Corrigendum to Sensitivity analysis and pattern-oriented validation of TRITON, a model with...

Ecological Modelling 274 (2014) 71

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Corrigendum

Corrigendum to “Sensitivity analysis and pattern-oriented validationof TRITON, a model with alternative community states: Insights ontemperate rocky reefs dynamics”[Ecol. Model. 258 (2013) 16–32]

Martin P. Marzloff a,b,∗, Craig R. Johnsona, L. Rich Littleb, Jean-Christophe Souliéc,Scott D. Linga, Stewart D. Frushera

a Institute for Marine and Antarctic Studies, University of Tasmania, Private Bag 129, Hobart, Tasmania 7001, Australiab CSIRO Wealth from Ocean Flagship/CSIRO Marine and Atmospheric Research, Castray Esplanade, Hobart, Tasmania 7000, Australiac CIRAD - BIOS “BIOlogical Systems” Department, UMR AGAP - TA-A 108/01, Avenue Agropolis Lavalette, 34398 Montpellier Cedex 5, France

The authors regret that throughout most of the article references to Fig. 5, 6, 7 or 8 do not always correspond to the actual figure describedin the text. Specifically, Fig. 5(a and b) (about the pattern-oriented validation of the mean behaviour of the model against large-scale reefsurvey data) is wrongly referred to as Fig. 6(a and b) throughout section “4.2. Pattern-oriented model validation”. Note also that Figs. 6and 7 are also sometimes wrongly referred to as Figs. 7 and 8, for instance in the “Results” section. Unfortunately, the mismatch betweencross-referencing in the text and actual figure numbers is not consistent throughout the whole article so readers will have to decide forthemselves whether references to any of Fig. 5, 6, 7 or 8 in the main text are actually correct or not.

The authors would like to apologise for any inconvenience caused.

DOI of original article: http://dx.doi.org/10.1016/j.ecolmodel.2013.02.022.∗ Corresponding author at: Institute for Marine and Antarctic Studies, University of Tasmania, Private Bag 129, Hobart, Tasmania 7001, Australia.

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

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Ecological Modelling 258 (2013) 16– 32

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Sensitivity analysis and pattern-oriented validation of TRITON, amodel with alternative community states: Insights on temperaterocky reefs dynamics

Martin P. Marzloff a,b,∗, Craig R. Johnsona, L. Rich Littleb, Jean-Christophe Souliéc,Scott D. Linga, Stewart D. Frushera

a Institute for Marine and Antarctic Studies, University of Tasmania, Private Bag 129, Hobart, Tasmania 7001, Australiab CSIRO Wealth from Ocean Flagship/CSIRO Marine and Atmospheric Research, Castray Esplanade, Hobart, Tasmania 7000, Australiac CIRAD - BIOS “BIOlogical Systems” Department, UMR AGAP - TA-A 108/01, Avenue Agropolis Lavalette, 34398 Montpellier Cedex 5, France

a r t i c l e i n f o

Article history:Received 6 December 2012Received in revised form 19 February 2013Accepted 22 February 2013Available online 1 April 2013

Keywords:Model calibrationParameter uncertaintyPhase shiftHysteresisRock lobsterSea urchin barren

a b s t r a c t

While they can be useful tools to support decision-making in ecosystem management, robust simulationmodels of ecosystems with alternative states are challenging to build and validate. Because of the possi-bility of alternative states in model dynamics, no trivial criteria can provide reliable and useful metricsto assess the goodness-of-fit of such models. This paper outlines the development of the model TRITON,and presents simulation-based validation and analysis of model sensitivity to input parameters. TRITONcaptures the local dynamics of seaweed-based rocky reef communities in eastern Tasmania, which nowoccur in two alternative persistent states: (1) either as dense and productive seaweed beds, (2) or as seaurchin ‘barrens’ habitat, i.e. bare rock largely denuded of macroalgae and benthic invertebrates due todestructive grazing by sea urchins. Pattern-oriented-modelling, i.e. comparing patterns in model dynam-ics across Monte–Carlo simulations with direct observations of Tasmanian reef communities over largescales, provides a valuable approach to calibrate the dynamics of TRITON.

Using the computationally efficient, model-independent extended Fourier amplitude sensitivity test,we identify fishing down of predatory lobsters, sea urchin recruitment rate, as well as seaweed growthrate as key parameters of influence on overall model behaviour. Through a set of independent sensitivitytests, we isolate different sets of drivers facilitating the ‘forward’ shift from the seaweed bed to theurchin-dominated state, and the reverse or ‘backward’ shift from denuded sea urchin barren to recoveryof seaweed cover. The model suggests that rebuilding populations of large rock lobsters, which predatethe urchins, will be effective in limiting ongoing formation of sea urchins barrens habitat, but that thechances of restoring seaweed beds from extensive barrens are relatively low if management relies solelyon rebuilding stocks of large rock lobsters. Moreover, even when it does occur, seaweed bed restorationtakes up to three decades in the simulations and so is arguably unrealistic to implement under short-termfishery management plans. The process of model validation provided both a better understanding of thekey drivers of community dynamics (e.g. fishing of predatory lobsters), and an assessment of priorityareas for future research.

© 2013 Elsevier B.V. All rights reserved.

1. Introduction

Models of ecological dynamics can be helpful to inform decision-making and improve the management of human activities that relyon natural resources (Clark et al., 2001; Smith et al., 2011). Morespecifically, simulation models can be useful decision-support

∗ Corresponding author at: Institute for Marine and Antarctic Studies, Universityof Tasmania, Private Bag 129, Hobart, Tasmania 7001, Australia.

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

tools to assess the effects of different management scenarios inecosystems with alternative community states, where anthro-pogenic effects can lead to dramatic and possibly irreversiblechanges in structure and function across entire landscapes (Esteset al., 2011; Firn et al., 2010; Fung et al., 2011; Melbourne-Thomaset al., 2010; Mumby et al., 2007; Scheffer et al., 2001). How-ever, building reliable simulation models requires a comprehensiveunderstanding of key processes and drivers of system dynamics,and the accuracy of simulations will depend on the robustness ofmodel parameterisation. Ecological processes, especially trophicinteractions, are by essence variable and the dynamics of systems

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M.P. Marzloff et al. / Ecological Modelling 258 (2013) 16– 32 17

can be sensitive to this variation. However, ecological processesare usually difficult to measure precisely (Novak, 2010) and evenin well-studied ecosystems, a complete and precise understand-ing and quantification of ecological processes is rarely possible.Thus, uncertainty arises as a major feature of ecological models,stemming from the variable nature of ecological processes, fromimperfect understanding of the mechanisms underpinning ecosys-tem dynamics, and limited ability to quantify complex naturalprocesses with precision (Saltelli et al., 2000).

In this context, useful ‘minimum realistic’ ecological modelsmust adequately address questions of interest to managementwhile accounting for the amount and reliability of the informa-tion available about the study system (Fulton et al., 2003). Theart of ecosystem modelling lies in making a series of assumptionsand, to a certain degree, an ecological model is only as reliableas the modeller’s understanding of system dynamics (Klepper,1997). Therefore, simulation models require objective assessmentprior to their application, and several approaches are available tovalidate and calibrate the dynamics of complex ecosystem mod-els (Duboz et al., 2010; Klepper, 1997; Turley and Ford, 2009).Model calibration is often undertaken by optimising the fit ofsimulated community dynamics to available empirical observa-tions. Snapshots or mean observations of the composition of thestudy system are often used as metrics for model validation (e.g.mean species biomasses; see Marzloff et al., 2009), although thesecriteria poorly characterise the variability of system dynamics,which may be of critical importance. In ecosystems that exhibitalternative states, ecologists can exhaustively study and describecommunities in one state or the other, while discontinuous shiftsin community dynamics are, by definition, swift and are thusrarely observed or monitored (Scheffer et al., 2001). Therefore, pre-cise information of a system with hysteresis (i.e. where a smallchange in parameters or species abundance can lead to a dra-matic shift to a new community state that persists even whenthe change is reversed; see Donahue et al., 2011) at its thresholdpoints is nearly always lacking. Lack of observations of commu-nity dynamics for systems that manifest hysteresis, and lack ofmeaning in mean observations in these systems, make validationof ecosystem models with alternative states particularly challeng-ing (Scheffer and Carpenter, 2003, but see Mumby et al., 2007;Lauzon-Guay et al., 2009; Fung et al., 2011 for examples of modelvalidation).

Given the inability to formally and comprehensively validatethe accuracy of ecosystem models against reality, predictionsfrom ecosystem models are inherently uncertain. Uncertaintyin simulation models can be broken down into three maincomponents:

(i) structural uncertainty, which refers to model structure and itsresolution, e.g. the extent to which species are aggregated, orthe nature of functional groups; the number and certainty oftrophic and other ecological interactions considered; and thespatial and temporal scales of relevant physical and ecologicalprocesses (Hosack et al., 2008; Laskey, 1996; Marzloff et al.,2011);

(ii) choice of model formulation, which includes programmingchoices (e.g. discrete versus continuous time (Deng, 2008),the timing of processes operating at different scales, andwhether the model is spatially explicit) as well as the particularrepresentation of ecological processes in the model (e.g. alter-native ways to account for density- dependence in functionalresponses; Skalski and Gilliam, 2001);

(iii) uncertainty in model parameterisation; uncertainty in indi-vidual parameter estimates, which can rapidly compounddepending on interactions in the model, contributes directly

to uncertainty in model outputs (Cariboni et al., 2007; Saltelliet al., 2000).

