A bio-physical coastal ecosystem model for assessing environmental effects of marine bivalve...

Ecological Modelling 183 (2005) 323–346

A bio-physical coastal ecosystem model for assessingenvironmental effects of marine bivalve aquaculture

Michael Dowd∗

Department of Mathematics and Statistics, Dalhousie University Halifax,Nova Scotia, Canada B3H 3J5

Received 30 July 2003; received in revised form 6 April 2004; accepted 9 August 2004

Abstract

A simple lower trophic level, bio-physical marine ecosystem model is developed for the purpose of assessing the environmentaleffects of bivalve aquaculture in coastal embayments. The ecosystem box model includes pelagic and benthic components anddescribes the cycling of a most-limiting nutrient. The pelagic compartment is comprised of phytoplankton, zooplankton, nutrientsand detritus. These populations interact following predator–prey dynamics and biogeochemical processes. Mixing processeswithin the bay, and exchange of waters with the adjacent open ocean, are included. The pelagic ecosystem is coupled to asimple benthos containing a dynamically active organic matter pool. Benthic–pelagic coupling includes episodic resuspension,remineralization, sinking, and permanent burial. A population of grazing bivalves is superimposed on this system as a diagnosticvariable. The model is applied to a coastal bay and used to determine how bivalve populations affect nutrient cycling in thee ted nutrient( roductionf ding andb thic detritalp turing thee parameteri©

K

f

ua-lams,

rine

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cosystem. This is done by examining changes in the standing stock of the various populations, as well as associamass) fluxes, for cases both with and without intensive bivalve culture. It was demonstrated that bivalves divert prom the pelagic to benthic food webs. Phytoplankton and detritus are depleted from the water by bivalve filter feeiodeposited to the benthos as fecal matter. This organic loading causes order of magnitude changes in the benool and the associated benthic–pelagic fluxes. It was also shown that water motion and mixing is important in struccological dynamics in the bay. To faciliate future applications and observational studies, a retrospective analysis of

dentifiability and uncertainty was also undertaken.2004 Elsevier B.V. All rights reserved.

eywords:Aquaculture; Benthic–pelagic coupling; Nutrient cycling; Box model; Differential equations; Regression

∗ Tel.: +1 902 494 1048;ax: +1 902 494 5130.E-mail address:[email protected].

1. Introduction

The global production of marine bivalve aqculture, including species such as mussels, coysters and scallops, continues to expand(New, 1999).Filter feeding bivalves depend on the coastal ma

304-3800/$ – see front matter © 2004 Elsevier B.V. All rights reserved.
doi:10.1016/j.ecolmodel.2004.08.018

324 M. Dowd / Ecological Modelling 183 (2005) 323–346

ecosystem to supply food in the form of suspendedparticulate matter, both living and detrital. Largebivalve populations can lead to a variety of ecosystemeffects. This includes the localized depletion ofsuspended particulate matter in the vicinity of denseaggregates of bivalves (Incze et al., 1981; Frechette etal., 1989). It has also been suggested that the grazingactivity of bivalve populations controls planktondynamics on the scale of entire embayments(Cloern,1982; Dame and Prins, 1998). Alpine and Cloern(1992)conclude that the establishment of large bivalvepopulations can significantly alter mass and energyflows in coastal ecosystems. Bivalves filter particulatematter suspended in the water column and biodepositit to the seabed in the form of large and rapidly sinkingfecal matter; this diverts production from the pelagic tobenthic food webs(Cloern, 1982; Noren et al., 1999).Such high rates of organic biodeposition have beenshown to result in anaerobic benthic environments(Hatcher et al., 1994), and change the benthic faunalcommunity(Crawford et al., 2003). Benthic nutrientremineralization may also increase near aquaculturesites due to increased organic matter sedimentation(Grant et al., 1995). Ammonia excretions associatedwith bivalve culture can influence nutrient levels inseawater in some coastal regions(Dame et al., 1991).Harvesting activities remove biomass directly andKaspar et al. (1985)suggests that this may contributeto nitrogen limitation in some systems.

Bivalve aquaculture depends on the biological pro-d ati-c ssingt atede hichc ma-r cesb2 atf ca-p n(c odela ffectsoe delsf ingb rel-a odel

for the purpose of systematically investigating the ef-fects of intensive marine bivalve culture on its support-ing ecosystem.

This study outlines the development and applicationof a simple lower trophic level ecosystem model. Thephysical situation is one of a shallow, semi-enclosedembayment exchanging its waters (and the freelyfloating ecosystem components) with the adjacentopen ocean. Seasonal time scales are emphasized. Thepelagic compartment is comprised of phytoplankon,zooplankton, nutrients and detritus. The pelagicecosystem is coupled to a very simple, but dynami-cally active, benthos. A population of grazing bivalvesis also included. InDowd (1997), it was shown thattreating the bivalve population as a prognostic variable(i.e. superimposing a bioenergetic model on one of thesupporting ecosystem) lead to a degradation in the pre-dictive skill of the model. Since the focus here is not onthe prediction of bivalve growth and carrying capacity,bivalve biomass is prescribed and interacts with theecosystem as a diagnostic variable or forcing function(Kremer and Nixon, 1978; Chapelle et al., 2000). Theprimary objective of this study is to develop and applya general mathematical modelling framework for theassessment of ecosystem effects of marine bivalve cul-ture. A secondary emphasis considers how one mightanalyse the model dynamics in order to identify im-portant parameters, and to determine which variablesshould be measured in an observational program.

This paper is organized as follows. Section2 de-s temm andb ft ay-m -s howb pec-t ken.A

2

rinee l di-a theiri -t x.P kton

uction of the coastal marine ecosystem. Mathemal models are useful for understanding and assehe potential interactions in these complex manipulcosystems. Bio-physical models are required wonsider both interacting populations in the coastaline ecosystem, as well as hydrodynamic influenrought about by water circulation and mixing(Dowd,003; Duarte et al., 2003). Ecosystem box models th

ocus on predicting bivalve growth and carryingacity have been proposed byRaillard and Menesgue1994)andDowd (1997). Chapelle et al. (2000)alsoonsiders an aquaculture ecosystem using a box mpproach with an emphasis on the ecosystem ef land runoff. BothPastres et al. (2001)andDuartet al. (2003)present sophisticated ecosystem mo

ully coupled to hydrodynamic models, and includivalve bioenergetics. Here, we offer a general, andtively simple, bio-physical coastal ecosystem m

cribes the lower trophic level bio-physical ecosysodel including a description of the various pelagicenthic processes. In Section3, a specific application o

he model is outlined for a semi-enclosed tidal embent located off eastern Canada. Section4 presents re

ults from model simulations with an emphasis onivalve culture can alter ecosystem fluxes. A retros

ive analysis of the model dynamics is also undertasummary and conclusions follows in Section5.

. Model

The model describes a simple coastal macosystem with bivalve aquaculture. A conceptuagram showing the ecological components and

nteractions is given inFig. 1. Population interacions occur within a finite volume of water, or a boelagic ecosystem components include phytoplan

M. Dowd / Ecological Modelling 183 (2005) 323–346 325

Fig. 1. Conceptual diagram for the ecosystem box model. Prognosticecosystem state variables are phytoplankton (P), zooplankton (Z),nutrients (N), pelagic detritus (D), and benthic detritus (B). Bivalves,M, are a diagnostic variable. Arrows represent mass, or nutrient,fluxes.

(P), zooplankton (Z), nutrients (N) and detritus (D).These are suspended or dissolved in seawater and aresubject to transport through water motion and mixing;by this mechanism they are exchanged with adjacentregions. The benthic compartment contains a detritalorganic matter pool (B). A population of spatially fixedgrazing bivalves (mussels orM) is superimposed onthe system. Note thatM is distinguished from the otherecosystem components since its biomass is prescribedat a steady state level. The other ecosystem compo-nents are prognostic variables which interact with oneanother and dynamically co-evolve.

In words, changes in ecological state variablesP, Z,N, D andB are given by,

dPdt = +growth− losses− Z,M grazing± mixing

dZdt = +growth− losses± mixing

dNdt = +sources+D,B remin+ Z,M excretion− P uptak

dDdt = +sources− remin− sed+ P,Z losses+ B resusp−dBdt = −resusp− remin− burial+D sed+M biodepositio

Here, d{·}/dt represents the time rate of change.Note that mixing acts only on the freely floating

pelagic state variables and can either increase or de-crease their concentration. Mussels are assumed tobe maintained at a constant biomass through harvest-ing activities. Mussels then participate in material cy-cling and interact with the ecosystem according tothe following

dM

dt= 0 = harvest− (ingestion− feces− excretion).

Mathematically, the prognostic ecoystem variableswithin a box evolve according to the following nonlin-ear system of ordinary differential equations:

dP

dt= f {N; kn}γpP − λpP − f {P ; kp}IzZ − ImP

+K(P∞ − P) (1)

dZ

dt= εzf {P ; kp}IzZ − λzZ +K(Z∞ − Z) (2)

dN

dt= Nin + φdD + η−1φbB + βzf {P ; kp}IzZ

+βmIm(P +D) − f {N; kn}γpP+K(N∞ −N) (3)

dD

dt= Din − φdD − λdD + λpP + (1 − εz − βz)

× f {P ; kp}IzZ + η−1r − ImD +K(D∞ −D)

T0 e net

r(

e± mixing

M grazing± mixing

n.

(4)

dB

dt= −r − φbB − αB + ηλdD

+ η(1 − εm − βm)Im(P +D). (5)

he steady state assumption forM growth (dM/dt =) means that the harvest rate is balanced by th

ate of production, both of which are given byεmImP +D).


All quantities used in this model are summarized inTable 1. The exception isB, which represents a lineartransformation ofB(t) and is defined in Section2.3.The nondimensional modulation functionsf are of the

Table 1Definition of quantities in the ecosystem model

Quantity Units Value Definition

(i) State variablesP gC m−3 Eq.(1) PhytoplanktonZ gC m−3 Eq.(2) ZooplanktonN gC m−3 Eq.(3) NutrientsD gC m−3 Eq.(4) Water column detritusB gC m−2 Eq.(5) Benthic detritus

(ii) ParametersK d−1 Appendix A Exchange/flushing

coefficientkn gC m−3 0.2 Half-saturation forN

uptake byPγp(t,Temp) d−1 Eq.(8) P growth rateλp d−1 0.1 Loss term forPIz d−1 0.5 Ingestion rate forZ on

Pkp gC m−3 0.1 Half-sat const forZ in-

gestion ofPεz – 0.3 Assim fraction forZ

ingested rationβz – 0.4 Excreted fraction forZ

ingested rationλz d−1 0.1 Loss term forZφd (Temp) d−1 (0.02–0.1) Remin rate ofD toNλd d−1 0.05 Sinking rate forDφb(Temp) d−1 0.01 Remin rate forB toDr(t) gC m−3 d−1 Eq.(9) Resuspension fluxα – 0.05 Burial fractionIm d−1 Section3 Ingestion rate of

bivalvesεm – 0.1 Assimilated fraction

for bivalvesβm – 0.1 Excreted fraction for

bivalvesη m 5 Water depth

(iii) External inputsP∞(t) gC m−3 Fig. 4a Far-fieldPZ∞(t) gC m−3 Fig. 4b Far-fieldZN∞(t) gC m−3 Fig. 4c Far-fieldND∞(t) gC m−3 Fig. 4d Far-fieldDNin(t) gC m−3 d−1 Section3 ExternalN inputDin(t) gC m−3 d−1 Section3 ExternalD input

Groupings are according to variable type. For each quantity the fol-lowing information is given: units, its numerical value (or its source),and a brief definition. Explicit functional dependence on time (t) ortemperature (Temp) is indicated. Here gC denotes grams carbon, mis metres, and d is days. See text for details.

form

f {X; kx} = X

kx +X(6)

whereX in the above is one ofP or N. The mixingterm for the pelagic state variables has been replacedwith a gradient-flux relation of the formK(X −X∞)and is discussed in Section2.1. State variables andmodel parameters are all non-negative; the sign of eachterm correponds to the direction of mass flow as indi-cated by the arrows inFig. 1. Pelagic state variablesare in concentration units, whileB is measured ona per unit area basis. Cycling of material is definedin terms of a most limiting nutrient which for manytemperate marine systems is considered to be nitro-gen(Parsons et al., 1977). Ecosystem components are,however, measured in units of carbon equivalents suchthat 1 g carbon = 10 mmol nitrogen(Steele and Hen-derson, 1981; Fasham, 1993; Edwards, 2001). The var-ious bio-physical processes in the model are explainedin detail below.

