a beta pdf, the distributional parameters are es-timated asβ(8.4,2.3), as compared toβ(5,2) for theprior. The mean and variance differ only slightly. Incontrast, the posterior pdf for the respiration rate,R,differs significantly from its prior. The mean for theposterior and prior are both near 0.01 day−1. How-ever, the variance of the posterior (3.5× 10−6 day−2)is reduced relative to the prior (3.1 × 10−5 day−2).

Most importantly, the shape of the posterior is moresymmetric compared to the prior. The additional infor-mation associated with the observations ofM have al-lowed refinement of the posterior pdf of the respirationparameter, but left the pdf of the ingestion rate almostunchanged. The inverse method ensures consistencybetween observed and predictedM by altering the res-piration terms in the scope for growth computation.

Fig. 6 shows the time evolution of the posterior es-timates for the state variablesM, P , andD. These arereported in terms of time series of the median of thestate variables along with their 95% confidence inter-vals (Fig. 6a, c, and e), as well as in terms of the timeevolution of the pdf for each of the state variables(Fig. 6b, d, and f). The posterior pdf ofM appearsto be nearly Gaussian with a time-dependent meanlevel and relatively small and constant variance. Thisis in sharp contrast to the corresponding results fromthe stochastic simulation, where variance spreadingwas strongly evident. The additional information con-tained in theM observations have constrained theposterior density to be much closer to the measure-ments, which is consistent with the assumption of asmall measurement error variance. Another feature ofnote is the growth in the variance ofM in the obser-vation void near mid-record. The envelope of 95%confidence is widest at the greatest distance awayfrom the observations; deviations from normality arealso most pronounced here. The median values forP

andS are similar to those from the stochastic simu-lation, except for differences near the end of the year.The additional information in theM observations hassubstantially reduced the overall variance of bothPandD during the latter half of the simulation period,when these food sources are being strongly influencedbyM grazing activity. Non-normality in the posteriorpdfs is generally lessened compared to the corre-sponding results from the stochastic simulation. TheBayesian inverse method appears to be an effectivemeans to blend the measurement information with anecosystem model to produce posterior estimates ofthe ecological state and model parameters.

4. Summary and conclusions

This study has investigated a probabilistic, orBayesian, approach to the ecosystem inverse problem.


The emphasis is on time-dependent models of inter-acting populations, such as those used in marine ecol-ogy. The goal is to combine (i) prior information fromthe model dynamics and substantive knowledge aboutunknown parameters with (ii) information stemmingfrom the observations. The use of inverse methods inecological modelling is developing rapidly. Applica-tions have focused mainly on parameter estimationusing both nonlinear optimization methods(Vezinaand Pace, 1994; Lawson et al., 1995; Vallino, 2000;Lehman et al., 2001), as well as Bayesian techniques(Dilks et al., 1992; Aldenberg et al., 1995; Steinberget al., 1997; Harmon and Challenor, 1997; Annan,2001; Borsuk et al., 2001). This study provides anextension of these methods by considering the si-multaneous joint estimation of both the time-varyingecological state and model parameters. Nonlinearand non-Gaussian ecological systems are treated herefrom a probabilistic, or Bayesian, perspective; a view-point well supported by the statistical time seriesliterature(Kitagawa, 1987; Carlin et al., 1992). Thestate space representation provides us with a flexi-ble framework for considering nonlinear ecologicalmodels and measurement operators having very gen-eral error processes, such as those with non-normaldistributions and entering the system multiplicatively.The end result of our Bayesian inverse approach isa complete characterization of both the time-varyingecological state and the models parameters throughobtaining posterior estimates for their pdfs.

The ideas behind the Bayesian ecosystem inverseapproach were illustrated using a simple biophysicalshellfish ecosystem model describing the co-evolutionof a mussel population and its food sources: plank-ton and detritus. This model is of the type usedto investigate coastal ecosystem effects of shellfishaquaculture and assess carrying capacity(Raillardand Ménesguen, 1994; Dowd, 1997; Chapelle et al.,2000). Application of the inverse method extendsthe utility of this subclass of ecological modelswhose predictive skill has thus far been limited byparameter uncertainty(Dowd, 1997). The nonlineartime-dependent ecosystem model was cast in a statespace framework, making use of an augmented statevector which included both the state variables, as wellas the model parameters(Kitagawa, 1998). The spec-ification of prior information is a contentious issuefor Bayesian applications. Our choices are based on