Assessing these different sources of model uncertainty is anessential ingredient of ecological modelling (Marzloff et al., 2011;Saltelli et al., 2000). An added complication for models with alterna-tive community states is that sensitivity analysis can be of limitedvalue (van Nes et al., 2003). This is because simulation outcomesmay only reflect whether the community reaches one state or theother and only partially depict hysteresis in model dynamics. Addi-tionally, the modelled community is more prone to shift to analternative state when parameter space is near bifurcation points,so linear and partial sensitivity tests are limited because they typ-ically neglect the influence of interactions between multiple inputparameters giving rise to complex non-linear dynamics (Saltelliet al., 1999; van Nes et al., 2003).

In this paper, we explore and validate the behaviour of amodel of subtidal seaweed-based reef community dynamics ineastern Tasmania, south east Australia. These temperate rockyreefs occur in two alternative community states: productive anddiverse stands of canopy macroalgae referred to as ‘seaweed bed’habitat; or as bare rocky expanses known as sea urchin ‘bar-ren’ habitat (Johnson et al., 2005; Ling et al., 2009a). On theeast coast of Tasmania, the climate-driven range extension ofthe long-spined sea urchin Centrostephanus rodgersii represents amajor threat to endemic seaweed bed communities (Ling, 2008)including high value commercial species such as rock lobster andabalone (Johnson et al., 2011). Within its new eastern Tasmanianrange, C. rodgersii forms and maintains extensive barrens habi-tat, i.e. areas of bare rock up to tens of hectares, following thedestruction of seaweed beds by its grazing activity. Large lobsters(carapace length > 140 mm) constitute the only efficient preda-tors of C. rodgersii in south eastern Australian waters (Ling et al.,2009a), so that commercial and recreational fishing of lobstersdirectly facilitates the formation of C. rodgersii barrens. Comparedto the seaweed beds, sea urchin barrens have dramatically lowerproductivity (Chapman, 1981), habitat complexity and speciesdiversity (Ling, 2008). Note that key commercial species do notoccur in commercially harvestable quantities on barrens habitat(Johnson et al., 2005, 2011). Thus, preventing the formation of fur-ther C. rodgersii barrens, and promoting the reverse shift back toseaweed beds where barrens now occur, is a priority for the man-agement of reef communities and fisheries in eastern Tasmania(Ling et al., 2009a). It is therefore important that managers under-stand the fundamentally different ecologies operating within eachalternative state, the ecological mechanisms that drive the shiftfrom dense seaweed bed to urchin barrens and vice versa, andthe circumstances in which these shifts are likely to occur. Here,we calibrate and validate model behaviour against observed pat-terns that describe community dynamics, including shifts betweenthese alternative states. Structural uncertainty has been compre-hensively tested in this model (Marzloff et al., 2011) and hencethis paper focuses on sensitivity to uncertainty in model formu-lation and parameterisation. Using Monte–Carlo simulations, weexplore the effects of parameter uncertainty on the behaviour ofthe model.

Our work comprises three steps: first, we quantify model sen-sitivity to alternative formulations and input parameters usingthe extended Fourier amplitude sensitivity test (FAST), a quantita-tive model-independent sensitivity analysis technique for complexsimulation models (Saltelli et al., 1999). The extended FAST assessesthe contribution to model output variance of each input parame-ter, including through interactions with other factors. We analysemodel global behaviour as well as specific components of itsdynamics; by decomposing overall model dynamics into ‘forward’shift (from seaweed bed to barren) and ‘backward’ shift (from

18 M.P. Marzloff et al. / Ecological Modelling 258 (2013) 16– 32

barren back to seaweed bed) components, the sensitivity testsovercome problems inherent to sensitivity analysis of models withhysteresis (van Nes and Scheffer, 2003). Second, we use sensitiv-ity analysis to identify sources of model uncertainty and select anadequate ‘minimum realistic’ model form that can adequatelytackle key management questions, i.e. estimate thresholds in com-munity dynamics and assess community-level effects of alternativemanagement scenarios. We compare the dynamics of Monte–Carlosimulations against large-scale patterns observed on Tasmanianreefs to validate model behaviour, and calibrate the propensityof the simulated community to shift from the seaweed bed tothe sea urchin barren state against the known probability of bar-rens formation in south eastern Australia. Finally, the sensitivityanalysis helps to both identify key ecological processes that driveTasmanian reef community dynamics, and highlight gaps in knowl-edge about processes of high influence on community dynamics.In this context the sensitivity analysis provides a valuable toolto guide and prioritise future research about temperate reefdynamics.

2. Materials and methods

We developed a simulation model of Tasmanian reef communi-ties, which we have called TRITON (Temperate Reefs in Tasmaniawith lObsters and urchiNs), to inform best management practicesagainst shifts in these ecological dynamics. If simulation mod-elling is to assist management of formation of barrens habitatby overgrazing by the urchins, the ability of TRITON to real-istically capture the potential for discontinuous shifts betweenthe two alternative states (seaweed bed versus sea urchin bar-ren) is essential. The following subsections describe the structureof the TRITON model, its parameterisation and the empiricaldata available to calibrate model dynamics. We then outlinethe extended Fourier amplitude analysis sensitivity test (FAST;Saltelli et al., 1999) used to test model sensitivity to parame-ter values, before specifying both the simulation characteristicsand the important output metrics screened for the sensitivitytests.

2.1. TRITON: local dynamics of Tasmanian rocky reefcommunities

TRITON represents the mean community dynamics of an indi-vidual patch of rocky reef (area 100 m2–10 ha; depth 8–35 m on

open exposed reef habitat where C. rodgersii barrens occur in Tas-mania). The dynamics of three functional groups or species arecaptured explicitly (Fig. 1), representing the dynamics of the sea-weed bed (SW) (Eq. (1)), the sea urchin C. rodgersii (CR) (Eq. (2))and rock lobsters (RL) (Eq. (3)). Size-structured dynamics for bothsea urchin and rock lobster populations are key for TRITON to real-istically capture both the effects of size-related fishing regulations(e.g. legal size), and the size-structured nature of lobster predationon the urchin (Ling et al., 2009a) (cf. Eq. (2)). Each is introduced inturn:

(i) The seaweed bed (SW) includes understorey algal assem-blages and all canopy-forming macroalgae dominated byEcklonia radiata at depth > 6 m, or Phyllospora comosa on shal-low reef (generally with small contribution < 5% covers of otherlarge phaeophytes, including representatives of the generaCystophora, Sargassum, Carpoglossum, Acrocarpia). The under-standing of both the dynamics of the different guilds of algaethat constitute the seaweed bed, and the details of overgraz-ing of these different algal species and groups by C. rodgersii isincomplete. Thus, in the model, the seaweed bed compartmentcorresponds to the current minimum realistic representationof temperate algal communities. Seaweed assemblage dynam-ics follow logistic growth (Eq. (1)), with parameters derivedfrom monitoring macroalgal recovery from a barren state overtwo years after experimental removal of the urchins (Figs.A1 and A2; Appendix A; Ling, 2008). Propagule supply isassumed to be constant and independent of the local state ofthe seaweed bed, as external supply from adjacent macroal-gal beds is not limiting (CR Johnson and SD Ling, personalobservations). Although a range of herbivorous species relyon macroalgae as part of their diet, TRITON focuses exclu-sively on the dynamics of exposed inshore reefs where onlyC. rodgersii has demonstrated the ability to overgraze Tas-manian seaweed beds. Thus, grazing by the long-spined seaurchin is the only explicit source of seaweed biomass lossin the model. Urchin grazing rate is assumed to be constant,dissimilar to northern hemisphere strongylocentroid urchinsthat destructively graze seaweeds by forming a grazing frontonce critical density and behavioural thresholds are reached(Lauzon-Guay et al., 2009). In Tasmania there is no evidenceof density-dependence of C. rodgersii grazing rate, and theeffects of individual grazers are additive. Across incipientand extensive barrens habitat, sea urchin destructive grazingshows a remarkably consistent ratio of ∼0.6 m2 of grazed areaper individual emergent (of test diameter > 70 mm; smallerindividuals stay cryptic, hiding in crevices) urchin irrespec-tive of the size of the barrens patch (Flukes et al., 2012).Although all size classes of emergent urchins consume sea-weed at the same rate for a given biomass of urchins (thelast term in Eq. (1)), larger urchin individuals have a higherper capita destructive impact on standing macroalgae in themodel since urchin population dynamics (see Eq. (2)) capturebiomass gain from one size class to the next due to indi-vidual growth. The equation for the seaweed assemblage isgiven as:

SWt+1 = max

⎛

⎜⎜⎜⎜⎝0, rSW︸︷︷︸

Recruitment

+ SWt ×

⎛

⎜⎜⎝1 + ˛SW × (KSW − SWt)KSW︸︷︷︸

Logistic population dynamics

⎞

⎟⎟⎠− ˇSW,CR ×NCR∑

s=1

CRs,t

︸︷︷︸Urchin grazing

⎞

⎟⎟⎟⎟⎠(1)

where SWt is the biomass density of seaweed at time t(g/200 m2); rSW, seaweed recruitment rate (g/year/200 m2);˛SW, seaweed intrinsic growth rate (/year); KSW, seaweed car-rying capacity (g/200 m2); ˇSW,CR, sea urchin grazing rate (g ofSW/g of /CR/year/200 m2); CRs,t, biomass density of sea urchinsin size class s at time t (g/200 m2).