2.1. Exchange and mixing

The model ecosystem(1)–(5) is considered in thecontext of a box model. All ecological interactions oc-cur within a finite volume, or box. Components are as-sumed to be distributed homogeneously within a box,and spatial gradients occur between boxes. Each of thepelagic components of the ecosystem are exchangedw ac-c m-b ivenb

X

w ra-t e ofP entb ntt e re-g is-tp es,s n in-v oft tm d to

ith its corresponding population in adjacent areasording to the motion of the fluid in which they are eedded. The horizontal exchange of material is gy

˙ = K(X∞ −X) (7)

hereX = dX/dt. Here,X represents the concention the pelagic ecosystem components, i.e. on, Z,N, orD. The corresponding value in the adjacox isX∞. The parameterK is an exchange coefficie

hat scales the concentration difference between thions. Eq.(7) describes a gradient-flux relation cons

ent with Fickian diffusion(Fischer et al., 1979). ThearameterK includes a variety of physical processuch as tidal mixing, and can be interpreted as aerse time scale for the flushing of the box. A termhe form(7) is appended to each of(1)–(4)to represenixing processes. Multiple boxes can be combine


add a spatial aspect to the time dependent ecosystemmodel.

An illustrative example of the use of(7), which an-ticipates the application in Section3, is the following.Consider a shallow, semi-enclosed embayment whichis well mixed in the vertical and horizontal, and therebycan be considered as a box. The far-field concentra-tions of the ecosystem components in the adjacentopen ocean areX∞. Their time-varying level mightbe specifed as a type of boundary condition, or forcingfunction. Ecological interactions described by(1)–(5)occur within the bay, and pelagic ecosystem compo-nents are exchanged between the ocean and the bay. Akey element of the box model is the control of the in-ternal ecosystem processes by the far-field conditions,which brings about a form of ecosystem closure. Theaddition of mixing terms has been shown to lend sta-bility to an ecosystem(Edwards et al., 2000; Edwards,2001). In more general ecological terms, it is an ex-ample of an open system which allows componentsto disperse, or exchange, outside the control volume(Nisbet et al., 1997).

2.2. Pelagic processes

The pelagic part of the model includes the bioticcomponents,P and Z, and the abiotic components,N and D. These are found in the water column assuspended (P,Z,D) or dissolved (N) material. Con-sider the pelagic ecosystem equations(1)–(4), ignor-iφ sent( icf l-i ssics -n anicm andP lc n bym uchP g-i )B her-e ta ag e cy-c ms,

a simple form of thePZNDmodel is used. The pelagicprocesses considered here include: (i) primary produc-tion, (ii) predation, and (iii) biogeochemistry. These arediscussed below.

2.2.1. Primary productionPhytoplankton growth is governed by the seasonally

varying maximum light limited photosynthetic rateγp.It is computed as a daily and depth averaged growthrate according to

γp(t,Temp)

= chl

C

1

η T

∫ T

0

∫ η

0g {I(z, t); θ(Temp)}× dzdt (8)

whereT here corresponds to one day,η is the depthof the water column, andchl/C is the chlorophyll-a tocarbon ratio of the phytoplankton. The photosynthesis–irradiance curve is given byg{·} and uses the functionalform reported inPlatt et al. (1980). It is a function ofthe seasonally and diurnally varying underwater lightfield I(z, t). The photosynthetic parametersθ are takenfrom Platt and Jassby (1976)and modified using thetemperature modulation ofEpply (1972).

The growth rateγp in (1) is reduced by nutrient lim-itation imposed by the saturation functionf {N; kn}.This is interpreted here in the context of Michaelis-Menten kinetics(Caperon, 1967; Dugdale, 1967). Thisautotrophic process results in the uptake of inorganicNand its conversion to organicP. Phytoplankton lossesor inTt hal-l on-t latem

2d bi-

v ac-cd eret eas-i lleg : (i)a a

ng the terms related to benthic–pelagic coupling (r =b = 0). Also suppose that no bivalves are preIm = 0). The model(1)–(4) then reduces to a basorm of the widely usedPZND model. This generazed predator–prey model has its roots in the clatudy of Riley (1947) and aPZN model was origially used to describe plankton dynamics in the oceixed layer(Steele and Henderson, 1981; Evansarslow, 1985). Fasham et al. (1990)added a detritaomponent to account for organic matter breakdowarine bacteria, or the so-called microbial loop. SZNDmodels are now being widely applied in biolo

cal oceanography(Kemp et al., 2001; Waniek, 2003.ecause of the complex dynamical properties innt in even the most simplePZNDmodels(Franks el., 1986; Edwards, 2001), as well as our goal ofeneral understanding of aquaculture effects on thling of matter in shallow coastal marine ecosyste

ccur through mortality at a constant rateλp and di-ectly enter the detrital pool; its numerical valueable 1 is taken fromLe Pape et al. (1999). Notehat we do not consider self-shading in these sow coastal systems where light attenutation is crolled mainly by non-plankton suspended particuatter.

.2.2. Secondary productionSecondary production includes zooplankton an

alve growth. Zooplankton consume phytoplanktonording to the ingestion functionf {P ; kp} which hereescribes a Holling type II functional response wh

he predator’s ingestion of prey saturates with incrng prey density(Holling, 1959). This process is westablished forZ grazing onP (Frost, 1975). The in-ested ration is partitioned into three componentsfractionεz incorporated into biomass growth, (ii)


fractionβz excreted into theNpool, and (iii) the remain-ing fraction, 1− εz − βz, lost through fecal productiontoD. A constant loss rateλz governs natural mortalityand higher order predation onZ. Numerical values forthese parameters inTable 1follows Edwards (2001)and references therein.

Bivalves are treated as a diagnostic variable whichaffect the ecosystem through the processes of ingestion,excretion and fecal production. Their population dy-namics is mainly under the control of aquaculturaliststhrough stocking and harvesting activities and a steadystate assumption has been made for their biomass, i.e.the instantaneousM production rateεmIm(P +D) isassumed to be exactly equal to the removal rate throughharvesting. Filter feeding bivalves consume bothPandD. The ingestion rateIm is based on their vol-umetric clearance of particulates in a water mass, aprocess consistent with a type I functional response(Hassell, 1978). Knowledge of the clearance rate (mea-sured in units of m3 d−1 kg−1) and theM biomass den-sity (kg m−3) allows computation of the ingestion rateIm (d−1). FollowingZ above, the total ingested rationIm(P +D) is partitioned into three components forgrowth, excretion and feces, as dictated by the parame-tersεm andβm. The fraction 1− εm − βm is comprisedof mussel feces and pseudofeces and, unlikeZ, theseparticle aggregates sink rapidly and are biodepositeddirectly into the benthic detrital poolB (Widdows etal., 1998). The growth rate ofM is constrained to bal-ance the harvest rate so that totalM biomass remainscf

2a-

t othP -t imes -t ltp -p ormφ ec rtedr gedw n asd

The nutrient pool,N, receives the remineralizedD.It also receives the excretion products fromZandM, asfractions (βz, βm) of their ingested ration. The mainNsink in the water column is its uptake byP as dictatedby primary production. The benthos also provides anN source through remineralization of benthic detritalorganic matter,B, and is discussed below.

2.3. Benthic processes

The time evolution of the benthic detrital pool,B,is given by(5). In shallow coastal ecosystems thereis a strong interaction between the water column andthe benthos. Here, a simple benthos is coupled to thepelagic model to account for the major bio-physicalprocesses which affect the water column. This view-point avoids detailed biogeochemical and sedimenttransport models in favor of a very simple descriptionof their emergent properties.

Note that in the model(1)–(5) the pelagic compo-nentsP,Z,N, andD are measured on a per unit vol-ume basis as concentrations (gC m−3), whereasB ismeasured as mass per unit area (gC m−2). The massfluxes associated with benthic–pelagic coupling mustbe modified accordingly: a flux from a pelagic com-ponent toB is multiplied by the water depthη, whilefluxes fromB to the pelagic components are divided byη.

2.3.1. Sinking, resuspension and burialn

t d re-s fluxf , orp encea nsityaa n-sat -t d byH

wa-ti hears setp ing

onstant. Values forεm andβm in Table 1are takenrom Griffiths and Griffiths (1987).

.2.3. Pelagic biogeochemistryParticulate organic matter,D, suspended in the w

er column receives inputs from the mortality of band Z. Breakdown of organicD to inorganic nu

rients,N, by marine bacteria takes place on a tcale of days(Heip et al., 1995). This remineralizaion process occurs at a rateφd which is proportionao the level ofD. Jones and Henderson (1986)re-ort a range forφd of 0.004–0.18 d−1. Here, a temerature dependent remineralization rate of the fd = const× exp(Q(Temp− Temp)) is used with thoefficients determined so as to fall into this repoange. Water column detrital matter is also exchanith the benthos through sinking and resuspensioiscussed in Section2.3.

The net input flux of detritus from the water columo the benthos is governed by sedimentation anuspension. Sinking of detrital matter represents aromD toB. The sinking rate of suspended particlesarticle aggregates, is affected by both water turbuls well as particle characteristics, such as their dend size. The representation of sinking in(4) and(5)ssumes it is proportional toD and occurs at a cotant rateλd . Settingλd = 0.05 d−1, D = 1 gC m−3

ndη = 5 m depth translates to an annual fluxηλdD tohe benthos of near 100 gC m−2 year−1. This is consisent with the coastal sedimentation rates compileeip et al. (1995).Resuspension represents a detrital flux into the

er column,D, from the benthos,B. Physically, this fluxs driven by the exceedance of a critical bottom stress, or near-bed erosional velocity, which canarticles in motion and entrain them into the overly


fluid. This process depends on large and small scalewater motions in the benthic boundary layer, as wellas on seabed characteristics. It can be considered inde-pendent of eitherD orB (except for the constraint thatB must remain non-negative). Resuspension in shal-low systems is often forced by wind driven currentsand waves. These irregular events occur on short timescales (<1 d), and take the form of large fluxes whichdisrupt the equilibrium sediment distribution. Resus-pension is therefore considered as an idealized eventbased process, i.e.

r(t) =∑i

c δ(t − ti) (9)

whereδ is the Dirac delta function,ti represents thetime of a resuspension event, andc is the event in-tensity. The flux

∫r(t) dt over single event cannot ex-

ceed the amount of material inB. Here, it is assumedthat the time between events,*ti, is random and dis-tributed according to an exponential distribution witha seasonally varying width parameter. This mimicswind driven resuspension, with*ti ranging from 10days in the summer to 4 days in the winter consis-tent with the event frequency in meteorological bandfrom our region, and with the observations reportedin Edelvang et al. (2002). The stochastic process de-scribed here resembles shot noise in electronic systemsand has been extensively analysed in this context (e.g.Gardiner, 1997, Section 1.4).