both parsimony and biological realism, as well as adesire to illustrate the flexibility of the method. Mul-tiplicative non-Gaussian system noise was used in thediscretized model equations. This took the form ofa lognormal distribution which acted to ensure thatpopulations remained non-negative. A subset of themodel parameters were also considered as randomvariables. These were restricted to non-negative inter-vals consistent with their literature values, and theirprior pdfs were based on beta distributions. Stochasticsimulation allowed us to first determine the time evo-lution of the prior pdf of the state, without referenceto any observations. A set of available measurementson one ecosystem component (mussels) was thenused to solve the inverse problem using Bayesianinference. In this manner, the time-varying posteriorpdfs of both the ecological state and parameters weresimultaneously determined. It was shown that pos-terior estimates of the parameters and state were allinfluenced by the information in the measurements,as dictated by the strength of their dynamic coupling.The application demonstrated how ecological dynam-ics link not only the mean state, but also quantitiesrelated to the uncertainties (via their joint pdfs).

While the theory of Bayesian inverse methods isstraightforward, practical applications in ecologicalmodelling must consider a number of factors, suchas the specification of priors for uncertain quantities(Qian et al., 2003). However, a key element, as withmany problems in ecology, is the complexity of thedynamic model considered(Wu and Marceau, 2002).In the context of inverse problems, the influenceof complexity is manifested in terms of the dimen-sionality of the forward model. This factor, togetherwith the number of random variables, determines ourability to estimate the ecological state and parame-ters using Monte Carlo based simulation techniques.Applications of Bayesian methods to ecosystem mod-els have to date been focused on the estimation ofrelatively small number of parameters(Omlin andReichert, 1999; Borsuk et al., 2001; Qian et al., 2003).In fact, Harmon and Challenor (1997)concluded thatit was impractical to determine second order proper-ties for more than about 10 parameters in their model.In this study, Bayesian posterior computation was per-formed using the Gibbs sampler and proved suitablefor determining the posterior density of 158 variables(156 associated with the time-varying state and 2 for


the parameters). The use of a simple, time-dependentecosystem box model with careful attention given tocontrolling the dimensionality clearly facilitated theapplication. However, further computational savingsare possible. For example,MacEachern and Berliner(1994)suggest sampling the Markov chain at intervalsless than its decorrelation timescale (as was used inour application). This implies many fewer iterationsof the model are needed, with consequent computa-tional savings. The shellfish ecosystem model usedhere, with minimal extensions such as multiple boxesand additional ecosystem components, can providea realistic description of a coastal marine ecosys-tem (Dowd, 1997), and yet still prove amenable toBayesian inverse methods.

The trend in ecosystem models of marine systemsis towards increasing complexity and high dimension-ality through their coupling with water circulationmodels (Hofmann and Lascara, 1998). Specifically,ecosystem models are incorporated as source andsink terms in the tracer equations of fluid dynamicalmodels(Huthnance et al., 1993). Pastres et al. (2001)provide an example for the case of a biophysicalmodel of a shellfish aquaculture ecosystem. The ex-tent to which probabilistic techniques can be adaptedto these very high dimension spatio-temporal modelsis unclear. It is suggested that the use of Bayesianinverse methods with complex ecological modelsmay require the development of specifically tailoredMCMC algorithms that are more efficient than theGibbs sampler. It is also encouraging to note thatcomplex, high-dimension physical models of oceanand atmospheric circulation have successfully adoptedapproximate methods based on probabilistic MonteCarlo methods(Evensen, 1994). These techniques arecurrently being adapted to marine ecological systems(Natvik et al., 2001). It is still an open question asto whether simplified ecological models with sophis-ticated inverse methods, such as our Bayesian ap-proach, prove to be more effective than simple inversemethods used with very complex dynamical models.

In summary, our Bayesian inverse approach for in-tegrating ecosystem models with observations appearspromising. Inverse methods are expected to becomewidely used as new, automated observing technologiesare developed and used in conjunction with ecologi-cal models. They offer great potential for improvingpredictions of the time evolution of biological popula-

tions. Another important use of inverse techniques isto provide guidance for the refinement of the mathe-matical structure of ecological models, or the problemof system identification(Omlin and Reichert, 1999).Bayesian analysis is particularly well suited to thisrole, having the capability to make inferences on bothparameters and the state, as well as to carry out hypoth-esis testing. However, a great deal more work must becarried out to assess the feasibility of Bayesian inversemethods as applied to ecological models. It is hopedthat this study provides stimulation for the further de-velopment of novel inverse techniques appropriate forecosystem models.

Acknowledgements

The authors would like to thank Dr. W. CarlisleThacker of the NOAA Atlantic Oceanographic andMeteorological Laboratory for stimulating discussionson the role of Bayesian methods in data assimila-tion. This work was supported by Fisheries & OceansCanada and the Royal Society of New Zealand Mars-den Fund. The authors also thank Dr. Svetlana Losa ofthe Bedford Institute of Oceanography for reviewingthe manuscript.