(ii) Population growth of C. rodgersii is size-structured (Eq. (2))and fitted against data from large-scale population surveyson the east coast of Tasmania (Fig. A3; Appendix A; Linget al., 2009b; Johnson et al., 2011). Despite its destructivegrazing of seaweed beds, sea urchin population dynamics is


Fig. 1. Conceptual diagram of TRITON, a model of local community dynamics on rocky reefs in eastern Tasmania. The boxes represent the three functional groups or speciesexplicitly interacting in TRITON, namely southern rock lobster, long-spined sea urchin and the seaweed assemblage. Each box lists all the parameters defining the dynamicsof each group. Interactions between the three groups are represented as arrows, where a full circle at the end of lines indicates a negative effect to the adjacent group whilean actual arrow head points to a group positively affected in this interaction. Photography credits: Scott D. Ling.

independent of seaweed consumption because sea urchinsforage on drift material, ephemeral filamentous algae andmicroalgae to subsist on barrens habitat in the absence ofattached canopy macroalgae (Ling and Johnson, 2009). InTRITON, the size structure of sea urchin individuals is dis-tributed across 21 size classes ranging from 40 to 120 mmtest diameter using 4 mm increments (Fig. A6; Appendix A).

The effect of habitat complexity on survival of juveniles (pro-vision of crevices to shelter from predation) is implicitlymodelled in the Monte–Carlo simulations through changesin mean recruitment rate. Only adult animals of test diam-eter > 70 mm are fully emergent in Tasmania and smallerindividuals largely stay cryptic in crevices, with virtually noeffect on standing macroalgae through grazing and likely very


limited interactions with rock lobster (Ling and Johnson, 2012;Ling et al., 2009a). Hence, only these larger animals affect sea-weed material and are exposed to lobster predation in themodel. Recruitment is stochastic and independent of local pop-ulation size given that C. rodgersii has a planktotrophic larvalstage of ca. 3 months duration that disperses with currents atscales of 102–103 km (Huggett et al., 2005; Banks et al., 2007).The southern rock lobster is the only effective predator of C.rodgersii in Tasmanian waters. Because a lobster’s ability tohandle a given size of sea urchin is determined by the size ofits front pair of walking legs (Ling et al., 2009a), predation ofC. rodgersii by rock lobster is constrained by the relative size ofprey and predator (Eq. (2)). Hence, size-structured predationby lobsters (third term of Eq. (2)) is the only explicit sourceof natural mortality on sea urchins in the model. The preda-tion rate ˇCR,RL accounts for density-dependence of C. rodgersiipredation following any of Holling’s Type I, II or III functionalresponses (Holling, 1966; cf. Fig. A11 and Tables A7 and A8 inAppendix A for further details about the definition and param-eterisation of Holling’s functional responses in TRITON). Theequation for urchin dynamics is given as:

CRs,t+1 = max

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0, rCR︸︷︷︸Recruitment to the first size class

(only if s = 1)

+CRs,t × exp(−MCRs )︸︷︷︸Biomass at time t affected

by natural mortality

+

j<s∑

j=1

(ı′s,j × CRj,t) −

(NCR∑

i>s

ıi,s

)× CRs,t

︸︷︷︸Growth between different size classes accounts

for individual weight gain

− ˇCR,RL

NRL∑

i=minCL

RLi,t

︸︷︷︸Size-structured predation

− CRs,t × (1 − exp(−FCRs ))︸︷︷︸Culling mortality

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(2)

where CRs,t is the biomass density of sea urchin in size classs (g/200 m2); rCR, urchin recruitment rate to the first sizeclass s = 1 (g/year/200 m2), and where the mean recruitmentrate !CR varies stochastically (see below); MCR, urchin naturalmortality (/year); FCR, urchin harvesting mortality (/year); ıi,s,abundance-based growth transition probability from size classs to i (/year); ı′

s,j, biomass-based growth transition probabilityfrom size class j to s (/year); ˇCR,RL, lobster predation rate on seaurchins of size class s (g of CR/g of /RL/year/200 m2), which fol-lows any of Holling’s Type I, II or III functional responses. Onlysize classes of lobsters larger than a critical carapace length(CLmin, in mm) can prey on urchins of class s; this minimumcarapace length CLmin for rock lobster to predate upon seaurchin individuals of a given test diameter (TD, in mm) can beexpressed after Ling et al. (2009a) as: CLmin = ˛1 log(TD) − ˛2where ˛1 and ˛2 are scalars defining the allometry of the size-structured interaction (cf. Appendix A, Section 3.2.2).

Recruitment to the smallest emergent size class of urchinsin a given year is determined in part by a binomial term whichdetermines whether a recruitment event will occur at all in anygiven year, which acknowledges that water temperatures insome years are not sufficiently warm to support larval devel-opment (Ling et al., 2008). When recruitment does occur, itsmagnitude is determined with a parameter ! from a uni-form distribution ranging between minimum and maximum

absolute values (and which reflects variability between reefs,with some reefs consistently receiving more recruits thanothers on average) modified by a lognormal scaling quantity(with mean "CR and standard deviation #CR) to capture annualstochastic variation (cf. Eq. (A4) and Fig. A5 in Appendix A fordetails).

(iii) The size-structured rock lobster (RL) population componentis derived from the Tasmanian rock lobster fishery stockassessment model (see McGarvey and Feenstra, 2001; Puntand Kennedy, 1997), and so TRITON represents the lob-ster population across 31 size classes ranging from 65 to215 mm of carapace length by 5 mm increments. This enablesa realistic representation of the effects of size-related fishingregulations (e.g. minimum legal sizes are 105 and 110 mmcarapace length for females and males, respectively). Thelobster size-structured population model was closely fittedagainst observed population recovery following protectionfrom fishing (Figs. A4 and A8 in Appendix A; Barrett et al.,2007). The natural mortality term accounts for sources of

mortality that are not explicitly captured elsewhere in theequation, e.g. through predation by sharks or cephalopods(Pecl et al., 2009).

The lobster population in the model relies on the localstate of the seaweed bed as an essential source of habi-tat and food. More specifically, abundances of juveniles arelower on barrens habitat than in adjacent kelp beds, whileobservations associated with experimental manipulation oflarge lobsters suggest that abundances of large supra-legalpredatory-capable lobsters are largely unaffected by barrenshabitat (Fig. A10; Johnson et al., unpublished data). Canopy-forming macroalgae can facilitate both, settlement of lobsterpuerulus by providing a complex three-dimensional structureand development of juvenile lobsters by supporting a broaddiversity of invertebrate prey species (Ling, 2008). Therefore,a constant coefficient, ranging from 0 (for no recruitment onbarren habitat) to 1 (for no effect of barrens on recruitment),scales lobster recruitment as a function of seaweed cover, i.e.extent of sea urchin barrens (cf. first term of Eq. (3); TableA5). Lobster recruitment rate rRL is (i) stochastic followinga lognormal stochastic function and (ii) independent of thelocal lobster population given that lobsters have an 18–24month pelagic larval stage, implying large-scale dispersal(Bruce et al., 2007). The equation for rock lobster dynamics isgiven as:


RLs,t+1 = max

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0, rRL

[1 − (1 − $RL,SW)

(1 − SWt

KSW

)]

︸︷︷︸Recruitment to the first size class (only if s = 1)

gets reduced as barren habitat expands

+ RLs,t × exp(−MRL)︸︷︷︸Biomass at t affected by

natural mortality

+

j<s∑

j=1

(ı′s,j × RLj,t ) −

(NRL∑

i>s

ıi,s

)× RLs,t

︸︷︷︸Growth between different size classes accounts

for individual weight gain

− (1 − exp(−FRLs )) × RLs,t︸︷︷︸Fishing mortality

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(3)

where RLs,t is the biomass density of rock lobsters in sizeclass s at time t (g/200 m2); rRL, lobster recruitment rate(g/year/200 m2), which derives from a mean recruitment rate!RL varied stochastically with a lognormal stochastic func-tion of mean "RL and standard deviation #RL (Fig. A7); $RL,SWis a scalar, ranging from 0 for no lobster recruitment onbarren grounds to 1 for no effect of barrens habitat on lob-ster mean recruitment; MRL, lobster natural mortality (/year);ı′

s,j, biomass-based transition probability from size class j tos, or element of the sth row, jth column of the transitionprobability matrix (/year or g/g/year); ıi,s, abundance-basedtransition probability from size class s to i (/year or individ-ual/individual/year); SWt, seaweed biomass density at time t(g/200 m2); FRLs, fishing mortality for lobster of class s (/year).