2

t rew itedo ma-t hisd ce ofb ponr fluxi rti-c con-c sses( -q ls tof lingi

de-c on-

centration. This follows the widely used one-G bio-geochemical model, originally due toBerner (1964),and corresponds to the level 3 model ofSoetaert et al.(2000). However, this representation is modified hereto better account for higher order effects. Specifically,theN efflux in (5) is given asφbB where

B(t) =∫ t

0B(t − τ)κ(τ) dτ. (10)

The quantityB(t) is a convolution ofB(t) with the ker-nelκ. In this study, we use a Gaussian kernel of the formκ(τ) = exp(−τ2/ζ2)/

∫ t

0 exp(−τ2/ζ2) dτ. Here,ζ has amean of 20 days but varies smoothly from 10 to 30 daysin phase with the temperature. The convolution integralin (10) acts to time shift and smoothB(t), making theN efflux a function of the time history of the inputDinflux. It is then less dependent on the episodic fluctu-ations imparted by theδ-forcing in r(t). The tempera-ture dependence ofζ means that the biogeochemistryresponds rapidly for smallζ (high temperature), andslowly for largeζ (low temperature). The representa-tion here provides an empirical accounting of the emer-gent effects of the reaction–diffusion processes whichboth transport and transform particulate organic mat-ter within the sediments. The remineralized efflux is asmoothed and time shifted version of the depositionalinflux, as is found in the fully coupled diagenetic modelof Soetaert et al. (2000, Fig. 3c). A similar rationaleapplies to the burial process in(5), wherein a smallf r-a thicd ent(t u-t riert

3

T theen -g ,2 het idi-

.3.2. Benthic remineralizationMineralization of the benthic detrital pool,B, re-

urns nutrients,N, to the water column. Sediment poater containing particulate organic matter deposn the sea floor is transported into the sediment

rix through advective and diffusive processes. Tetrital organic matter then undergoes a sequeniogeochemical reactions in the sediment whereuemineralized products are released as a diffusivento the water column. Diagenetic models are veally resolved sediment biogeochemical modelserned with a detailed accounting of these proceBoudreau, 1997). Soetaert et al. (2000)reviews a seuence of approximations to these complex mode

acilitate the incorporation of benthic–pelagic coupn oceanic biogeochemical models.

Here, it is assumed that benthic organic matteromposes at a rate directly proportional to its own c

raction of the organic materialαB escapes reminelization and enters the refractory part of the benetrital pool to be permanently buried in the sedimMiddleburg et al., 1997). Note that use ofB rendershe system(1)–(5)non-Markovian, however convolion can be efficiently implemeted using the fast Fouransform (Press et al., 1996, Chapter 12).

. Application

The ecosystem box model(1)–(5) was applied toracadie Bay, located on Prince Edward Island offast coast of Canada (Fig. 2). This is a shallow (<7 m)early-enclosed tidal embayment (Fig. 3). Oceanoraphic overviews are given inDowd et al. (2001002). Currents are both tidal and wind driven. T

idal regime is comprised of mixed diurnal and sem


Fig. 2. Location map for the study area.

urnal tides, with a range<1 m. Instantaneous exchangeof bay waters with the adjacent Gulf of St. Lawrence isup to 500 m3 s−1. Freshwater inputs are small, with cli-matological summer values<1 m3 s−1 (Gregory et al.,1993). Tracadie Bay supports extensive bivalve aqua-culture in the form of long-line culture of the blue mus-selMytilus edulis. There is ongoing concern that thebay has reached its carrying capacity for mussel cul-ture (Page et al., 1999). There is also speculation onpossibles changes in nutrient cycling in the ecosys-tem, as well as shifts in community structure, result-ing from intensive bivalve aquaculture (Cranford et al.,2002). Dowd (2003)calculates that in Tracadie Bay thecharacteristic time scale for: (i) residence time of thewater in the bay, (ii) the filtration time for the bivalvepopulation to clear this water, and (iii) the productiontimescale for the primary producers, are all around 5days. This implies that water transport processes, in-ternal biological production within the bay, and bivalvegrazing activity all play comparable roles in controllingin the cycling of matter in the ecosystem of the bay.

Tracadie Bay was divided into three boxes as shownin Fig. 3. The boundaries were based on circulationand water properties of the bay as reported inDowdet al. (2001, 2002). The implication is that the regionshave distinct and relatively homogenous water prop-erties and gradients occur roughly on the boundaries.The inner regions (boxes 2 and 3) exchange their waterswith the open ocean through the central bay (box 1);there is little direct exchange between boxes 2 and 3.G m

Fig. 3. Map of Tracadie Bay including the bathymetry in grayscaleand the contour associated with the intertidal zone (depth in metres).The three boxes and their boundaries are given. Arrows indicatemixing between the boxes, and with the adjacent open ocean.

model(1)–(5)yields a three box system of the form,

X1 = h{[X1, B1]′} + ν1 +K∞(X∞ −X1)

−K12(X1 −X2) −K13(X1 −X3) (11)

X2 = h{[X2, B2]′} + ν2 +K12(X1 −X2) (12)

X3 = h{[X3, B3]′} + ν3 +K13(X1 −X3). (13)

Here, theXi = (Pi, Zi, Ni,Di)′ is a vector containingconcentrations of the pelagic state variables in boxi. The superscript′ denotes matrix transpose. Thevector functionh contains the homogenous terms inthe ecosystem dynamics(1)–(5). This operator acts onthe full ecosystem state, both pelagic and benthic, foreach box. The state independent vectorνi contains thenonhomogeous terms of(1)–(5), such asNin andDin,for box i. There are three exchange coefficients:K∞represents mixing between the ocean and box 1;K12represents mixing between box 1 and box 2; andK13represents mixing between boxes 1 and 3.

Numerical values are required for the three ex-change coefficients. These were determined by inver-sion of a heat budget model using observed temperaturetime series for each of the boxes. Details are given in
eneralizing the relation in(7)and using the ecosyste


Appendix A. The following estimates for the exchangecoefficients were obtained:K∞ = 1.3,K12 = 0.4 andK13 = 0.5 in units d−1. The expected time for the ma-terial to remain in each of three boxes are 0.6, 2.5 and2.2 days, respectively. For 90% of the material to re-moved, the corresponding timescales are 1.2, 7.6 and6.4 days. The interested reader is referred toTakeoka(1984)for more details on computing residence times.

The general mathematical structure for the ecosys-tem dynamics was described in Section2. Applicationof this model to a specific situation, such as that of Tra-cadie Bay, requires a number of quantities be specified.These include: initial conditions for each of the statevariables; numerical values for the model parameters;internal source or sink terms; and far-field values of thepelagic state variables. Time integration of the modelproduces predictions for each of the state variables. Therationalization for model parameter values was givenin Section2 and numerical values are found inTable 1.Specification of the remaining model inputs is outlinedbelow.

Idealized annual cycles for each of the far-fieldpelagic state variables are given inFig. 4. These ex-ternal source terms correspond to concentrations ofthe pelagic state variables in the adjacent Gulf of St.Lawrence. They were obtained as follows.

Fig. 4. Annual cycle of the far-field concentrations of the pelagic state ear.

• P∞(t): The annual cycle of chlorophyll concen-tration from the region adjacent to Tracadie Baywas determined from biweekly composite SeaWIFSsatellite images compiled at the Bedford Instituteof Oceanography. These were converted to carbonunits by assuming a carbon:chlorophyll-a ratio of50:1.

• Z∞(t): Zooplankton are assumed to be a reducedmagnitude, smoothed and lagged version of thephytoplankton time series. This assumption is con-sistent with observed Gulf of St. Lawrence cope-pod biomass (Fisheries & Oceans Canada, AtlanticZonal Monitoring Program).

• N∞(t): A compilation of nitrate and ammo-nium from the adjacent Gulf of St. Lawrenceyielded mean winter values near 5 mmole N m−3

(or 0.5 gC m−3) and summer values near1 mmole N m−3 (0.1 gC m−3), but with a highdegree of scatter (Dr. Peter Strain, Bedford Instituteof Oceanography, personal communication). Thesevalues motivate the idealized seasonal nutrientcycle.

• D∞(t): Detritus was simply assumed to be constantwith a level of 0.6 gC m−3. Time series of point mea-surements made in and near Tracadie Bay exhibitedhigh variability and an apparent lack of a seasonal

variablesP∞, Z∞, N∞ andD∞. The time axis covers the calendar y


cycle. A representative annual mean value was cho-sen.

The temperature cycle is also required for its rolein modulating the remineralization rates, as well asthe phytoplankton growth rate. A sine curve was fitto the annual cycle of observed temperature yieldingTemp(t) = 10+ 11 sin(ωt + 4π/3), wheret is in days,ω = 2π/365 and Temp(t) is in degrees Celsius.

Internal source terms,Nin, for nutrients were con-sidered to be non-zero only for box 2, which receivesthe bulk of the nutrient inputs from land-based sourcesvia the only substantial freshwater source, WinterRiver. Nitrogen values measured near Winter River inJuly 1999 were near 30 mmole N m−3 (or 3 gC m−3).Gregory et al. (1993)reports a mean freshwater flux of1 m3 s−1 for Tracadie Bay. Assuming this inputN fluxis uniformly distributed over box 2, which has a volumenear 107 m3, yields a input flux of 0.025 gC m−3 d−1.A detrital input termDin was also included in(4), andin Tracadie Bay results from land-runoff and the decayof macrophyte detritus. A value of 0.03 gC m−3 d−1

was assumed for all boxes. This allows interior valuesof D to match far-field valuesD∞, consistent with therelative lack of spatial gradients in detritus as reportedin Dowd (2003).

The phytoplankton growth rateγp was computedaccording to(8). A carbon to chlorophyll ratio of 50 wasassumed. Surface solar irradiance observations fromthe region were used along with astronomical relations.T g ano ionsds htl ivelyw anb area

ra-c i-fi d ust adieB cko s-s ay.I )r urei 1 L

Fig. 5. Annual cycle of the maximum light limited phytoplanktongrowth rateγp. The time axis covers the calendar year.

per individual mussel per hour(Griffiths and Griffiths,1987). These figures, together with the estimated stand-ing stock and the volume of the bay (36× 106 m3),yields an estimate forIm near 0.1 d−1.

Numerical implementation of the ecosystem boxmodel proceeds as follows. The governing Eqs.(1)–(5) were discretized and solved using an Euler method(Boyce and DiPrima, 1986, Section 8.2). The systemis Markovian except for the convolution in(10)whichis associated with benthic remineralization; this termis evaluated every time step and requires past valuesof the model state. The time step of the model was0.1 d and the simulation starts at the beginning of thecalendar year. Initial conditions for the pelagic statevariablesP, Z, N, andD are set based on their cor-responding far-field values inFig. 4. This is reason-able for winter periods when the internal ecology inthe bay is inactive and the system is controlled by thefar-field conditions; pelagic variables equilibrate withthe far-field conditions on a time-scale set by the mix-ing processes (days to weeks). The benthic detrital poolB has a much longer response timescale (months) dueto storage effects and remineralization processes. Theinitial condition forB is a zero field, and the modelis integrated for three years until a seasonally varyingperiodic steady state forB is reached. Model variablesare then recorded for one calendar year (January 1 toDecember 31).

In more general terms, the numerical version of theecosystem model can be represented by following dis-

he underwater irradiance field was computed usinptical depth of 3m, as measured in various locaturing summer 1999. The resultant time series ofγp ishown inFig. 5. It is maximal in late August, when ligevels are high and water temperatures are relatarm; a doubling time for phytoplankton of 1 day ce achieved during this period if sufficient nutrientsvailable.

To apply the model to the present situation in Tadie Bay, the mussel grazing rate,Im, must be speced. Government and industy communications leao estimate production of mussel biomass in Tracay at 3× 106 kg wet weight and the standing stof mussels at 2× 106 kg wet weight. Biomass is aumed to be uniformly distributed throughout the bm is determined as follows.Carver and Mallet (1990eport 80 mussels per kg wet weight in longline cultn the region. Assume also a mean filtration rate of


crete equation

X = f {θ} (14)

whereX is aq × 1 vector andq = nmT . Hence,Xcon-tains then = 5 ecosystem state variables for them = 3boxes and for all model time stepsT. Here, theq × 1vector functionf represents the discretized model dy-namics. It is a function of thep× 1 vector of modelparametersθ which includes all quantities which mustbe specified to carry out a model run, including the timedependent forcing functions. The model output is a de-terministic function of the parameters as dictated bythe mapping described byf. To isolate a particular statevariable in a particular box denoteX(a,b) as the subsetof X containing the time series for the ecosystem statevariablea located in boxb. For example,X1,2 wouldrepresents ecosystem component 1 (i.e.P) in box 2.This T × 1 vector is determined asX(a,b) = H (a,b)X,whereH (a,b) is a sparseT × q matrix which picks outthe appropriate elements ofX. This is used for modelanalysis in the next section.