References

Aldenberg, T., Janse, J.H., Kramer, P.R.G., 1995. Fitting thedynamics model PCLake to a multi lake survey using Bayesianstatistics. Ecol. Model. 78, 83–90.

Alspach, D.L., Sorenson, H.W., 1972. Nonlinear Bayesianestimation using Gaussian sum approximations. IEEE Trans.Autom. Control AC-17, 439–448.

Annan, J.D., 2001. Modelling under uncertainty: Monte Carlomethods for temporally varying parameters. Ecol. Model. 136,297–302.

Batchelder, H.P., Van Keuren, J.R., Vaillancourt, R., Swift,E., 1995. Spatial and temporal distributions of acousticallyestimated zooplankton biomass near the Marine Light-MixedLayers station (59 degree 30′N, 21 degree 00′W) in the NorthAtlantic in May 1991. J. Geophys. Res. 100 (C4), 6549–6563.

Bennett, A.F., 1992. Inverse Methods in Physical Oceanography.Cambridge University Press, New York, 346 pp.

Borsuk, M.E., Higdon, D., Stow, C.A., Reckhow, K.H., 2001. ABayesian hierarchical model to predict benthic oxygen demandfrom organic matter loading in estuaries and coastal zones.Ecol. Model. 143, 165–181.

Boyce, W.E., DiPrima, R.C., 1986. Elementary Differential Equa-tions and Boundary Value Problems. Wiley, New York, 654 pp.


Bryson, A.E., Ho, Y., 1969. Applied Optimal Control. Blaisdell,Waltham, 481 pp.

Carlin, B.P., Polson, N.G., Stoffer, D.S., 1992. A Monte Carloapproach to nonnormal and nonlinear state space modelling. J.Am. Stat. Assoc. 87, 493–500.

Chapelle, A., Menesguen, A., Deslous-Paoli, J.-M., Souchu, P.,Mazouni, N., Vaquer, A., Millet, B., 2000. Modelling nitrogen,primary production and oxygen in a Mediterranean lagoon.Impact of oysters farming and inputs from the watershed. Ecol.Model. 127, 161–181.

Cowles, M.K., Carlin, B.P., 1996. Markov Chain Monte Carloconvergence diagnostics: a comparative review. J. Am. Stat.Assoc. 91, 883–904.

Cranford, P.J., Hill, P.S., 1999. Seasonal variation in foodutilization by the suspension-feeding bivalveMytilus edulisandPlacopeten magellanicus. Mar. Ecol. Prog. Ser. 190, 223–239.

Cullen, J.J., Ciotti, A.M., Davis, R.F., Lewis, M.R., 1997. Opticaldetection and assessment of algal blooms. Limnol. Oceanogr.42, 1223–1239.

Dilks, D.W., Canale, R.P., Meier, P.G., 1992. Development ofBayesian Monte Carlo techniques for water quality modeluncertainty. Ecol. Model. 62, 149–162.

Doucet, A., de Freitas, N., Gordon, N., 2001. Sequential MonteCarlo Methods in Practice. Springer, New York, 581 pp.

Dowd, M., 1997. On predicting the growth of cultured bivalves.Ecol. Model. 104, 113–131.

Dowd, M., Martin, J.L., LeGresley, M.M., Hanke, A., Page,F.H., 2003. Interannual variability in a plankton time series.Environmetrics 14 (1), 73–86.

Edwards, A.M., 2001. Adding detritus to a nutrient-phytoplankton-zooplankton model: a dynamical systems approach. J. PlanktonRes. 23 (4), 389–413.

Ennola, K., Sarvala, J., Devai, G., 1998. Modelling zooplanktonpopulation dynamics with the extended Kalman filteringtechnique. Ecol. Model. 110, 135–149.

Evensen, G., 1994. Sequential data assimilation with a nonlinearquasi-geostrophic model using Monte Carlo methods to forecasterror statistics. J. Geophys. Res. 99, 10143–10162.

Fruehwirth-Schnatter, S., 1994. Data Augmentation and DynamicLinear Models. J. Time Ser. Anal. 15, 183–202.

Gamerman, D., 1997. Markov Chain Monte Carlo: StochasticSimulation for Bayesian Inference. CRC Press, Boca Raton,264 pp.

Gangnery, A., Bacher, C., Buestel, D., 2001. Assessing theproduction and impact of cultivated oysters in the Thau lagoon(Mediterranee, France) with a population dynamics model. Can.J. Fisheries Aquat. Sci. 58, 1012–1020.