Recruitment rates vary stochastically for both lobster and seaurchin populations (See Eqs. (2) and (3)), while propagule supplyis assumed constant for the seaweed bed (Eq. (1)). Recruitment isindependent of local spawning population densities: indeed, for allthree modelled groups, larval or propagule settlement occurs overmuch larger spatial scales than individual reefs (Banks et al., 2010;Johnson, unpublished data; Banks et al., 2007; Coleman et al., 2011;Linnane et al., 2010).

2.2. Parameterisation

Variables are expressed as fresh weight biomass density witha default parameterisation for a reef area of 200 m2 (variablesin g/200 m2). Biomass density allows for weight-based (ratherthan abundance-based) trophic interactions and was derived fromexperimental or other empirical observation (Table 1; see AppendixA for details). All modelled processes were parameterised fromin situ observations or measurements (Barrett et al., 2009; Ling,2008; Ling and Johnson, 2009; Ling et al., 2009b; Redd et al., 2008),field- or laboratory-based experiments (Hill et al., 2003; Ling et al.,2009a), or well-validated models (McGarvey and Feenstra, 2001;Punt and Kennedy, 1997; Punt et al., 1997). For each parameter,Table 1 summarises data sources and the estimated distributionof each parameter (i.e. mean and standard deviation for normaldistributions; minimum and maximum bounds for uniform distri-butions). For normally distributed parameters, values within the90% confidence interval (bounded by the 5 and 95% quantiles) wereexplored during the sensitivity analyses. As well as envelopinguncertainty in modelled processes, these ranges implicitly encom-pass the span of environmental conditions (e.g. habitat, depth)and anthropogenic forcing (e.g. fishing pressure) encountered on

Tasmanian rocky reefs. Appendix A comprehensively describes alldata sources and the estimation of model parameters.

2.3. Global sensitivity analysis with the extended Fourieramplitude sensitivity test (FAST)

The extended FAST, available within the R ‘sensitivity’ package(R Development Core Team, 2010), does not assume linearity ormonotony between model inputs and outputs, and hence providesa robust quantitative and model-independent sensitivity analysismethod for models of complex systems dynamics (Saltelli et al.,1999). In the absence of sufficient empirical data to derive distri-butions, we assumed uniform distributions for input parameterswithin the bounds given in Table 1. The experimental plan ofthe extended FAST follows a Latin hypercube sampling designwhere a unique frequency is assigned to each input parameter.These frequencies define the cyclic exploration of each parame-ter’s range through successive Monte–Carlo simulations so as tocomprehensively explore parameter space whilst optimising thecontrasts between the phases of each input signal. Thus, across allMonte–Carlo simulations, a multidimensional frequency decom-position of model outputs based upon a Fourier analysis assessesthe contribution of each input to the variance of the output. Thecontribution of each input is expressed as a total sensitivity indexincluding both the main effect attributable to that parameter, andhigher degree effects from interactions with other parameters(Saltelli et al., 1999). Following preliminary tests, each parame-ter range was divided into 500 levels. This sampling resolutionbrought the total number of Monte–Carlo simulations per test to500 n (where n refers to the number of input parameters screened).

2.4. Types of simulations and key outputs screened for sensitivityanalysis

We used FAST sensitivity analysis tests to dissect the influenceof model formulation and input parameters on TRITON’s generalbehaviour and, more specifically, on its ability to shift from sea-weed bed to sea urchin barren and back (see following subsections).Model outputs were saved monthly for each 50-year-long simu-lation, and the extended FAST applied to several output metrics.The first of these was the mean simulated biomass of each modelgroup over the last 10 years of simulation. Note that the relativebiomass of the seaweed bed is directly convertible to percentagecover of seaweed. We also used the first axis of a principal com-ponent analysis on the three normalised biomass densities as a


Table 1Parameter estimates and confidence intervals used in Monte–Carlo simulations with TRITON. Data sources used to define (a) seaweed bed logistic growth, (b) sea urchinsize-structure dynamics, (c) rock lobster size-structured dynamics, (d) lobster dependency on the seaweed bed, (e) urchin grazing rate, (f) rock lobster predation and (g)allometric relationships are also specified.

(a) Seaweed bed logistic growth with ˛, intrinsic growth rate; K, carrying capacity; !, mean annual recruitment rate(Fitted against observations of seaweed bed recovery following the removal of grazers; Ling, 2008)Parameter Units Estimate Std. error Conf. interval

˛SW /year 4.43 1.65 1.72–7.14KSW g SW/200 m2 3.4e+05 3.6e+04 2.8e+05 to 4e+05!SW g SW/200 m2/year 5000 2500–10,000

(b) Sea urchin size-structured population growth with a growth transition matrix derived from an inverse logistic growth function (Ling et al., 2009b); MCR, annualnatural mortality; !CR, mean annual recruitment rate. The annual stochastic recruitment function follows a binomial distribution with a 0.4 probability of success, whichis combined with a lognormal distribution of mean "CR = −0.15 and standard deviation #CR = 0.5(Fitted against large-scale population surveys; Johnson et al., 2005; Ling et al., 2009b)Parameter Units Estimate Conf. interval

MCR /year 0.11 0.1–0.15!CR g CR/200 m2/year 4100 2500–10,000

(c) Lobster size-structured population growth with a growth transition matrix derived from polynomial growth functions (McGarvey and Feenstra, 2001); ˇRL, annualnatural mortality; !RL, mean annual recruitment rate. The annual stochastic recruitment function follows a lognormal distribution of mean "RL = −0.15 and standarddeviation #RL = 0.6.(Fitted against observation of population recovery following protection from fishing; Barrett et al., 2009)Parameter Units Estimate Conf. interval

MRL /year 0.23 0.20–0.26!CR g CR/200 m2/year 350 200–800

(d) Lobster dependency on the state of the seaweed bed. Lobster recruitment is scaled by: (1 − $RL,SW) (1 − BSW/KSW) with BSW, seaweed bed biomass density; KSW,seaweed bed carrying capacity(Johnson and Ling, unpublished data)Parameter Units Estimate Std. error Conf. interval

$RL,SW Constant 0.64 0.11 0.46–0.83

(e) Urchin grazing rate(After in situ experiments by Hill et al., 2003)Parameter Units Estimate Std. error Conf. interval

ˇSW,CR g SW/g CR/year 5.94 1.10 4.13–7.75

(f) Functional responses of lobster predation on urchin with BCR, urchin biomass density (g 200 m−2)(Fitted against predation estimates from Ling et al., 2009a; K. Redd, unpublished data)• Holling Type I as ˇCR,RL = min( ̌ N, ˇ′)

Parameter Units Estimate Std. error Conf. intervalˇ /g RL/year 6.68e−04 2.27e−05 6.31e−04 to 7.05e−04ˇ′ g CR/g RL/year 9.40 3.00 4.46–14.33

•Holling Type II as ˇCR,RL = ̌ N/(1 + ˇ′ N)Parameter Units Estimate Std. error Conf. intervalˇ /g RL/year 11.09e−04 1.68e−04 8.34e−04 to 13.85e−04ˇ′ /g CR 1.10e−04 0.20e−04 7.76e−05 to 14.19e−05

•Holling Type III as ˇCR,RL = ̌ N2/(1 + ˇ′ N2)Parameter Units Estimate Std. error Conf. intervalˇ /g CR/g RL/year 2.35e−07 0.55e−07 1.46e−07 to 3.25e−07ˇ′ /g CR/g CR 2.50e−08 0.60e−08 1.47e−08 to 3.60e−08

(g) Allometric and other size-based relationshipsLength-weight relationship for the long-spined sea urchin (Ling, unpublished data)B = 0.00267 × TD2.534 with B, urchin individual weight (g); TD, urchin test diameter (mm)

Length–weight relationship for the southern rock lobster (Punt and Kennedy, 1997)B = 0.000271 CL3.135 with B, lobster individual weight (g); CL, lobster carapace length (mm)

Size-structured predation of lobster on urchin (after Ling et al., 2009a): TDmax = ˛exp(0.023 CL) with ̨ in [3.08:5.12] orCLmin = 43.5 log(TD) − ˇ, with ̌ in [48.91:71.01]; CL, lobster carapace length (mm); TD, urchin test diameter (mm)

one-dimensional summary of community state (accounting for 73%of the total variance).

2.4.1. Sensitivity to the formulation of rock lobster predation onsea urchins

We specifically tested for sensitivity of TRITON’s generalbehaviour to alternative formulations of the lobster predationrate (simulations with random initial condition; see Table 2).Density-dependence of lobster predation rate on urchin density

was represented as a Holling Type I, II or III functional response(Fig. A11; Holling, 1966), and the effects on overall model behaviourcompared using the FAST method (Fig. 2). The effects of alterna-tive formulations of lobster predation rate were also examined bycomparing the scores on the first two axes of the PCA of simula-tion outcomes with each of the Holling Type I, II or III functionalresponses (Fig. 3). The comparison of the projection of simula-tion outcomes with each functional response on the first two PCswas both qualitative (based on the visual inspection; Fig. 3), and


Table 2Initial conditions for the different sets of Monte–Carlo simulations, where the modelled community can be initialised in either the seaweed bed or in the sea urchin barrenstate (biomass densities in g/200 m2). Unconstrained initial conditions are used for global sensitivity test. The values of seaweed biomass densities associated with 10% and50% of canopy cover are also used to define presence (1) or absence (0) of a shift to the alternative state at the end of a simulation: based upon TRITON mean behaviour(Fig. 5a), a persistent shift to sea urchin barrens is assumed if the seaweed bed cover drops below 10%, while recovery of seaweeds from the barren state corresponds to theseaweed bed re-growing above a 50% of cover.