4. Results

4.1. Ecosystem simulations

Model simulations are concerned with predictingthe ecosystem effects of intensive bivalve culture. Twoc Sec-t elg faC thep g toI sti-g dings howi

a iesft agics ect-i theo nd 3)d ker

coupling with the far-field and the relatively greater im-portance of internal ecosystem dynamics. TheZ followclosely the far-fieldZ∞ for all boxes. TheD time seriesexhibits irregular fluctuations as a result of episodic re-suspension events, followed by slower exponential set-tling. The mean level ofD is close to the time-invariantD∞. The dynamics ofP indicate an enhanced but nearsynchronous spring bloom relative toP∞. This resultsfrom highN levels coupled with a high growth potentialγp. After the bloom,P levels fall. However, through thesummer and fallP is enhanced by a factor of 2 to 4 rel-ative toP∞ due to the highγp. Eventually,P declineswith low light levels and temperatures. On short timescales, irregularities inN andP are associated with re-suspension of detritus, its subsequent remineralizationto N, and eventual uptake byP. In box 2, enhancedNandP occur due to riverine nutrient inputs,Nin. Rem-ineralization ofD increasesN levels elsewhere. Sum-merN levels are controlled largely byPuptake and ex-change processes. The winter period effectively resetsall the pelagic state variables to their far-field values,with exchange processes dominating over the relativelyinactive biological processes.

Fig. 7 shows predicted time series of the pelagicstate variables for the case for which mussels werepresent (M = 0). (Model predicted mussel harvest,whose computation was outlined in Section2, wasclose to independent estimates, thereby providing aconsistency check for the mussel mass balance param-eters.) In comparison with theM = 0 case, the fol-l l-a in-n s re-m n ber -p ol.T vari-a edl -e -i tionb gZ ods

i ne eent ying

ases are considered. These are set up followingions 2 and 3, and differ only in the level of mussrazingIm. (i) Case 1:M = 0. This simulation is oculture free system with no mussels andIm = 0. (ii)ase 2:M = 0. This simulation take place underresent level of aquaculture activity correspondinm = 0.1 d−1. The two cases are contrasted to inveate how intensive bivalve culture changes the stantock of the ecosystem state variables, as well ast affects mass flows within the ecosystem.

Results from a model simulation for theM = 0 casere given inFig. 6. This shows predicted time ser

or the pelagic state variablesP, Z, N andD for thehree boxes in Tracadie Bay. Physical control of peltate variables is evident with the standing stock reflng the corresponding far-field values, particularlyuter bay (box 1). The innermost areas (boxes 2 aeviate more from their far-field values due to wea

owing patterns are evident.D levels are depleted retive toD∞. This effect is most pronouced in theer bay and results from the filter feeding musseloving particulates at a rate faster than they ca

enewed by water mixing. IngestedPandD are biodeosited toB leading to an elevated benthic detrital pohis drives enhanced resuspension and greaterbility in the water columnD. It also leads to increas

evels ofN through greaterB remineralization. Howver, in spite of the elevatedN, phytoplankton stand

ng stock actually decreases with any new produceing rapidly grazed down byM. The correspondinlevels are also lower due to depletion of their fo

ource.Time series of the benthic detrital pool,B, are shown

n Fig. 8for both theM = 0 andM = 0 cases. Withiach case, there is relatively little difference betw

he boxes and the figure reports only the time-var


Fig. 6. Simulated time series of the pelagic state variables for theM = 0 case. Columns correspond to three boxes. Rows correspond to statevariables. Solid lines refer toP,Z,N andD in the box, while dotted lines are the reference far-field (P∞, Z∞, N∞,D∞). The time axis coversthe calendar year.

mean ofB over all three boxes. TheB time series isdominated by high frequency fluctuations associatedwith resuspension and settling, but also influenced bythe processes of benthic remineralization and burial.The benthic detrital pool is an order of magnitude largerfor theM = 0 case, as compared to theM = 0 case.For the caseM = 0, little permanent detrital accumu-lation is evident, only temporary storage ofB whichis frequently resuspended. For theM = 0 case, musselbiodeposition leads to accumulation of up to 7 gC m−2,with the low frequency portion ofB reaching a quasi-periodic steady state for its annual cycle. The seasonalcycle inB shows a rapid increase following the springphytoplankton bloom, after which a steady decay oc-

curs as sediment biogeochemistry becomes active withincreasing summer temperatures.

Table 2shows the annual fluxes for all the ecosys-tem components for both theM = 0 and M = 0cases. These are reported as areal measures (ingC m−2 year−1) to allow direct comparison of the ben-thic and pelagic components. (Note that each of thesemass fluxes corresponds to one of the arrows inFig.1). Fluxes between the pelagic components for theM = 0 case are dominated by nutrient uptake asso-ciated with primary production (N → P), and watercolumn remineralization (D → N). With M = 0 sig-nificant amounts ofD are also removed from the watercolumn by bivalve filter feeding (D → M), an effect


Fig. 7. Simulated time series of the pelagic state variables for theM = 0 case. Columns correspond to three boxes. Rows correspond to statevariables. Solid lines refer toP,Z,N andD in the box, while dotted lines are the reference far-field (P∞, Z∞, N∞,D∞). The time axis coversthe calendar year.

which is most pronounced in the inner boxes. Benthic–pelagic fluxes in theM = 0 case are such that sedi-mentation slightly exceeds resuspension with most ofthe difference being buried. WithM = 0, biodeposi-tion of mussel feces (M → B) dominates the organicmatter flux to the benthos. As a consequence theBpool increases, and order of magnitude changes areseen in benthic remineralization (B → N) and burial.The resuspension flux is also doubled. These effects aresimilar between boxes.Hargrave (1985)reports naturalsedimentation rates of 0.1 to 1 gC m−2 d−1. By sum-ming theD → B andM → B fluxes, it is seen thattheM = 0 case falls at the lower end of this range,

while theM = 0 case falls at the mid to upper end ofthe range. Examination of the source and sink termsindicate that with an increased bivalve population thesystem becomes a strong importer ofP andD, and anexporter ofN. The flux ofZ to higher trophic levelsseems little affected byM levels.

4.2. Model analysis

In order to facilitate further application of theecosystem model, an analysis of some of its dynam-ical properties was undertaken. These were assessed inorder to better understand model structure, and to help


Fig. 8. Time series for the benthic detrital poolB for the two caseswith (M = 0) and without (M = 0) mussels present. The values ofBshown in each case are averages over all three boxes. The time axiscovers the calendar year.

Fig. 9. Observed and predicted temperature records for summer 1999for the three boxes. Observations were low pass filtered with 100 hcutoff. Predicted temperature is based on the inverse method usingthe heat equation.

Table 2Annual fluxes in the ecosystem model in gC m−2 year−1

Flux Box 1 Box 2 Box 3

M = 0 M = 0 M = 0 M = 0 M = 0 M = 0

(i) Pelagic fluxesN → P 72.1 55.2 104.0 65.7 83.3 54.1P → Z 38.3 30.7 42.3 27.8 39.3 26.5P → D 20.7 13.6 28.9 13.2 24.6 12.0Z → N 15.3 12.3 16.9 11.1 15.7 10.6Z → D 11.5 9.2 12.7 8.3 11.8 8.0D → N 90.1 73.5 88.2 57.4 87.3 58.9P → M 0.0 27.3 0.0 26.4 0.0 23.9D → M 0.0 141.6 0.0 114.3 0.0 117.1M → N 0.0 16.9 0.0 14.1 0.0 14.1

(ii) Benthic fluxesM → B 0.0 135.1 0.0 112.6 0.0 112.9D → B 43.4 35.4 43.5 28.6 43.1 29.3B → N 1.9 17.3 1.8 12.2 1.8 12.4B → D 32.9 67.2 33.0 67.2 32.7 67.2Burial 8.9 88.5 9.0 63.9 8.8 64.8

(iii) Sources and sinksP export 13.1 −16.4 32.8 −1.7 19.4 −8.3Z export −1.6 −3.3 −0.3 −3.4 −1.1 −3.8N export 35.1 64.6 46.6 72.5 21.5 41.6D export −9.9 −102.0 71.5 −54.5 −2.7 −61.1N input 0.0 0.0 43.8 43.8 0.0 0.0D input 58.4 58.4 58.4 58.4 58.4 58.4Z output 13.1 12.5 13.0 11.7 12.9 11.8M output 0.0 16.9 0.0 14.1 0.0 14.1

These are reported for both theM = 0 andM = 0 cases for all threeboxes. The notationX → Y indicates mass flux from ecosystemcomponentX toY. Fluxes are grouped by type as indicated.

guide the collection of observations which can be usedfor validating the model. Towards these ends, two com-plementary aspects are considered: (i) the sensitivity ofthe predicted ecological state variables to variations inthe parameters, and (ii) the extent to which observationsof the state variables allow us to determine the param-eters.Omlin et al. (2001)further discusses aspects ofparameter estimation in ecological prediction.

The ecosystem model is governed by the determin-istic map given by(14). The central quantity used hereto examine properties of the ecological model is theJacobian matrix of partial derviatives,

J = ∂f

∂θ

∣∣∣∣θ=θ

(15)


Table 3Sensitivity of the ecosystem state variables in each of the boxes (columns) to variations in model parameters (rows)