Gardiner, C.W., 1997. Handbook of Stochastic Methods. Springer,Berlin, 442 pp.

Gelfand, A.E., Smith, A.F.M., 1990. Sampling-based approachesto calculating marginal densities. J. Am. Stat. Assoc. 85, 398–409.

Gelman, A., Carlin, J., Stern, H.S., Rubin, D.B., 1995. BayesianData Analysis. CRC Press, Boca Raton, 552 pp.

Geman, S., Geman, D., 1984. Stochastic relaxation, Gibbsdistributions and the Bayesian restoration of images. IEEETrans. Pattern Anal. Mach. Intell. 6, 721–741.

Gilks, W.R., Richardson, S., Spiegelhalter, D.J. (Eds.), 1996.Markov Chain Monte Carlo in Practice. Chapman and Hall,London.

Grant, J., Dowd, M., Thompson, K.R., Emerson, C., Hatcher. A.,1993. Perspectives on field studies and related biological modelsof bivalve growth. In: Dame, R. (Ed.), Bivalve Filter Feedersand Marine Ecosystem Processes. Springer-Verlag, New York,pp. 371–420.

Griffiths, C.L., Griffiths, R.J., 1987. Bivalvia. In: Pandian, T.J.,Vernberg, F.J. (Eds.), Animal Energetics. Vol. 2. Bivalviathrough Reptilia. Academic Press, New York, pp. 2–87.

Harmon, R., Challenor, P., 1997. A Markov chain Monte Carlomethod for estimation and assimilation into models. Ecol.Model. 101, 41–59.

Hastings, W.K., 1970. Monte Carlo sampling methods usingMarkov Chains and their applications. Biometrika 57, 97–109.

Hodges, P.E., Hale, D.F., 1993. A computational method forestimating densities of non-Gaussian nonstationary univariatetime series. J. Time Ser. Anal. 14, 163–178.

Hofmann, E.E., Lascara, C.M., 1998. Overview of interdisciplinarymodeling for marine ecosystems. In: Brink, K.H., Robinson,A.R. (Eds.), The Sea: Ideas and Observations on Progress inthe Study of the Seas. Volume 10. The Global Coastal Ocean:Processes and Methods. Wiley, New York.

Huthnance, J.M., Allen, J.I., Davies, A.M., Hydes, D.J., James,I.D., Jones, J.E., Millward, G.E., Prandle, D., Proctor, R.,Purdie, D.A., Statham, P.J., Tett, P.B., Thompson, S., Wood,R.G., 1993. Towards water quality models. Philos. Trans. R.Soc. Lond. A 343, 569–584.

Jazwinski, A.H., 1970. Stochastic Processes and Filtering Theory.Academic Press, New York, 376 pp.

Jorgensen, S.E., 2001. Parameter estimation and calibration by theuse of exergy. Ecol. Model. 146, 299–302.

Kahru, M., Mitchell, B.G., 2001. Seasonal and nonseasonalvariability of satellite-derived chlorophyll and colored dissolvedorganic matter concentration in the California Current. J.Geophys. Res. 106, 2517–2529.

Kitagawa, G., 1987. Non-Gaussian state-space modeling ofnonstationary time series. J. Am. Stat. Assoc. 82, 1032–1041.

Kitagawa, G., 1998. A self-organizing state space model. J. Am.Stat. Assoc. 93 (443), 1203–1215.

Kitagawa, G., Gersch, W., 1996. Smoothness Priors Analysis ofTime Series. Lecture Notes in Statistics No. 116. Springer-Verlag, New York, 261 pp.

Lawson, L.M., Sptiz, Y.H., Hofmann, E.E., Long, R.L., 1995. Adata assimilation technique applied to a predator-prey model.Bull. Math. Biol. 57, 593–617.

Lehman, J.T., Foy, R., Lehman, D.A., 2001. Inverse model methodfor estimating assimilation by aquatic invertebrates. Aquat. Sci.63, 168–181.

MacEachern, S.N., Berliner, L.M., 1994. Subsampling the Gibbssampler. Am. Statistician 48, 188–190.

Marion, G., Renshaw, E., Gibson, G., 2000. Stochastic modellingof environmental variation for biological populations. Theor.Popul. Biol. 57, 197–217.

May, R.M., 1972. Limit cycles in predator-prey communities.Science 177, 900–902.


Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H.,Teller, E., 1953. Equation of state calculations by fast computingmachines. J. Chem. Phys. 21, 1087–1091.