Initial simulation state

Dense seaweed cover Sea urchin barrens Unconstrained

Seaweed assemblage 2 × 105–4 × 105 (more than 50% cover) 0–4 × 104 (less than 10% cover) 0–4 × 105

Sea urchins 0–4000 7 × 104–1.4 × 105 0–1.4 × 105

Rock lobsters 0–1.4 × 104 0–1.4 × 104 0–1.4 × 104

statistical (using a MANOVA with the type of functional responseas a factor).

2.4.2. Global sensitivity and pattern-oriented model validationWe investigated the influence of input factors on the

general behaviour of TRITON with a global sensitivity test(Figs. 2 and 4) in which all parameters varied and initialconditions were unconstrained (Table 2). Monthly outputs fromthese simulations were used to investigate both model com-munity composition and the dynamics of the TRITON model(Fig. 6a), and to assess the model’s ability to mimic observedpatterns (Fig. 6b) of seaweed percentage cover and sea urchindensity from large surveys of reef habitat and reef species abun-dance around Tasmania during the period 2000–2011 (Johnsonet al., 2005, 2011; Johnson et al., submitted report), which weconverted to biomass densities directly comparable to model out-puts. The frequency of occurrence of community states alongthe Tasmanian coastline, which encompasses a gradient of localcontexts in terms of fishing pressure, habitat complexity andurchin invasion history, could then be compared to the pat-terns emerging from Monte–Carlo simulations with TRITON(Fig. 6).

2.4.3. Sensitivity analysis of the forward and backward shiftsWe also focused on the effect of input parameters on the ‘for-

ward’ (kelp bed to urchin barren state) and ‘backward’ (seaweedrecovery from the barren state) shifts. In each of these cases,initial conditions were constrained to mimic either an urchin-free seaweed bed (for the forward shift) or a well-establishedsea urchin population on extensive barrens habitat (for the back-ward shift; see Table 2). For the sensitivity tests on the forward(Fig. 7) and backward (Fig. 8) shifts, we also measured the timefor the community to shift to the alternative state as an impor-tant feature of model dynamics. A shift to the barren state wasdefined as seaweed bed cover dropping below 10%, while sea-weed bed recovery corresponded to >50% seaweed cover (seeTable 2).

2.5. Choice and calibration of a minimum-realistic model

In marine ecosystem models, recruitment rates are often themost uncertain parameters and are commonly used as calibrationfactors (e.g. Marzloff et al., 2009). In TRITON we adjusted C. rodgersiirecruitment to ensure both that simulations could achieve real-istic sea urchin biomass densities, and that across Monte–Carlosimulations the model’s propensity to shift ‘forward’ (from theseaweed bed to the sea urchin barren state) agrees with large-scale surveys of barren habitat across reefs where C. rodgersiioccurs.

No meaningful optimisation could be designed to calibrate thegoodness-of-fit of the model against multiple quantitative criteria(Duboz et al., 2010; e.g. Klepper, 1997). In particular, because ofthe occurrence of alternative states, consideration of model mean

dynamics to capture mean community composition is not mean-ingful. Also, because of the model complexity an interpretableanalytical solution could not be derived to formally validate theoccurrence of alternative stable states within the estimated param-eter space as was achieved, for example, by Fung et al. (2011).Accordingly, we used pattern-oriented modelling, proposed as ameans to calibrate agent-based models (Grimm et al., 2005), as aneffective way to validate and calibrate the behaviour of TRITONagainst the data available for Tasmanian reef dynamics.

In the context of pattern-oriented modelling, we note that inregions where C. rodgersii has been present for several decadesand where key reef predators have been depleted by fishing (e.g.New South Wales, the Furneaux group and north-eastern Tasma-nia), about 50% of coastal rocky reef habitat is reported as seaurchin barrens (Andrew and O’Neill, 2000; Johnson et al., 2011).Thus, we focused on the ability of TRITON to reproduce acrossa set of Monte–Carlo simulations initialised in the seaweed bedstate (see Table 2) these large-scale patterns of barren formationemerging across reef habitat where C. rodgersii occurs. In thesesimulations, fishing mortality was set to mimic historical fish-ing mortalities derived from the rock lobster stock assessmentmodel for eastern Tasmania (FRL within 1–1.8/year; Klaas Hart-mann, pers. comm.), and size-structured predation of lobsters onsea urchins, which notably influences TRITON’s behaviour and itsability to shift to sea urchin barrens (Figs. 4 and 7), was set basedonly on field observations and ignoring information from tank pre-dation experiments in which lobster predation on sea urchin isartificially enhanced (Ling et al., 2009a) (˛2 = 49 in: CLmin = ˛1 log(TD) – ˛2; where CLmin is the minimum carapace length (in mm)for lobster to prey on sea urchins of test diameter TD (in mm); cf.Section 3.2.2 in Appendix A). The proportion of simulations shift-ing ‘forward’ (from seaweed bed to sea urchin barren habitat) wasspecifically examined as a function of sea urchin mean recruitmentrate (!CR; Eq. (2)) so as to calibrate the probability of sea urchinbarrens formation across Monte–Carlo simulations with TRITON(Fig. 8).

3. Results

We first consider sensitivity of model behaviour to the variousparameters before considering the calibration and validation of themodel.

3.1. Sensitivity analysis and identifying parameters that mostinfluence model behaviour

3.1.1. Functional response for lobster predationChecking that the particular formulation of density dependence

in lobsters’ predation on urchins has minor influence on modelbehaviour can be taken as a component of model validation. Foreach of Holling’s Type I, II or III functional responses, the two param-eters defining the shape of the response had no more influence onmodel behaviour than did most of the other 14 input factors (cf.


Fig. 2. Extended FAST indices quantifying the contribution of input parameter val-ues to model output variance, using the first principal component from the PCA(accounting for 73% of the total variance) on mean-centred normalised biomassdensity outputs, under alternative formulations of the functional response of lobsterpredation on sea urchin, assuming either Holling Type I (a), II (b) or III (c) relation-ships. Fig. 3 provides a graphical summary of final model state across the three modelgroups.

FAST sensitivity indices in Fig. 2). Indeed, the influence of thesetwo parameters was relatively small compared to parameters withgreatest influence on model behaviour (i.e. lobster fishing mortal-ity, sea urchin recruitment, initial sea urchin population density,seaweed growth rate), and also smaller than the influence of thecoefficient defining the allometry of rock lobster size-structuredpredation on sea urchins. The projection of simulation outcomeson the first two PCs also suggests that the nature of the functionalresponse has little influence on model behaviour (Fig. 3) in that thepattern of scores on the first two PCs (capturing 87.4% of the totalvariability) are similar for all three functional responses. Althoughresults from MANOVA suggest significantly different mean scores(P value < 10−15; F2,23997 = 67.5 from MANOVA Pillai’s Trace statis-tic) on the first two PCs for each type of functional response, thisis likely to reflect the very large number of replications (8000simulations, which ensures extremely small multivariate standarderrors and large power) rather than ecologically meaningful dif-ferences. Given that overall model behaviour was not sensitive toeither the choice of functional response or to its parameterisation,we adopted the Type III functional response, which is consistentwith most models of predation behaviour in decapods based onfield observations (see Table A8 for complete list of references;Appendix A).

3.1.2. Global sensitivity analysisThe sensitivity to input parameters of final abundances (after

50 years of community development) of seaweed, sea urchins andlobsters, and of overall community structure, was examined across8000 Monte–Carlo simulations with unconstrained initial condi-tions (Figs. 2c and 4). Total extended FAST indices quantify inputparameters’ relative contribution to model output variance for agiven sensitivity test (but their absolute values are not comparableacross different extended FAST tests). Overall, the most influentialvariables were similar for each component of community struc-ture we examined, namely fishing mortality of lobsters, sea urchinrecruitment rate, sea urchin initial abundance and seaweed growthrate (although some other variables were moderately influentialfor some components). However, the rank order of influence dif-fered depending on whether it was final densities of seaweed,sea urchins or lobsters that were examined. Final biomass densityof seaweed is predominantly determined by, in order of impor-tance: the initial density of sea urchins; urchin recruitment rates;seaweed growth rate; size-structured lobster predation on seaurchin; lobster fishing mortality and initial biomass (cover) of sea-weed (Fig. 4a). The two most influential parameters on final seaurchin biomass densities are sea urchin recruitment rate and lob-ster fishing mortality (Fig. 4b). Not surprisingly, the final biomassdensity of lobsters is mostly determined by lobster fishing mor-tality and, to a lesser extent, lobster recruitment rate (Fig. 4c).In comparison, other input parameters defining lobster popula-tion dynamics (e.g. initial biomass, natural mortality, the extentof dependency on the state of the seaweed bed) have a marginalinfluence.