P1 P2 P3 Z1 Z2 Z3 N1 N2 N3 D1 D2 D3 B1 B2 B3

P∞ 8 (10) 6 (8) 7 (9) 1 (1) 1 (2) 1 (1) 1 (1) 2 (2) 2 (2) 0 (0) 1 (1) 1 (1) 1(3) 2(3) 1(3)Z∞ 3 (3) 5 (7) 5 (6) 9 (9) 9 (8) 9 (9) 1 (1) 2 (2) 2 (2) 0 (0) 0 (0) 0 (0) 0(1) 0(2) 0(2)N∞ 2 (2) 3 (3) 3 (3) 0 (0) 0 (0) 0 (0) 8 (8) 6 (6) 7 (7) 0 (0) 0 (0) 0 (0) 0(0) 1(1) 1(1)D∞ 3 (2) 5 (4) 5 (4) 0 (0) 1 (1) 1 (1) 1 (2) 2 (2) 2 (2) 8 (8) 7 (5) 7 (6) 11(11) 10(9) 9(10)γp 5 (5) 7 (11) 7 (9) 1 (1) 1 (2) 1 (2) 3 (3) 5 (5) 4 (4) 0 (0) 1 (1) 1 (1) 1(1) 1(5) 1(3)Q10 3 (2) 5 (4) 5 (3) 0 (0) 1 (1) 1 (0) 1 (1) 2 (2) 2 (2) 1 (1) 3 (2) 3 (2) 2(1) 4(2) 3(2)K∞ 3 (2) 2 (1) 3 (1) 0 (1) 1 (1) 1 (1) 2 (2) 1 (1) 1 (1) 1 (2) 1 (1) 1 (1) 1(2) 1(2) 1(2)K12 0 (0) 3 (2) 0 (0) 0 (0) 1 (1) 0 (0) 0 (0) 2 (2) 0 (0) 0 (0) 1 (1) 0 (0) 0(0) 1(2) 0(0)K13 0 (0) 0 (0) 2 (1) 0 (0) 0 (0) 0 (1) 0 (0) 0 (0) 1 (1) 0 (0) 0 (0) 1 (1) 0(0) 0(0) 1(2)λp 2 (2) 4 (3) 4 (3) 0 (0) 1 (0) 0 (0) 1 (0) 1 (1) 1 (1) 0 (0) 1 (0) 1 (0) 0(0) 1(1) 1(0)Iz 5 (4) 8 (9) 7 (8) 2 (1) 3 (2) 3 (2) 2 (2) 3 (3) 3 (2) 0 (0) 0 (0) 0 (0) 0(1) 1(3) 1(2)kp 2 (2) 2 (4) 2 (4) 0 (0) 1 (1) 1 (1) 1 (1) 1 (1) 1 (1) 0 (0) 0 (0) 0 (0) 0(0) 0(1) 0(1)λz 1 (1) 2 (3) 2 (2) 2 (2) 4 (4) 4 (3) 0 (0) 1 (1) 1 (1) 0 (0) 0 (0) 0 (0) 0(0) 0(1) 0(1)εz 2 (1) 3 (3) 3 (2) 2 (1) 4 (3) 4 (3) 0 (0) 1 (1) 1 (0) 0 (0) 1 (0) 1 (0) 1(0) 1(1) 1(1)βz 1 (0) 1 (1) 1 (1) 0 (0) 0 (0) 0 (0) 0 (0) 0 (0) 0 (0) 0 (0) 1 (0) 1 (0) 0(0) 1(0) 1(0)λd 0 (0) 0 (0) 0 (0) 0 (0) 0 (0) 0 (0) 0 (0) 0 (0) 0 (0) 1 (0) 1 (1) 1 (1) 12(3) 14(2) 12(2)φd 3 (2) 5 (4) 5 (3) 0 (0) 1 (1) 1 (0) 1 (1) 2 (2) 2 (2) 1 (1) 3 (2) 3 (2) 2(1) 4(2) 3(2)Din 1 (1) 1 (1) 1 (1) 0 (0) 0 (0) 0 (0) 0 (0) 1 (1) 1 (1) 1 (1) 2 (2) 2 (2) 1(1) 3(3) 3(3)r(t) 0 (0) 0 (1) 0 (1) 0 (0) 0 (0) 0 (0) 0 (0) 0 (0) 0 (0) 1 (2) 1 (3) 1 (3) 5(5) 7(5) 5(5)Nin 1 (0) 2 (2) 1 (1) 0 (0) 0 (0) 0 (0) 0 (0) 1 (1) 0 (0) 0 (0) 0 (0) 0 (0) 0(0) 0(1) 0(0)kn 2 (2) 3 (5) 3 (4) 0 (0) 1 (1) 1 (1) 1 (1) 2 (2) 2 (2) 0 (0) 0 (0) 0 (0) 0(1) 1(2) 1(2)α 0 (0) 0 (1) 0 (1) 0 (0) 0 (0) 0 (0) 0 (0) 0 (0) 0 (0) 0 (0) 0 (0) 0 (0) 3(8) 4(8) 3(8)φb 0 (0) 0 (1) 0 (1) 0 (0) 0 (0) 0 (0) 0 (0) 0 (0) 0 (0) 0 (0) 0 (0) 0 (0) 1(2) 1(1) 1(1)Im – (3) – (7) – (6) – (0) – (1) – (1) – (1) – (1) – (1) – (2) – (4) – (4) –(9) –(6) –(6)εm – (0) – (0) – (0) – (0) – (0) – (0) – (0) – (0) – (0) – (0) – (0) – (0) –(2) –(2) –(2)βm – (0) – (1) – (1) – (0) – (0) – (0) – (0) – (0) – (1) – (0) – (0) – (0) –(1) –(2) –(2)

Results for theM = 0 andM = 0 cases are given, withM = 0 in brackets. For presentation purposes, values forSare scaled by a factor of 10,and rounded to the nearest integer.

which is evaluated for a specific parameter setθ = θ.It takes the forms of anq × p matrix where thei, jthelement is∂fi/∂θj, for i = 1, . . . , q andj = 1, . . . p.It provides a linearization of the ecological dynamicsaboutθ. Appendix Bprovides further details. Here,Jwas evaluated for both theM = 0 andM = 0 cases.

Sensitivity analysis identifies influential parametersin the nonlinear ecosystem model. The idea is to look atchanges in the ecological state variables brought aboutby small perturbations in the parameters. Thekth col-umn ofJ, denotedJk, measures the sensitivity of theecosystem componentsX to variations in thekth param-eter,θk. For the purposes of sensitivity analysis, a scaledJacobian, denotedJ is used whereJij = (θj/Xi)Jij.The measure of sensitivity is then

S(a,b)k = 〈|H (a,b)

k Jk|〉. (16)

Here, | · | represents the absolute value and〈·〉 pro-vides for an averaging of the time series. The quantity

S(a,b)k measures the ratio of the percentage change in the

model solution to a percentage change in a single pa-rameter, following the usual definition (e.g.,Jorgensen,1994). It is computed for all 5 ecosystem components(superscript a) in all three boxes (superscript b) for eachof the parameters contained inθ.

Results from the sensitivity analysis are shown inTable 3. For theM = 0 case, pelagic ecosystem com-ponents in all boxes are sensitive to their respective far-field concentrations. The far-fieldD∞ also influencesthe level of the benthic detrital poolB. P in the innerboxes 2 and 3 are sensitive toγp as well as toIz. Thesinking rateλd , and to a lesser extent resuspensionr(t),are sensitive parameters in setting the level ofB.Table 3also shows results for theM = 0 case. Here, parametersensivity for the pelagic components remains relativelyunchanged. The exception isγp which is now impor-tant forP in the inner boxes, reflecting the increasingimportance of internal production in the poorly flushedinner regions as grazing activity is increased. However,


Table 4Estimated parameter precision reported as the ratio of the mean parameter value to its standard deviation

ALL P Z N D B

P∞ 2.99(1.69) 0.31 (0.71) 0.00 (0.00) 0.09 (0.16) 0.02 (0.08) 0.39 (0.07)Z∞ 1.08(1.03) 0.00 (0.01) 0.27 (0.57) 0.01 (0.01) 0.00 (0.00) 0.04 (0.00)N∞ 4.45(3.32) 0.03 (0.07) 0.00 (0.00) 1.50 (1.91) 0.01 (0.08) 0.41 (0.05)D∞ 8.64(7.19) 0.00 (0.02) 0.00 (0.00) 0.04 (0.04) 1.64 (3.12) 1.69 (0.28)γp 1.77(1.04) 0.06 (0.17) 0.01 (0.00) 0.10 (0.12) 0.01 (0.08) 0.30 (0.08)K∞ 2.85(0.94) 0.07 (0.23) 0.03 (0.03) 0.21 (0.22) 0.22 (0.17) 0.89 (0.10)K12 2.73(0.85) 0.08 (0.19) 0.02 (0.03) 0.16 (0.20) 0.12 (0.11) 0.73 (0.07)K13 2.95(0.79) 0.09 (0.19) 0.02 (0.02) 0.17 (0.20) 0.14 (0.13) 0.79 (0.07)λp 0.43(0.87) 0.00 (0.01) 0.00 (0.00) 0.01 (0.03) 0.01 (0.06) 0.07 (0.03)Iz 0.69(0.61) 0.00 (0.01) 0.00 (0.00) 0.01 (0.01) 0.00 (0.00) 0.04 (0.00)kp 0.50(0.31) 0.01 (0.04) 0.00 (0.00) 0.04 (0.03) 0.00 (0.01) 0.07 (0.01)λz 0.29(0.27) 0.01 (0.02) 0.03 (0.03) 0.01 (0.01) 0.00 (0.00) 0.04 (0.00)εz 0.37(0.35) 0.00 (0.01) 0.00 (0.00) 0.01 (0.01) 0.00 (0.02) 0.14 (0.01)βz 0.39(0.35) 0.01 (0.03) 0.00 (0.00) 0.02 (0.03) 0.00 (0.01) 0.10 (0.01)λd 0.67(2.36) 0.00 (0.01) 0.00 (0.00) 0.00 (0.02) 0.00 (0.29) 0.11 (0.23)φd 3.47(1.12) 0.00 (0.02) 0.00 (0.00) 0.03 (0.04) 0.37 (0.43) 1.24 (0.15)Din 1.01(0.57) 0.00 (0.00) 0.00 (0.00) 0.02 (0.02) 0.05 (0.04) 0.28 (0.02)r(t) 13.02(6.37) 0.00 (0.03) 0.00 (0.00) 0.04 (0.04) 1.90 (1.02) 8.86 (4.96)Nin 1.28(0.45) 0.02 (0.08) 0.00 (0.00) 0.13 (0.18) 0.01 (0.05) 0.63 (0.05)kn 0.92(0.59) 0.02 (0.07) 0.00 (0.00) 0.12 (0.27) 0.01 (0.03) 0.20 (0.02)α 1.37(0.24) 0.00 (0.02) 0.00 (0.00) 0.02 (0.02) 0.00 (0.03) 0.53 (0.01)φb 0.26(0.05) 0.00 (0.01) 0.00 (0.00) 0.02 (0.01) 0.00 (0.01) 0.10 (0.00)Im 2.34(0.01) 0.00 (0.00) 0.00 (0.00) 0.01 (0.00) 0.01 (0.00) 0.60 (0.00)εm 0.21(0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00) 0.06 (0.00)βm 0.26(0.00) 0.00 (0.00) 0.00 (0.00) 0.01 (0.00) 0.00 (0.00) 0.06 (0.00)

The column ‘ALL’ refers to observations available for all ecosystem state variables. The remaining columns assume that observations only onthe variable indicated are available. Results are reported for both theM = 0 andM = 0 cases, withM = 0 in brackets.

it is the benthosB which shows the greatest changes.There is increased sensitivity inB to burialα, but muchless sensitivity to sinkingλd . The importance of biode-position is made evident by the factIm strongly influ-encesP in the inner boxes, and is also important indeterminingB.

To determine how the ecosystem state variables arerelated to the parameters the asymptotic error covari-ance matrix,9, for the parameters was determined.This used the Jacobian matrix in(15) together withsome basic principles in nonlinear regression, as out-lined in Appendix B. Examining9 shows the ex-tent to which information on the ecosystem state vari-ables allows us to determine model parameters, andhence predict the ecosystem state. The diagonal ele-ments of9 provide an estimate for the variance ofthe model parametersθ. The off-diagonal elementsprovides information on how they are related, or co-vary. It is important to note that we do not need ac-tual observations for this exercise; only informationon which variables are measured and their error statis-

tics is required (seeAppendix B). Here, the covari-ance matrix9 was computed assuming a completeset of observations on the ecosystem state,X, as wellas for limited sets of observations, i.e.X(a,b). Forsimplicity, a unit variance for the observation errorswas assumed, which is reasonable starting point forvariables having the same order of magnitude. How-ever, in reality, observation errors will be unequaland depend on sampling variability and measurementprecision.