Meyer, R., Christensen, N.L., 2000. Bayesian Reconstruction ofChaotic Dynamical Systems. Phys. Rev. E 62, 3535–3542.

Meyer, R., Millar, R., 1999. BUGS in Bayesian stock assessments.Can. J. Fisheries Aquat. Sci. 56. 1078-1087

Meyer, R., Yu, J., 2000. BUGS for a Bayesian analysis of stochasticvolatility models. Econometrics J. 3, 198–215.

Millar, R.B., Meyer, R., 2000. Bayesian State-Space Modeling ofAge-Structured Data: Fitting a Model is Just the Beginning.Can. J. Fisheries Aquat. Sci. 57, 43–50.

Natvik, L.J., Eknes, M., Evensen, G., 2001. A weak constraintinverse for a zero dimensional marine ecosystem model. J. Mar.Syst. 28, 19–44.

Nisbet, R.M., Diehl, S., Wilson, W.G., Cooper, S.D., Donaldson,D.D., Kratz, K., 1997. Primary productivity gradients andshort-term population dynamics in open systems. Ecol. Monogr.67 (4), 535–553.

Omlin, M., Reichert, P., 1999. A comparison of techniques for theestimation of model prediction uncertainty. Ecol. Model. 115,45–59.

Omlin, M., Brun, R., Reichert, P., 2001. Biogeochemical model ofLake Zurich: sensitivity, identifiability and uncertainty analysis.Ecol. Model. 141, 105–123.

Pastres, R., Solidoro, C., Cossarini, G., Melaku Canu, D., Dejak,C., 2001. Managing the rearing ofTapes philippinarumin thelagoon of Venice: a decision support system. Ecol. Model. 138,231–245.

Qian, S., Stow, C.A., Borsuk, M.E., 2003. On Monte Carlo methodsfor Bayesian inference. Ecol. Model. 159, 269–277.

Ross, A.H., Nisbet, R.M., 1990. Dynamic models of growth andreproduction of the musselMytilus edulisL. Funct. Ecol. 4,777–787.

Raillard, O., Ménesguen, A., 1994. An ecosystem box modelfor estimating the carrying capacity of a macrotidal shellfishecosystem. Mar. Ecol. Process Ser. 115, 117–130.

Reichert, P., Omlin, M., 1997. On the usefulness of overpara-meterized ecological models. Ecol. Model. 95, 289–299.

Robert, C.P., Casella, G., 1999. Monte Carlo Statistical Methods.Springer-Verlag, New York.

Sieracki, C.K., Sieracki, M.E., Yentsch, C.S., 1998. An imaging-in-flow system for automated analysis of marine microplankton.Mar. Ecol. Prog. Ser. 168, 285–296.

Soudant, D., Beliaeff, B., Thomas, G., 1997. Dynamic linearBayesian models in phytoplankton ecology. Ecol. Model. 99,161–169.

Spiegelhalter, D.J., Thomas, A., Best, N.G., Gilks, W.R., 1995.BUGS: Bayesian Inference Using Gibbs Sampling, Version0.50. MRC Biostatistics Unit, Cambridge.

Steinberg, L.J., Reckhow, K.H., Wolpert, R.L., 1997. Characteri-zation of parameters in mechanistic models: a case study of aPCB fate and transport model. Ecol. Model. 97, 35–46.

Steele, J.H., Henderson, E.W., 1981. A simple plankton model.Am. Nat. 117, 676–691.

Tarantola, A., 1987. Inverse Problem Theory: Methods for DataFitting and Model Parameter Estimation. Elsevier, New York,613 pp.

Thacker, W.C., Long, R.B., 1988. Fitting dynamics to data. J.Geophys. Res. 93, 1227–1240.

Thompson, K.R., Dowd, M., Lu, Y., Smith, B., 2000. Oceano-graphic data assimilation and regression analysis. Environ-metrics 11, 183–196.

Vallino, J.J., 2000. Improving marine ecosystem models: use ofdata assimilation and mesocosm experiments. J. Mar. Res. 58,117–164.

Vezina, A.F., Pace, M.L., 1994. An inverse model analysis ofplanktonic food webs in experimental lakes. Can. J. FisheriesAquat. Sci. 51, 2034–2044.

West, M., Harrison, P.J., Migon, H.S., 1985. Dynamic generalizedlinear models and Bayesian forecasting (with discussion). J.Am. Stat. Assoc. 80, 73–97.

Wu, J., Marceau, D., 2002. Modeling complex ecological systems:an introduction. Ecol. Model. 153, 1–6.

A Bayesian approach to the ecosystem inverse problem

Documents

Transcript of A Bayesian approach to the ecosystem inverse problem