Given these results, it is not surprising that overall communitystructure described by the first principal component of the mean-centred normalised simulated biomasses of the three groups (andaccounting for 73% of the total variance; Fig. 2c) is most influencedby, in order of importance: lobster fishing mortality; sea urchinrecruitment rate; initial sea urchin abundance; seaweed growthrate; and finally the three parameters defining lobster predationon sea urchins. Across all four outputs considered in this sensi-tivity analysis, the carrying capacity and recruitment rate of theseaweed assemblage; sea urchin natural mortality and their graz-ing rate; initial abundance and natural mortality of lobsters; andthe coefficient of lobster dependency on the state of the seaweedbed, have relatively marginal influence on the end point community


Fig. 3. Effect of different formulations of lobster predation rate on the scores of simulation outcomes on the first two axes of the PCA, which capture 87.4% of the totalvariance. Scores are plotted for all functional responses (a) then respectively for Holling Type I (b), II (c) and III (d) functional responses.

structure in the simulations. Fig. 6a depicts the general behaviour ofTRITON, i.e. the range and frequency of model community composi-tion and mean trajectory (fortnightly change in biomass) emergingfrom these 8000 Monte–Carlo simulations with random initial con-ditions.

The final two sets of sensitivity tests focus respectively on the‘forward’ shift from the seaweed assemblage to sea urchin bar-ren habitat (Fig. 7), and the ‘backward’ shift from extensive seaurchin barrens to recovery of dense seaweed cover (Fig. 8). Seaurchin recruitment rate, lobster fishing mortality, seaweed growthrate and the three parameters defining lobster predation rate mostinfluenced the tendency to shift from dense seaweed assemblageto sea urchin barrens (Fig. 7a). TRITON’s ability to shift from anestablished sea urchin barren state back to dense seaweed coveris essentially driven by the values of lobster fishing mortality andrecruitment rate (Fig. 8a). Formation of extensive sea urchin bar-rens becomes more likely and the time to destructive grazing ofseaweed beds becomes shorter in an essentially linear manner withincreasing lobster fishing mortality and sea urchin mean recruit-ment rate (Fig. 7b). Conversely, as fishing mortality on lobstersdecreases and their recruitment rate increases, the time to recov-ery of a dense seaweed cover from the barren state decreases in

an approximately linear fashion (Fig. 8b). Note, however, that thelikelihood of seaweed bed recovery from extensive sea urchin ‘bar-rens’ is small (less than 10%), even as fishing mortality of lobstersis reduced and their recruitment increased.

A final important point to emerge for all sensitivity analyses(Figs. 2, 4, 7 and 8) is that interaction terms contribute consistentlymore – and in most cases very much more – to the variance of modeloutputs than first order ‘main’ effects due to single input parame-ters acting directly on their own. Across all input parameters andall output variables considered in global sensitivity analysis tests(Figs. 2 and 4), interaction terms contribute to over 80% of variancein model outputs. The total influence on model output variance ofall input parameters is greater than the sum of their direct individ-ual influence. This highlights the dominant contribution of complexnon-linear interactions between modelled processes to TRITON’soverall dynamics.

3.1.3. Pattern-oriented validation and calibration of TRITONSensitivity analyses proved useful to explore model behaviour,

to assess sources of model uncertainty, and to define aparsimonious and reliable version of TRITON for applica-tion to management questions. To calibrate model behaviour


Fig. 4. Sensitivity analysis based on extended FAST indices quantifying the contri-bution of all model input parameter values to model output variance. Final biomassdensities of (a) seaweeds, (b) sea urchins and (c) rock lobsters at the end of 50-yearsimulations with unconstrained initial conditions are used as model outputs.

to empirical observations, we scrutinised the parameters ofhigh influence on the ‘forward’ shift, i.e. lobster fishingmortality, sea urchin recruitment, seaweed growth rate andallometry of lobster size-structured predation on sea urchin(Fig. 7). Lobster size-structured predation was based on field obser-vations indicating that only large lobsters (>140 mm carapacelength) can prey on emergent sea urchins (Ling et al., 2009a).Smaller lobsters may occasionally predate smaller urchins largelyconfined to the interstices of the reef matrix but this is likely to beoffset by our assumption that any lobster >140 mm CL can predateany emergent sea urchin it encounters. Lobster fishing mortalityfor these calibration simulations was restricted to historical lev-els experienced in eastern Tasmania. The influence of seaweedgrowth rate and sea urchin recruitment rate on the risk of bar-ren formation is non-linear (Fig. 8a) and the likelihood of barrensforming increases dramatically when sea urchin mean recruitmentrate exceeds a threshold of about 7000 g/200 m2/year.

Monte–Carlo simulations with TRITON, in which combina-tions of input parameters are comprehensively tested, adequatelyencompass the diversity and spatial heterogeneity encounteredon Tasmanian rocky reefs. For example, patterns emerging acrosssimulations suggest that some reefs are more prone to barrens for-mation than others depending on local seaweed productivity andlocal recruitment of urchins (Fig. 8a). The proportion of simulationsshifting to sea urchin barrens increases non-linearly from about 15%up to 80% as the maximum value of the range of sea urchin meanrecruitment rate is increased from 2000 to 10,000 g/200 m2/year(Fig. 8b). The two grey dashed horizontal lines (Fig. 8b) delimit thebulk range of sea urchin barrens habitat extent (∼50% of reef area)in New South Wales (Andrew and O’Neill, 2000) and northeast-ern Tasmania (Johnson et al., 2005, 2011) where C. rodgersii is longestablished. Consequently, maximum sea urchin recruitment ratewas set to 6000 g/200 m2/year to ensure that the probability of theTRITON model shifting to barrens is in line with large-scale obser-vations of the extent of sea urchin barrens in reef areas where C.rodgersii has been long established.

At a holistic level, the capacity of the model to demonstrateshifts (in either direction) between seaweed and sea urchin dom-inated reefs represents a validation of the observed dynamics. Weaggregated monthly outputs from the 8000 Monte–Carlo simula-tions to compare patterns emerging from simulations with TRITONto patterns observed in large-scale surveys (Johnson et al., 2005,2011; Johnson et al., submitted report) of Tasmanian temperatereef communities (Fig. 6). Fig. 6a describes the frequency of thedifferent community states in terms of seaweed bed versus seaurchin biomass densities with overlaid arrows representing themodel mean trajectory (i.e. fortnightly change in biomass densitythrough simulations) at different points of reef state. In Fig. 6b, datafrom large-scale surveys describes the frequency of reef communi-ties on the east coast of Tasmania in 2000–2011 being in any givenstate. Importantly, both the modelled and observed reef communi-ties identify two persistent and dominant states representing (i) theseaweed bed state with a high cover of seaweed and a low densityof sea urchin and (ii) the sea urchin barren state with virtually noalgal cover and a high density of sea urchins. This indicates broadagreement of the behaviour of the model with the occurrence ofthe two community states as observed in the field. Note that thevolume of output from the TRITON model enables a much more con-tinuous picture of the range of community states encountered onTasmanian reefs than can be obtained by direct diver-based mea-surements. Moreover, the model can provide insight on aspects ofreef dynamics that have not been able to be documented from fieldobservations, in particular the point at which recovery from exten-sive sea urchin barrens commences as urchin density falls (bottomleft region of Fig. 5a). The conspicuous ‘hole’ of low frequency ofobservations of this state – i.e. very low urchin density and seaweed


Fig. 5. Frequency (logarithmic scale) of community states as a function of sea urchin versus seaweed bed biomass densities from (a) the 8000 Monte–Carlo simulationswith TRITON and from (b) large-scale surveys on the east coast of Tasmania (Johnson et al., 2005, 2011; Johnson et al., submitted report). Arrows in (a) represent the meansimulation trajectory in terms of fortnightly change in sea urchin and seaweed bed biomass densities.

biomass at ∼105 g/200 m2 – in eastern Tasmania (Fig. 5b) reflectsthat there is no evidence of recovery of seaweed cover on any ofthe extensive barrens monitored thus far.

4. Discussion

4.1. Model sensitivity to input parameters

Breaking down the sensitivity analysis into a series of testsscreening for different model outputs and different aspects ofmodel behaviour (i.e. ‘forward’ and ‘backward’ shifts) is a means torobustly identify input parameters that have a consistently smallor large influence on simulation outcome (Klepper, 1997). Overall,the identity of variables most influential in accounting for vari-ance in simulation outcomes is similar across the different typesof sensitivity tests we conducted (unconstrained initial conditions,or a constrained focus on the ‘forward’ or ‘backward’ shift), andwhether we considered final abundances of individual groups (sea-weed, sea urchins and lobsters) or of the community as a whole(Figs. 4, 7 and 8). These analyses identified lobster fishing mortal-ity, lobster and urchin recruitment rates, size-structured predationof lobsters on urchins, as well as initial urchin densities as thekey drivers of model dynamics. Conversely, seaweed recruitmentrates, initial cover and carrying capacity (i.e. the upper limit of sea-weed biomass density); sea urchin natural mortality and grazingrate; and the initial biomass and natural mortality rates of rocklobsters, all have relatively marginal influence on the simulationoutcomes.