Table 4 shows information on parameter uncer-tainty. These are reported as the ratio of the param-eter values to their estimated standard deviation, andcan be interpreted as a measure of the relative pre-cision of the parameters. First, consider the situationwhere a complete set of observations ofX is available(the ‘ALL’ column in Table 4). For both theM = 0andM = 0 cases, it is seen that the far-field pelagiccomponents are well determined, as are the parametersfor resuspension and water column remineralization.In contrast, parameters associated withZ tend to be

M.Dowd/EcologicalModelling183(2005)323–346

Table 5Estimated correlation matrix for the parameters for both theM = 0 case (above main diagonal) and theM = 0 case (below main diagonal)

P∞ Z∞ N∞ D∞ γp K∞ K12 K13 λp Iz kp λz εz βz λd φd Din r(t) Nin kn α φb Im εm βm

P∞ • 2 0 0 −4 −1 −4 −2 0 2 2 1 −1 1 0 0 6 0 −2 −4 1 −1 6 7 −2Z∞ 2 • −1 0 −1 −1 0 0 0 0 1 7 1 −1 0 0 1 0 0 0 0 0 1 1 0N∞ −3 0 • 0 −2 4 0 0 −1 4 4 −5 −7 6 −1 1 −3 0 1 −3 2 −2 −4 1 −7D∞ −2 0 −1 • 0 0 0 0 0 0 0 0 0 0 −2 0 −3 0 0 0 0 0 0 0 0γp −1 −2 −1 −1 • 2 4 4 6 −4 −5 1 2 −2 2 3 0 0 2 7 0 0 0 −4 5K∞ −3 −1 2 −3 −1 • 5 4 4 1 −1 0 −1 −1 0 8 0 1 3 −2 1 −1 −4 −2 −2K12 −2 −1 −1 5 −1 1 • 7 4 −1 −3 1 2 −3 1 6 −2 0 7 1 0 0 −5 −4 0K13 −2 0 −1 5 −1 1 9 • 4 −1 −3 2 2 −3 0 6 0 1 2 0 0 1 −3 −4 1λp 5 0 −2 −3 5 −1 −1 −1 • −2 −4 1 1 0 4 4 0 0 2 2 0 0 −2 −3 2Iz 3 2 3 −1 −3 0 0 −1 −3 • 9 −2 −6 5 0 0 0 0 −1 −7 2 −2 0 1 −2kp 1 2 4 0 −5 1 0 0 −4 9 • −3 −5 5 0 −2 0 0 −1 −6 2 −2 0 2 −3λz 3 8 −4 0 −1 −1 1 1 1 −1 −2 • 7 −6 0 1 1 0 0 1 −2 2 1 −2 4εz 0 1 −6 1 0 0 2 2 3 −6 −6 6 • −8 0 1 1 0 0 3 −3 3 0 −4 5βz 2 −1 5 −1 1 −1 −1 −1 3 3 4 −5 −7 • 1 −4 −2 0 −1 −3 2 −3 0 3 −4λd 8 1 −3 −4 1 −2 −2 −2 6 2 −1 2 1 1 • −1 −2 −2 1 0 0 0 −3 −2 2φd −3 −1 −2 0 1 4 2 2 −3 −2 −2 0 2 −5 −2 • 3 1 3 −1 2 −1 −1 −1 −1Din 2 −1 −2 −3 4 −1 −4 −4 1 −2 −4 −1 0 −2 2 5 • 0 −3 −1 2 −2 8 7 1r(t) 1 0 −1 −1 0 −2 −1 −1 0 −1 −2 1 2 −2 3 −1 1 • 0 0 0 0 0 0 0Nin 1 −2 −1 4 −3 −1 6 7 0 1 1 −1 1 −1 0 0 −4 0 • 1 1 −1 −5 −2 −3kn −5 −1 −1 1 7 −1 −1 0 0 −4 −4 −1 1 0 −3 1 2 0 −3 • −1 1 0 −3 4α −2 0 −1 1 −2 0 1 1 −3 3 3 −2 −4 2 −2 1 0 −2 1 −2 • −9 1 1 1φb 1 −1 1 −1 2 1 −1 −1 3 −3 −3 2 4 −2 2 −1 1 3 −1 1 −9 • −1 −1 −1Im −4 −2 −1 1 3 1 0 1 −3 −4 −4 −2 0 −3 −4 6 7 0 −2 4 1 0 • 7 2εm 2 −1 3 −1 3 −1 −2 −2 2 −1 −2 −2 −1 1 2 2 6 2 −1 1 −6 6 4 • −5βm 1 1 −6 0 0 0 1 1 0 0 0 2 2 −2 1 1 0 −1 0 −1 7 −7 0 −8 •For presentation purposes, correlations have been multiplied by a factor of 10, and the main diagonal is represented by a “•”.

339


poorly determined. Comparing theM = 0 andM = 0cases indicates that the precision of nearly all param-eters increases with greater bivalve biomass. This isparticularly evident for the exchange coefficients, themussel parameters, and all parameters associated withbenthic–pelagic coupling. The exceptions areλd andλp. The remaining columns inTable 4report precisionestimates where observations on only one ecosystemcomponent is available. As expected, the precision isreduced with the tendency that only parameters directlylinked to the observables are well determined. Obser-vations of pelagic variables are more important for theM = 0 case, while observations of the benthic detritalpool are relatively more important for theM = 0 case.

The estimated correlation matrix for the parametersis shown inTable 5. Correlation measures the lineardependence amongst the parameters. A value close tozero indicates parameters are nearly independent. Ab-solute values near one indicates parameters may not beindependent, i.e. varying these parameters can lead to asimilar model predictions for the time evolution of thestate variables. Correlations were computed for caseswithM = 0 (above the main diagonal) andM = 0 (be-low the main diagonal), assuming a full observation set.For both cases, the majority of parameters have rela-tively low correlations (< 0.5); high correlations areevident only amongst theZ parameters, between theD source/sink terms (Im,Din), and between theB lossterms (α, φb). For theM = 0 case, high correlations areevident amongst theM parameters, the source/sink andl -to ancea asa den-c

5

stemm siveb tedb de-p posi-t t al.,1 ys-t -

tem effects have been widely documented for caseswhere bivalve populations have exploded as invasivespecies(Nichols, 1985; Nichols et al., 1990), or havebeen decimated as a result of disease(Ulanowicz andTuttle, 1992). Bivalves have been termed “ecosystemengineers” as a result of this ability to alter environ-mental conditions in their favour(Jones et al., 1994;Seed, 2000).

Quantitative approaches are useful for examiningaquaculture ecosystems and environmental manage-ment(Kishi et al., 1994).Brando et al. (2004)presents amass balance model of trophic flows for management ofa lagoon system as an extensive aquaculture operation.Congleton et al. (1999)considers a spatial modellingframework to take account habitat variability in sitingbivalve aquaculture. A population dynamics model ofclam culture is developed bySolidoro et al. (2003)andcoupled to one of individual bio-energetics; this wasapplied to the identification of rearing strategies to bal-ance ecological and economic considerations. Ecosys-tem models of bivalve aquaculture ecosystems havebeen focused mainly on predicting bivalve growth andcarrying capacity. One approach is to couple bivalvebioenergetic models with models of the supporting ma-rine ecosystem (Raillard and Menesguen, 1994; Dowd,1997). Increasingly the trend is to integrate complexecosystem and bioenergetic models with physical mod-els of hydrodynamic processes(Solidoro et al., 2000;Duarte et al., 2003). In our study, we emphasize envi-ronmental effects of bivalve culture. Towards this end,w e-w ua-c

ibest mi-e ons sea-s ing ab ono ing.T nk-t ithc bio-g da om-p hos.N dese on a

oss terms forZ (Z∞, λz) andP (P∞, λp), and the inerior exchange coefficients (K12,K13). Examinationf the eigenvalues and eigenvectors of the covarind correlation matrices did not prove informativemeans of identifying important parameter depenies.

. Summary and conclusions

This study has presented a bio-physical ecosyodel for assessing environmental effects of intenivalve culture in shallow coastal bays. It is motivay the potential of dense aggregates of bivalves tolete suspended particulates and increase biode

ion(Cloern, 1982; Dame and Prins, 1998; Noren e999). This can alter nutrient cycling in coastal ecos

ems(Hily, 1991; Alpine and Cloern, 1992). Ecosys

e identify a general mathematical modelling framork for quantifying ecosystem effects of bivalve aqulture from a nutrient cycling perspective.

The bio-physical model used in this study descrhe dynamics of interacting populations in a senclosed embayment. The focus is on variabilityubtidal time scales, with a particular emphasis ononal effects. Spatial aspects are incorporated usox model framework which allows for redistributif ecosystem components by water motion and mixhe pelagic compartment is comprised of phytopla

on, zooplankton, nutrients and detritus in keeping wurrent representations used in modelling pelagiceochemistry(Fasham, 1993). The model was targetet shallow coastal systems by coupling the pelagic conents to a simple, but dynamically active, bentovel aspects for the benthic compartment inclupisodic resuspension of benthic detritus based


stochastic (Poisson) process consistent with wind forc-ing; the model ecosystem appears to be resilient to theseshort time scale perturbations, in the sense defined byStone et al. (1996). In addition, remineralization of ben-thic detritus was represented using a convolution inte-gral which mimics the effects of finite reaction timesand sediment pore water diffusion as represented incomplex diagenetic models such as those inSoetaert etal. (2000). Considerable dynamical simplification wasalso achieved by superimposing a fixed population ofgrazing bivalves which interacts with the ecosystem asa forcing function.

A specific case study was undertaken for TracadieBay, a shallow semi-enclosed tidal embayment off theeast coast of Canada. The purpose was to illustrate ap-plication of the general model to a specific situation,as well as to generate and test some hypotheses on po-tential ecosystem effects resulting from intensive bi-valve culture. Recently, an extensive bio-physical fieldstudy of Tracadie Bay was initiated, and this workprovides an initial application to help focus the fieldprogram and interpret results. At this stage we makeuse of preliminary observations, information from re-gional databases and reports, and literature values tospecify model inputs, as well as to calibrate the modeloutput.

Effects of intensive bivalve culture were investi-gated by applying the aquaculture ecosystem modelto cases both with and without a population of graz-ing bivalves. Zooplankton play only a minor role inn bej iona ys-t d int ana ioni odw m-i to-p col-u udo-f idlyb n ofo lumnn ivesi allyn ares heir

standing stock actually decreases. Coastal eutrophica-tion thus appears to be partially mitigated by bivalvemediated sinks for excess nutrients, mainly the stor-age of biodeposited organic matter in the sedimentsand its burial (note that only a relatively small amountof nutrients are removed directly through harvesting).However, the major nutrient sink is through export tothe offshore, consistent with the findings ofLe Papeet al. (1999). The phytoplankton to detritus ratio maybe thought of as a measure of food quality and is animportant determinant of bivalve growth(Penney etal., 2001). Simulations suggest that bivalve popula-tions increase this ratio and therefore alter the envi-ronmental conditions in their favor, consistent withtheir hypothesized role as ecosystem engineers. Asthe fall period progresses, internal biological dynam-ics in the bay are weak, the system moves from netautotrophic to heterotrophic, and bivalves increasinglyrely on offshore production for their maintenance foodration.

In a semi-enclosed coastal embayment, ecologi-cal dynamics are determined both by local processeswithin the bay, as well as by the dynamics of thelarger scale oceanic ecosystem. These local and re-mote forcings are linked through water motion, whichredistributes and exchanges freely floating ecosystemcomponents. The importance of this physical oceano-graphic aspect in structuring the ecological system ledus to propose a unique means of determining the ex-change coefficients based on an inversion of observedt f them sedoo .,2 elu di-t ix-i stemd ov-e gice n ofd -t s oft rnale rtantr

thisp ions

utrient cycling and, from this perspective, couldustifiably neglected (but are important if productt higher trophic level is of interest). Major ecos

em effects of bivalve populations are concentratehe benthos where bivalve biodeposition leads toccumulation of organic matter. Biological product

s in fact diverted from the pelagic to the benthic foeb. In the biologically active summer period, the do

nant nutrient cycle is the following. Seston (phylankton and detritus) is depleted from the watermn by grazing. Bivalves produce feces and pse

eces from this ingested matter and which are rapiodeposited to the seabed. Benthic accumulatiorganic detritus on the seabed enhances water coutrients due to benthic remineralization. This dr

ncreased phytoplankton production in this normutrient limited system. However, phytoplanktonimultaneously grazed down by bivalves so that t

emperature records. This data-driven estimation oixing coefficients contrasts with other methods ban tidal prism calculations(Dowd, 1997)or analysisf numerical circulation model output (Chapelle et al000; Thompson et al., 2002). The limited-area modsed in this study clearly shows how far-field con

ions influence the internal biological dynamics. Mng processes may also lend stability to the ecosyynamics through the gradient-flux relation which grns exchange, similar to the way in which a pelacosystem model can be stabilized by the additioiffusive physics(Edwards et al., 2000). Water mo

ion and mixing also means that the inner portionhe bay are decoupled from the far-field, and intecosystem processes play a relatively more impoole there(Dowd, 2003).