At a more detailed level, independent sensitivity tests were usedto identify differences in the key variables influencing the differentindividual components of community structure, and in comparingthe influence of each input variable on particular groups (seaweed,sea urchins or lobsters) with the influence on overall communitystructure (as described by the first principal component from thePCA). While input parameters that most influence model dynam-ics are broadly similar for each component of the community, thedetailed differences between these four different tests (Fig. 4) areinformative. They show that:

(i) Seaweed biomass density is the only component for whichdynamics is driven primarily by the initial state of thesea urchin population rather than lobster fishing mortality(Fig. 4a). This occurs as a result of the hysteresis in modeldynamics, with initial sea urchin biomass density sitting eitherhigher or lower than the threshold above which the seaweedbed gets depleted by grazing (cf. Fig. 6a). Note that seaweedgrowth rate also exerts relatively high influence on seaweeddynamics, suggesting that rocky reefs where seaweed pro-ductivity is low (due to shading, unsuitable substratum ornutrient-poor conditions) will be more prone to sea urchinbarren formation for the same level of sea urchins. Decliningnitrate levels as a result of a changing ocean climate increas-ingly influenced by nitrate-poor waters of the East AustralianCurrent (Johnson et al., 2011) may play a key role in this con-text.

(ii) Sea urchin dynamics in the model is essentially affected byinput factors related to lobster predation pressure (the recruit-ment rate of lobsters, fishing mortality and the coefficients ofthe Type III functional response), as well as recruitment ratesof the sea urchins themselves. This implies that the poten-tial for C. rodgersii population to build up at the scale of anindividual rocky reef can be limited by: (1) exposure tolarge-scale oceanographic features transporting urchin larvae(Banks et al., 2007); (2) the suitability of the reef substra-tum (e.g. appropriate settlement cues, complexity of crevicestructure; Andrew, 1993) for metamorphosis and settlementof urchin larvae; or (3) exposure to predation (Ling andJohnson, 2012). Of all of these variables, lobster fishing mortal-ity is clearly the key variable amenable to ready managementintervention.

(iii) The rock lobster population is influenced largely by fish-ing and the mean recruitment rate to the population in themodel.

(iv) Not surprisingly, sensitivity indices focused on effects on thevariance in overall community structure (as described by thefirst principal component; Fig. 2c) identify all the parametersimportant to each of the three components of communitystructure when they are examined separately (Fig. 4a–c).


Fig. 6. Sensitivity of the ‘forward’ shift (from high seaweedbiomass to sea urchin barrens habitat) to model input parameters(i.e. this analysis was restricted to those simulations in which the ‘forward’shift occurred). Initial conditions correspond to the seaweed bed state withseaweed cover at >50%, low initial sea urchin density (<40,000 g/200 m2) andrandom rock lobster biomass density. (a) Extended FAST indices quantifying thecontribution of input parameters to model output variance in overall communitystructure (described as the first PC from the PCA on mean-centred normalisedbiomass density outputs of all groups) for 50-year simulations. (b) 3D plot repre-senting both the probability of (z axis) and the time for (colour scaling) barrensestablishment (in months) as a function of the two parameters most influential inaffecting the likelihood of the transition to barrens, viz. sea urchin recruitment rate(in g/200 m2/year) and lobster fishing mortality (in /year).

Conducting independent sensitivity tests to dissect model sensi-tivity in the ‘forward’ and ‘backward’ shifts separately (Figs. 7 and 8)also proved useful. This approach overcomes concerns about sen-sitivity analyses of models with multiple equilibria (van Nes andScheffer, 2003). It identified that model shifts from high sea-weed cover to sea urchin barren habitat, and the reverse shiftrealising recovery of seaweeds, are both driven predominantlyby lobster fishing mortality, lobster and sea urchin recruitmentrates, as well as lobster predation rates. Note that, surprisingly,the reverse shift is not so sensitive to the coefficient that scaleslobster recruitment to the level of canopy cover. A strong depend-ency of lobster recruitment on the seaweed canopy reinforces

Fig. 7. Sensitivity of the ‘backward’ shift (from sea urchin barrens to recovery ofdense seaweeds) to model input parameters (i.e. this analysis was restricted to thosesimulations in which the ‘backward’ shift occurred). Initial conditions correspondto sea urchin barrens habitat, with seaweed cover <10% of carrying capacity, ini-tial urchin density > 70,000 g/200 m2 and random rock lobster biomass density. (a)Extended FAST indices quantifying the contribution of input parameters to modeloutput variance in overall community structure (described as the first PC from thePCA on mean-centred normalised biomass density outputs of all groups) for 50-year simulations. (b) 3D plot representing both the probability of (z axis) and thetime to (colour scaling) seaweed bed recovery from sea urchin barrens (in months)as a function of the two parameters most influential in affecting the likelihood ofthe transition from established barrens back to dense seaweed cover, viz. lobsterrecruitment rate (in g/200 m2/year) and lobster fishing mortality (in /year).

the positive feedback between seaweeds, urchins and lobstersonce the macroalgal canopy is lost and contributes to the highresilience of the urchin barren state (Marzloff et al., 2011). Thus,this result, along with the marginal likelihood of seaweed recov-ery from a fully established barren state (Fig. 8b), suggests that,in TRITON, the barren state is highly resilient irrespective of thestrength of the dependency of lobster recruitment on the state ofthe seaweed cover. The high stability of this ‘deteriorated’ com-munity state, i.e. of sea urchin barrens, is a common feature of thedynamics of ecosystems with alternative community states (e.g.deforestation of tropical dry forest; Lawrence et al., 2007). Param-eters of high influence identify the key ecological processes that


Fig. 8. Results from 50-year-long Monte–Carlo simulations used to calibrate ranges in sea urchin recruitment from the model’s propensity to shift to sea urchin barrensunder historical rock lobster fishing conditions. (a) Probability of barren formation as a function of the two most influential input parameters, sea urchin recruitment rateand seaweed growth rate; (b) Probability of the shift from seaweed bed to sea urchin barren as a function of sea urchin maximum recruitment rate with fixed coefficientfor size-structured allometric relationship. The dashed horizontal lines mark the observed range of sea urchin barren cover across rocky reefs in New South Wales (Andrewand O’Neill, 2000) and Tasmania (Johnson et al., 2005, 2011, this study) where C. rodgersii is long established and where populations of reef predators have been depleted byfishing.

drive community dynamics in the model. Given that the emergentdynamics of the model broadly matches observations of commu-nity state on eastern Tasmania reefs (Fig. 6), we can have someconfidence that sensitivity analysis of TRITON helps to identifythe likely key drivers of the dynamics in nature, and thus assistswith both prioritising ongoing work in the field and in identifyingoptions for improved management of Tasmanian reef communitydynamics.

Transition times in shifting from one state to the other areimportant characteristics of the dynamics of systems with alter-native states and hence, are a key element in exploring modelsensitivity (Figs. 7b and 8b). For models with hysteresis, simula-tion outcomes are essentially binary, which can prove problematicwhen conducting sensitivity analyses, in particular in undertak-ing partial sensitivity tests to one input parameter at a time (vanNes and Scheffer, 2003, 2004). In the case of TRITON, the simu-lated community ultimately moves either towards the barren orthe seaweed-dominated state (Fig. 6), and so quantifying the influ-ence of parameters on the time for the model to shift ‘forward’ fromseaweed bed to sea urchin barrens, or ‘backward’ to effect seaweedrecovery, provides valuable insight into the detailed dynamics.Notably, the ‘forward’ shift (22.1 years ± 0.19 standard error) occurson average about ten years more quickly than the ‘backward’ shift(31.8 + - 0.92 standard error). These mean transition times, whichprovide another illustration of hysteresis in model dynamics inthe sense that ‘forward’ and ‘backward’ shifts are independentdynamics with different transition times, have major implicationsfor management of rocky reef communities in Tasmania. Prevent-ing the further spread of extensive sea urchin barrens appears asthe most realistic and time-efficient (and therefore cost efficient)management option. If solely relying on predation by rock lobstersto deplete sea urchin populations, the time frame for restorationto the seaweed dominated state from C. rodgersii barrens is ofthe order of three decades, and even then the predicted proba-bility of seaweed recovery is low, even under the most drastic

measures for the lobster fishery (Fig. 8b). Note that, given that sea-weed beds can recover in about two years in the absence of grazers(Appendix A; see also Ling, 2008), the lengthy time to seaweedbed recovery over ∼3 decades in the simulations again empha-sises the high resilience of the sea urchin barren state. Note thatthe predicted timeframe for seaweed recovery exceeds by far thetime span of current management plans for the Tasmanian lobsterfishery.

The sensitivity analysis highlights that interactions betweeninput parameters, rather than direct effects, have a major influ-ence on simulation outcomes. The dominant influence on modelbehaviour of interactions between input parameters is commonin models of complex dynamics (Saltelli et al., 2009, 1999). In thecontext of the dynamics of Tasmanian rocky reef, strong interac-tions between input parameters highlight the value of ecologicalmodels to inform managers of natural resources about non-trivialeffects of management interventions and environmental change onecosystem state. While qualitative modelling can track the influ-ence of indirect effects and the contribution of high level feedbackto community dynamics (Marzloff et al., 2011), simulation-basedsensitivity analysis of the quantitative model TRITON captures non-trivial interactions between modelled processes, and can providevaluable insights about indirect responses of the reef communityto perturbations or management intervention.