The aquaculture ecosystem model presented inaper is intended to be applied to similar situat


with minor modification to its basic structure. Towardsthis end, an brief analysis of some model propertieswas undertaken. Since the model provides a determin-istic map from the parameters to the ecosystem statevariables, we examined aspects of parameter estima-tion using quantities derived from the Jacobian matrixof partial derivatives. Overall, the parameter sensitivitywas found to be fairly uniform. More sensitive param-eters were associated with the major pathways of massflow in the system, i.e. the mussel energy balance pa-rameters and the parameters governing benthic–pelagiccoupling. More sophisticated sensitivity analysis forecological models considers interactions, or dependen-cies, amonst the parameters(Omlin et al., 2001; Beresand Hawkins, 2001; Barlund and Tattari, 2001). To ad-dress such parameter identification and uncertainty is-sues, we computed the asymptotic covariance matrixfor the parameters, assuming availability of observa-tions on some or all of the ecological state variables.The covariance matrix is the inverse of the Hessian ma-trix, which was used byMarsili-Libelli et al. (2003)todetermine confidence regions for ecological parame-ters. Not surprisingly, it was found that parameter vari-ances, or precision, estimates were often dependenton having direct observations of the associated statevariable. The majority of parameters turned out to bequasi-independent for this estimation problem whichis encouraging for future applications. Only minor dif-ferences in the covariance structure were evident forthe two bivalve population levels considered.

emsa nd-i osys-t om-p feredi -i ame-t eal-i ad-v cificc ted,t anda stedS theb ls ine -o bio-l sti-

gate emerging areas in predictive ecosystem modellingwhich are presently characterised by formidable com-puational requirements. In ecological modelling, theseinclude stochastic simulation(Marion et al., 2000)andinverse problems(Vallino, 2000; Dowd and Meyer,2003). In marine ecology specifically, a major thrust isthe coupling of biological dynamics to high dimension,fluid dynamical models of ocean circulation(Hofmannand Lascara, 1998). It is hoped that the bio-physicalecosystem model proposed in this study proves use-ful, both for its present practical application, as well asfor future developments towards the goal of predictivemarine ecosystem modelling.

Acknowledgements

The author would like to thank Dr. Alain Vezina ofthe Bedford Institute of Oceanography, Dr. Jon Grantof Dalhousie University and Dr. Cedric Bacher ofIFREMER for discussions on the modelling of coastalecosystems. Drs. Peter Cranford, Peter Strain, PaulKepkay and Fred Page of Fisheries & Oceans Canadaare also acknowledged for their contributions and sup-port in the ongoing field activities in Tracadie Bay. Thiswork was supported by Fisheries & Oceans Canadathrough their Environmental Sciences Strategic Re-search Fund. The author was supported by an NSERCDiscovery Grant.

Ac

r ngec edu f thebr cyt 9 inT passfi s.T nlys jorf atialg re ino y

Quantitative descriptions of aquaculture ecosystre important tools to further the scientific understa

ng and the management of these manipulated ecems. Parsimonious models of such inherently clex ecological systems, such as the approach of

n this study, offer simplicity by including only domnant processes and by using reliable, robust parerizations and approximations. Although some rsm is sacrified for generality, there are importantantages over their highly complex and case-speounterparts: simple models are more easily validaheir dynamical properties can be better understoodnalysed, and hypotheses can be systematically teuch models thus both complement, and provideasis for, the more complex systems-based modecology (seeWu and Marceau, 2002). However, anther reason exists for using simple, but non-trival,

ogical models: they provide a practical basis to inve

.

ppendix A. Determining the exchangeoefficients

To implement the ecosystem box model(11)–(13)equires numerical values for the three exchaoefficients:K∞,K12,K13. These were determinsing observed temperature time series for each ooxes, designated Temp1(t), Temp2(t) and Temp3(t),espectively.Fig. 9ashows the observed low frequenemperature variations recorded in summer 199racadie Bay. These have been zero-phase, lowltered with 100 h cutoff to eliminate tidal variationhis procedure is consistent with considering oubtidal variability in the box model. The maeature in the temperature regime of the bay is a spradient of a few degrees Celsius: the temperatuffshore waters (Temp∞) is lower than the central ba


(Temp1), which is in turn cooler than the inner regions(Temp2 and Temp3).

The time evolution of temperature can be modelledby transforming(11)–(13)into a heat equation as fol-lows. LetXi = Tempi andh{·} = 0. The termν is in-terpreted as a heating rate common to all boxes. A typ-ical summer value for the net heat flux for the regionis 150 W m−2 according to the NCEP/NCAR 40 yearglobal heat flux analysis for this area in the months Julyand August, as reported inKalnay et al. (1996). Theheating rate isν = Φ/(HρCp), whereΦ = 150 W m−2

is the input heat flux,Cp = 4000 J kg−1 ◦C−1 is thespecific heat of seawater, andρ = 1025 kg m−3 is thedensity of seawater. Using a water column of depthη = 5 m yields a heating rate of 0.75◦C d−1. An in-verse method was used to determine the exchange co-efficients. Specifically, the scalar quantity

Ψ =∑i

{∫t

(Tempi(t) − Tempi(t))2 dt

}(17)

was minimized with respect toK∞,K12,K13. Thisminimization is subject to the predictedTempi satis-fying the heat equation. In other words, the exchangecoefficients are chosen such that they provide pre-dicted temperatures which best fit, in a least squaressense, the observations. Carrying out the above pro-cedure resulted in the following estimates for the ex-change coefficients:K∞ = 1.3 d−1, K12 = 0.4 d−1,an K13 = 0.5 d−1. These values were used in the boxm giveni

Ar

y

w ntra sys-tm t thec ys-t ps).

Thek × 1 error vectorε is assumed, for simplicity, tobe independent, zero-mean and normally distributedwith common varianceσ2. Ordinary least squares non-linear regression is concerned with choosing an esti-mate forθ so as to minimize the error sum of squaresS(θ) = ‖y −Hf {θ}‖2. The interested reader is referredto Chapter 24 ofDraper and Smith (1998)for a moredetailed introduction to this problem.

Here, we are concerned with an examination of theproperties of this estimation problem independent ofany particular set of observations. That is, we use onlyknowledge of the observing array (H) and the errorstatistics (ε ∼ N(0, σ2I)). Suppose then we have an es-timateθ obtained by either minimizingS(θ), or simplyas a prescribed baseline parameter set. A linearization(truncated Taylor series) off {θ} aboutθ yields

f (θ) = f {θ} + ∂f

∂θ

∣∣∣∣θ=θ

(θ − θ). (19)

The Jacobian matrix of partial derivatives is given bytheq × pmatrixJ = ∂f/∂θ. Itsi, jth element is definedas

∂fi

∂θj= lim

*θj→0

1

*θj(fi{θ +*θ(j)} − fi{θ}) (20)

where*θ(j) is ap× 1 vector with all zero elementsexcepting thejth element,*θj. Using the formula(20),J is readily approximated using finite differences (e.g.G

θ

c

δ

w

9

a TheqI forθ

v

v

odel. The associated predicted temperatures aren Fig. 9b.

ppendix B. Model analysis using nonlinearegression

Consider the nonlinear regression equation

= Hf {θ} + ε (18)

herey represents ak × 1 vector of observations ohe ecosystem state variables andθ is thep× 1 pa-ameter vector. Theq × 1 vector functionf {·} oper-tes onθ and corresponds to the discretized eco

em dynamics as in(14). Thek × q matrixH picks outodel counterparts to the observations (note tha

ase whereH = I implies measurements of all ecosem components in all boxes at all model time ste

ill et al., 1986, Section 8.6).Define the incrementsδy = y − f {θ} andδθ = θ −

ˆ. The linear regressionδy = HJδθ + δε, which is lo-alized aboutθ, yields

θ = 9J ′H ′δy (21)

here

= (J ′H ′HJ)−1.

ssuming the quantity in brackets is non-singular.uantityσ29 represents the covariance matrix forδθ.

t also provides an approximate covariance matrixˆ asymptotically valid in the vicinity ofθ. Thep× 1ector of parameter variances is then

ar(θ) = σ2 diag(9). (22)


A linearized correlation matrix@ for theθ can also bedefined. Thisp× p matrix has elements

@ij = 9ij

(9ii ×9jj)1/2(23)

for i, j = 1, . . . , p. The eigenvalues and eigenvectorsof 9, or@, may prove useful in identifying parameterssets which are poorly determined by available obser-vations.

References

Alpine, A.E., Cloern, J.E., 1992. Trophic interactions and directphysical effects control phytoplankton biomass and productionin an estuary. Limnol. Oceanogr. 37, 946–955.

Barlund, I., Tattari, S., 2001. Ranking of parameters on the basisof their contribution to model uncertainty. Ecol. Modell. 142,11–23.

Beres, D.L., Hawkins, D.M., 2001. Plackett-Burman technique forsensivity analysis of many parametered models. Ecol. Modell.141, 171–183.

Berner, R.A., 1964. An idealized model of dissolved sulphate dis-tribution in recent sediments. Geochim. Cosmochim. Acta 28,1497–1503.

Boudreau, B.P., 1997. Diagenetic Models and Their Implementa-tion. Modeling Transport and Reactions in Aquatic Sediments.Springer, Berlin 414.

Boyce, W.E., DiPrima, R.C., 1986. Elementary Differential Equa-tions and Boundary Value Problems. John Wiley & Sons, NewYork 654.

B . As-water232.

C d by

C city.

C Ma-ri-pactdell.

C mass–202.

C ari-ell.

C , S.,e. In:ef-hnical–84.

Crawford, C.M., MacLeod, C.K.A., Mitchell, I.A., 2003. Effects ofshellfish farming on the benthic environment. Aquaculture 224,117–140.

Dame, R.F., Dankers, N., Prins, T., Jongsma, H., Smaal, A., 1991.The influence of mussel beds on nutrients in the West WaddenSea and Eastern Scheldt estuaries. Estuaries 14, 130–138.

Dame, R.F., Prins, T.C., 1998. Bivalve carrying capacity in coastalecosystems. Aquat. Ecol. 31, 409–421.

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

Dowd, M., 2003. Seston dynamics in a tidal embayment with shell-fish aquaculture: a model study using tracer equations. EstuarineCoast. Shelf Sci. 57, 523–537.

Dowd, M., Meyer, R., 2003. A Bayesian approach to the ecosysteminverse problem. Ecol. Modell. 168, 39–55.

Dowd, M., Page, F., Losier, R., 2002. Time series analysis of tem-perature, salinity, chlorophyll and oxygen data from TracadieBay, PEI. Canadian Technical Report of Fisheries and AquaticSciences, vol. 2441, pp. 86.

Dowd, M., Page, F., Losier, R., McCurdy, P., Budgen, G., 2001.Physical oceanography of Tracadie Bay, PEI: Analysis of sealevel, current, wind and drifter data. Canadian Technical Reportof Fisheries and Aquatic Sciences, vol. 2347, pp. 80.

Draper, N.R., Smith, H.S., 1998. Applied Regression Analysis. Wi-ley, New York 706.

Duarte, P., Meneses, R., Hawkins, A.J.S., Zhu, M., Fang, J., Grant, J.,2003. Mathematical modelling to assess the carrying capacity formulti-species culture within coastal waters. Ecol. Modell. 168,109–143.

Dugdale, R.C., 1967. Nutrient limitation in the sea: dynamics, iden-tification and significance. Limnol. Oceanogr. 12, 685–695.

Edelvang, K., Lund-Hansen, L.C., Christiansen, C., Peterson, O.S.,Uhrenholdt, T., Laima, M., Berastegui, D.A., 2002. Modellingsuspended matter transport from the Oder River. J. Coastal Res.18, 62–74.

E ton-nkton

E ility. J.

E the

E cles.

F , M.57–

F ener. J.

F .H.,ess,

F ur ofrbi-

rando, V.E., Ceccarelli, R., Libralatos, S., Ravagnan, G., 2004sessment of environment management effects in a shallowbasin using mass balance models. Ecol. Modell. 172, 213–

aperon, J., 1967. Population growth in micro-organisms limitefood supply. Ecology 48, 715–722.

arver, C.E.A., Mallet, A.L., 1990. Estimating the carrying capaof a coastal inlet for mussel culture. Aquaculture 88, 39–53

hapelle, A., Menesguen, A., Deslous-Paoli, J.-M., Souchu, P.,zouni, N., Vaquer, A., Millet, B., 2000. Modelling nitrogen, pmary production and oxygen in a Mediterranean lagoon. Imof oysters farming and inputs from the watershed. Ecol. Mo127, 161–181.

loern, J.E., 1982. Does the benthos control phytoplankton bioin southern San Francisco Bay? Mar. Ecol. Prog. Ser. 9, 191

ongleton, W.R., Pearce, B.R., Parker, M.R., Beal, B.F., 1999. Mculture siting: a GIS description of intertidal areas. Ecol. Mod116, 63–75.

ranford, P., Dowd, M., Grant, J., Hargrave, B., McGladdery2002. Ecosystem level effects of marine bivalve aquaculturVolume I: A scientific review of the potential environmentalfects of aquaculture in aquatic acosystems. Canadian TecReport of Fisheries and Aquatic Sciences, vol. 2450, pp. 51

dwards, A.M., 2001. Adding detritus to a nutrient-phytoplankzooplankton model: a dynamical systems approach. J. PlaRes. 23, 389–413.

dwards, C.A., Powell, T.A., Batchelder, H.P., 2000. The stabof an NPZ model subject to realistic levels of vertical mixingMar. Res. 58, 37–60.

pply, R.W., 1972. Temperature and phytoplankton growth insea. Fish. Bull. 70, 1063–1085.

vans, G.T., Parslow, J.S., 1985. A model of annual plankton cyBiol. Oceanogr. 3, 327–347.

asham, M.J.R., 1993. Modelling the marine biota. In: Heimann(Ed.), The Global Carbon Cycle. Spinger-Verlag, Berlin 4504.

asham, M.J.R., Ducklow, H.W., McKelvie, S.M., 1990. A nitrogbased model of plankton dynamics in the oceanic mixed layMar. Res. 48, 591–639.

ischer, H.B., List, E.J., Koh, R.C.Y., Imberger, J., Brooks, N1979. Mixing in Inland and Coastal Waters. Academic PrNew York 483.

ranks, P.J.S., Wroblewski, J.S., Flierl, G.R., 1986. Behavioa simple plankton model with food-level acclimation by hevores. Mar. Biol. 91, 121–129.


Frechette, M., Butman, C.A., Geyer, W.R., 1989. The importance ofboundary layer flow processes in supplying phytoplankton to thebenthic suspension feederMytilus edulisL. Limnol. Oceanogr.34, 19–36.

Frost, B.W., 1975. A threshold feeding behaviour in Calanus pacifi-cus. Limnol. Oceanogr. 20, 259–262.

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

Gill, P.E., Murray, W.E., Wright, M.H., 1986. Practical Optimization.Academic Press, London 401.

Grant, J., Hatcher, A., Scott, D.B., Pocklington, P., Schafer, C.T.,Honig, C., 1995. A multidisciplinary approach to evaluating ben-thic impacts of shellfish aquaculture. Estuaries 18, 124–144.

Gregory, D., Petrie, B., Jordan, F., Langille, P., 1993. Oceanographic,geographic, and hydrological parameters of Scotia-Fundy andsouthern Gulf of St. Lawrence inlets. Canadian Technical Reportof Hydrography and Ocean Sciences, vol. 143, pp. 248.

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

Hargrave, B.T., 1985. Particle sedimentation in the ocean. Ecol. Mod-ell. 30, 229–246.

Hassell, M.P., 1978. The Dynamics of Arthropod Predator–Prey Sys-tems. Princeton University Press, Princeton, pp. 237.

Hatcher, A., Grant, J., Schofield, B., 1994. Effects of suspended mus-sel culture (Mytilus spp.) on sedimentation, benthic respirationand sediment nutrient dynamics in a coastal bay. Mar. Ecol. Prog.Ser. 115, 219–235.

Heip, C.H.R., Goosen, N.K., Herman, P.M.J., Kromkamp, J., Mid-dleburg, J.J., Soetaert, K., 1995. Production and consumption ofbiological particles in temperate tidal estuaries. Oceanogr. Mar.Biol.: Annu. Rev. 33, 1–149.

Hily, C., 1991. Is the activity of benthic suspension feeders a factorcontrolling water quality in the Bay of Brest? Mar. Ecol. Prog.Ser. 69, 179–188.

H ary.R.Studyesses

H by awfly.

I itiesorld

J cosys-

J egen-l du

J . El-

K D.,, Y.,.C.,e, R.,

Joseph, D., 1996. The NCEP/NCAR 40 year reanalysis project.Bull. Am. Meteorol. Soc. 77, 431–437.

Kaspar, H.F., Gillespie, P.A., Boyer, I.C., MacKenzie, A.L., 1985.Effects of mussel aquaculture on the nitrogen cycle and ben-thic communities in Kenepuru Sound, Marlborough Sound, NewZealand. Mar. Biol. 85, 127–136.

Kemp, W.M., Brooks, M.T., Hood, R.R., 2001. Nutrient enrichment,habitat variability and trophic transfer efficiency in simple modelsof pelagic ecosystems. Mar. Ecol. Prog. Ser. 223, 73–87.

Kishi, M.J., Uchiyama, M., Iwata, Y., 1994. Numerical simulationmodel for quantitative management of aquaculture. Ecol. Modell.72, 21–40.

Kremer, J., Nixon, S.W., 1978. A Coastal Marine Ecosystem: Simu-lation and Analysis. Springer-Verlag, New York, pp. 217.

Le Pape, O., Jean, F., Menesguen, A., 1999. Pelagic and benthictrophic chain coupling in a semi-enclosed coastal system, theBay of Brest (France): a modelling approach. Mar. Ecol. Prog.Ser. 189, 135–147.

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

Marsili-Libelli, S., Guerrizio, S., Checchi, N., 2003. Confidence re-gions of estimated parameters for ecological systems. Ecol. Mod-ell. 165, 127–146.

Middleburg, J.J., Soetart, K., Herman, P.M.J., 1997. Empirical rela-tionships for use in global diagenetic models. Deep Sea Res. 44,327–344.

New, M.B., 1999. Global aquaculture: current trends and challengesfor the 21st century. World Aquacult. 30, 8–13.

Nichols, F.H., 1985. Increased benthic grazing: an alternative expla-nation for low phytoplankton biomass in northern San FranciscoBay during the 1976–1977 drought. Estuarine Coastal Shelf Sci.21, 379–388.

Nichols, F.H., Thompson, J.K., Schemel, L.E., 1990. Remarkableinvasion of San Francisco Bay (California, USA) by the Asian

rmer

N on,ort-r. 67

N nktonSer.

O l ofsis.

P orne,ano-s andeanog-ull.

P ano-

P ejak,the

ofmann, E.E., Lascara, C.M., 1998. Overview of interdisciplinmodeling for marine ecosystems. In: Brink, K.H., Robinson, A(Eds.), The Sea: Ideas and Observations on Progress in theof the Seas. Volume 10: The Global Coastal Ocean: Procand Methods. Wiley, New York.

olling, C.S., 1959. The components of predation as revealedstudy of small mammal predation of the European pine saCan. Entomol. 91, 293–320.

ncze, L.S., Lutz, R.A., True, E., 1981. Modeling carrying capacfor bivalve molluscs in open suspended-culture systems. J. WMaricult. Soc. 12, 143–155.

ones, C.G., Lawton, J.H., Shachak, M., 1994. Organisms as etem engineers. Oikos 69, 373–386.

ones, R., Henderson, E.W., 1986. The dynamics of nutrient reration and simulation studies of the nutrient cycle. JournaConseil 43, 216–236.

orgensen, S.E., 1994. Fundamentals of Ecological Modellingsevier, Amsterdam 687.

alnay, E., Kanamitsu, M., Kistler, R., Collins, W., Deaven,Gandin, L., Iredell, M., Saha, S., White, G., Woollen, J., ZhuChelliah, M., Ebisuzaki, W., Higgins, W., Janowiak, J., Mo, KRopelewski, C., Wang, J., Leetmaa, A., Reynolds, R., Jenn

clam Potamocorbula amurensis. II. Displacement of a focommunity. Mar. Ecol. Prog. Ser. 66, 95–101.

isbet, R.M., Diehl, S., Wilson, W.G., Cooper, S.D., DonaldsD.D., Kratz, K., 1997. Primary productivity gradients and shterm population dynamics in open systems. Ecol. Monog(4), 535–553.

oren, F., Haamer, J., Lindahl, O., 1999. Changes in the placommunity passing a Mytilus edulis bed. Mar. Ecol. Prog.19, 187–194.

mlin, M., Brun, R., Reichert, P., 2001. Biogeochemical modeLake Zurich: sensitivity, identifiability and uncertainty analyEcol. Modell. 141, 105–123.

age, F., Greenberg, D., Bugden, G., Losier, R., Shore, J., HE., Robinson, S., Chang, B., Sephton, T., 1999. Ocegraphic component of the Canadian Department of FisherieOceans science strategic research program on Coastal Ocraphy for Sustainable Aquaculture Development (COSAD). BAquacult. Assoc. Can. 99–2, 14–16.

arsons, T.R., Takahashi, M., Hargrave, B., 1977. Biological Ocegraphic Process. Permagon, Oxford 220.

astres, R., Solidoro, C., Cossarini, G., Melaku Canu, D., DC., 2001. Managing the rearing of Tapes philippinarum in


lagoon of Venice: a decision support system. Ecol. Modell. 138,231–245.

Penney, R.W., McKenzie, C.H., Mills, T.J., 2001. Assessment of theparticulate food supply available for mussel (Mytilus spp.) farm-ing in a semi-enclosed northern inlet. Estuarine Coastal ShelfSci. 53, 107–121.

Platt, T., Gallegos, C.G., Harrison, W.G., 1980. Photoinhibition ofphotosynthesis in natural assembalges of marine phytoplankton.J. Mar. Res. 38, 687–701.

Platt, T., Jassby, A.D., 1976. The relationship between photosynthe-sis and light for natural assemblages of coastal marine phyto-plankton. J. Phycol. 12, 421–430.

Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P., 1996.Numerical Recipes in Fortran 90. Cambridge University Press,Cambridge 576.

Raillard, O., Menesguen, A., 1994. An ecosystem box model for esti-mating the carrying capacity of a macrotidal shellfish ecosystem.Mar. Ecol. Process Ser. 115, 117–130.

Riley, G.A., 1947. A theoretical analysis of the zooplankton popula-tion on Georges Bank. J. Mar. Res. 6, 104–113.

Seed, R., 2000. Mussels: space monopolisers or ecosystem engineers.J. Shellfish Res. 19, 611–612.

Soetaert, K., Middelburg, J.J., Herman, P.M.J., Buis, K., 2000. On thecoupling of benthic and pelagic biogeochemical models. EarthSci. Rev. 51, 173–201.

Solidoro, C., Melaku Canu, D., Rossi, R., 2003. Ecological and eco-nomic considerations on fishing and rearing of Tapes phillip-inarum in the lagoon of Venice. Ecol. Modell. 170, 303–318.

Solidoro, C., Pastres, R., Melaku Canu, D., Pellizzato, M., Rossi, R.,2000. Modelling the growth of Tapes philippinarum in northernAdriatic lagoons. Mar. Ecol. Prog. Ser. 199, 137–148.

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

Stone, L., Grbric, A., Berman, T., 1996. Ecosystem resilience, sta-bility and productivity: seeking a relationship. Am. Nat. 108,892–903.

Takeoka, H., 1984. Fundamental concepts of exchange and trans-port time scales in a coastal sea. Continental Shelf Res. 3, 311–326.

Thompson, K.R., Dowd, M., Shen, Y., Greenberg, D.A., 2002. Prob-abilistic characterization of mixing in a coastal embayment:A Markov chain approach. Continental Shelf Res. 22, 1063–1614.

Ulanowicz, R.E., Tuttle, J.H., 1992. The trophic consequences ofoyster stock rehabilitation in Chesapeake Bay. Estuaries 15, 298–306.

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

Waniek, J.J., 2003. The role of physical forcing in initiation of springblooms in the northeast Atlantic. J. Mar. Syst. 39, 57–82.

Widdows, J., Brinsley, M.D., Salkeld, P.N., Elliott, M., 1998. Useof annular flumes to determine the influence of current velocityand bivalves on material flux at the sediment–water interface.Estuaries 21, 552–559.

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

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