4.2. Pattern-oriented model validation

A prime objective of this paper was to validate and calibratethe dynamics of the TRITON model. Mean observed communitystate, which is commonly used as an objective benchmark to cal-ibrate complex ecosystem models (e.g. Marzloff et al., 2009), isneither a reliable or meaningful criterion to assess the realismof a model for a system characterised by alternative commu-nity states. Further, the set of difference equations comprisingthe TRITON model are too complex to be analytically tractable to


identify the parameter space in which the model shifts discontinu-ously from the seaweed-dominated state to the sea urchin barrenstate and back as has been possible for relatively simple mod-els of coral reef community dynamics with hysteresis (see Funget al., 2011). Thus, calibration and validation of TRITON relies ontwo large-scale features that emerge from the dynamics of rockyreef communities under the threat of C. rodgersii destructive graz-ing.

First, following the long-term establishment of C. rodgersii inregions where key urchin predators are ostensibly at relativelylow abundances, about 50% of inshore rocky reef habitat occursas extensive C. rodgersii barrens habitat (Andrew and O’Neill, 2000;Johnson et al., 2005). Accordingly, we restricted mean recruitmentrate of C. rodgersii to allow the model to simulate both realisticbiomass densities of sea urchins on barrens habitat, and to realisethe shift from seaweed bed to sea urchin barrens with a probabil-ity of ∼0.5 under historical depletion of large lobsters by fishing(Fig. 8).

Second, simulations with TRITON accurately reproduce therange and frequency of community states observed in large-scaleintensive surveys of seaweed cover, barren habitats and densitiesof reef species on the east coast of Tasmania (Fig. 6). In particular,model dynamics demonstrates an ecological hysteresis (Donahueet al., 2011; Scheffer and Carpenter, 2003), showing two stablecommunity states, corresponding to either, dense seaweed coverwith low sea urchin abundance, or sea urchin barren habitat withvirtually no macroalgal cover (i.e. the zones where simulation tra-jectories converge), and an unstable equilibrium zone, where thetrajectories diverge, developing towards one state or the other(Fig. 6a). The mean simulation trajectories confirm the occurrenceof two alternative community states sensu Petraitis and Dudgeon(2004) such that under identical environmental conditions, initialconditions determine whether the simulated community developsto sea urchin barrens or the seaweed-dominated state. We concludethat pattern-oriented modelling, originally proposed to validateagent-based models (Grimm et al., 2005), can provide a valuableapproach for a simulation-based assessment of the dynamics ofecological models that manifest alternative community states. Thecapacity to investigate mean simulation trajectories (Fig. 6a) illus-trates the value of TRITON to explore features of reef communitydynamics that are at best challenging or, more likely, impossibleto observe or measure with sufficient precision in the field (e.g. theexistence of thresholds points leading to shifts in community state).

It is also worth mentioning that, as a first step, comprehensiveanalyses of model sensitivity to alternative formulation and inputparameters helped to tackle sources of model uncertainty priorto model calibration. Despite the lack of observations to deter-mine the form of density dependence on lobster predation rate(Fig. A11 in Appendix A), given marginal influence of the lobsterfunctional response on model behaviour (Figs. 2 and 3) we are con-fident to adopt Holling’s Type III functional response as the mostoften used response describing predation behaviour in decapods(e.g. Griffen and Delaney, 2007; Wong and Barbeau, 2006; Wonget al., 2010; cf. Table A7 in Appendix A for the complete referencelist).

4.3. Model limitations and guidance for future research

Derivation of all parameter estimates was based upon thebest available information at time of model development (seeAppendix A for further details). However, the results presentedhere are only as useful as the precision and accuracy of the param-eter estimates, and so it is worthwhile to acknowledge areaswhere parameter definition or the relative coarseness in represent-ing ecological processes may limit the realism of TRITON. Someof the ecological processes of seaweed-urchin-lobster dynamics

on subtidal rocky reefs in eastern Tasmania are captured rathercoarsely in TRITON and would benefit from further field-basedresearch. In particular it may be useful to have quantitative esti-mates of the size-dependent vulnerability of macroalgae to grazersand the magnitude of any size-structured dynamics of seaweedbeds; density-dependence in sea urchin grazing rates; importanceof seaweed habitat to the recruitment, productivity and carryingcapacity of lobster population; lobster predation rates at mediumand high sea urchin densities (i.e. density dependence in preda-tion impact); the strength of predatory interactions between smallcryptic sea urchins living in the reef matrix and rock lobsters; andeffects of habitat, depth and reef profile on all of the modelled pro-cesses. Storms and wave action can abrade seaweed cover (e.g. Reedet al., 2011) but these effects are likely to be marginal on E. radiatabeds in eastern Tasmania (CR Johnson, personal observation); cur-rent evidence and observation suggest that none of these effects islarge relative to the important parameters identified in the model,but if any of these effects did prove to be large, then the detail ofmodel dynamics may be different to that presented here. Nonethe-less, given current knowledge, we are comfortable to suggest that itis unlikely that any of these effects would materially influence thequalitative dynamics of the phase shifts and hysteresis in broadterms.

5. Conclusions

Communities with the potential for multiple stable states andecological hysteresis offer particular challenges and higher stakesfor managers because one of the alternative states is usually poorlyproductive and less desirable (Johnson and Mann, 1988; Lawrenceet al., 2007; Melbourne-Thomas et al., 2011; Strain and Johnson,2012; van de Koppel et al., 1997). Thus it is often of critical impor-tance to avoid transition to the less desirable state, in particularwhen management intervention to facilitate the return shift maybe unpractical. In this context, and particularly because it is notusually possible to identify tipping points from field based exer-cises (Hastings and Wysham, 2010; Osman et al., 2010, but seeCarpenter et al., 2011 for a quite unique “whole-ecosystem” exper-iment), models of ecological communities with alternative statesare essential to inform key thresholds in system dynamics andtest the effects of alternative management strategies (McClanahanet al., 2011; Mumby et al., 2007). However, validating this kindof model remains challenging, not the least reason for which isthat transitions between states are rarely if ever observed with anyprecision. Here, we have presented a comprehensive simulation-based exploration of the TRITON model that captures the potentialfor Tasmanian seaweed–sea urchin–lobster community dynam-ics to shift between two alternative states, dense seaweed bed orsea urchin barrens habitat. The series of Monte–Carlo simulationsdepicts the model’s overall behaviour, and pattern-oriented-modelling, i.e. comparison of patterns emerging from simulationsto large-scale patterns observed in the field, provided an efficientway to assess the robustness and calibrate the broad dynamicsof TRITON prior to its application. The extended FAST routine(Saltelli et al., 1999) provides a unique, computationally efficientframework to design robust model-independent sensitivity tests.Using the extended FAST, we identified parameters that most influ-ence both overall model dynamics, and particular independentfeatures of model hysteresis, i.e. the ‘forward’ and ‘backward’ shiftsbetween the alternative states. This enabled assessment of whethermanagement intervention in this system is practicable, and to iden-tify the nature of the intervention that is likely to have most effectin influencing community dynamics. Of the relatively small suiteof parameters to which the model is most sensitive, fishing mor-tality of lobsters emerges as the single factor to which the model


is particularly sensitive and on which human behaviour has a largeand direct effect.

Acknowledgements

If this modelling work only captures a coarse version of Tas-manian rocky reef local dynamics, it would have no grounding inreality at all without all the data referred to in this document:experimental studies (e.g. Ling et al., 2009a), empirical observations(e.g. Johnson et al., 2005), or monitoring of community recovery fol-lowing protection from fishing (e.g. Barrett et al., 2007). We thankNeville Barrett and Graham Edgar for generously sharing their20-year-long dataset of monitoring reef community responses fol-lowing protection from fishing; Stewart Frusher, Klaas Hartmannand Caleb Gardner for informing some of the parameters definingrock lobster dynamics; Kevin Redd for sharing the DNA-based esti-mates of lobster predation; Malcolm Haddon for useful discussionsabout size-structured modelling and recruitment stochasticity; PipBricher, the Mount family and Bruny for the writing retreat. Thefirst author was supported by a Ph.D. scholarship co-funded bythe joint CSIRO-University of Tasmania programme in QuantitativeMarine Science (QMS) and the CSIRO Wealth from Ocean flagship,and was also the recipient of an Endeavour International Postgrad-uate Research (EIPR) scholarship and a Tasmanian Marine Sciencefellowship to collaborate with Jean-Christophe Soulié.

Appendix A. Supplementary data

Supplementary data associated with this article can befound, in the online version, at http://dx.doi.org/10.1016/j.ecolmodel.2013.02.022. These data include Google map ofthe most important areas described in this article.

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Appendix A to Marzloff et al. “Sensitivity analysis and pattern-oriented validation of TRITON, a model with alternative community states: insights on temperate rocky reefs dynamics”: Parameterisation of the TRITON model

1

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29

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X4%-2)+)-+)34+G.5)2,&',/+c-=42/$4/)+S4%,2)$4/)+-1+O&'$,)4+O3,/04"+

U4)2,')'5B+U"H"+,/6+S.604-/B+H"X"B+FLL!"+S4)4()'-/+-1+,&)42/,)'=4+5),9&4+5),)45+'/+$,2'/4+

(-$$./')'45"+[-.2/,&+-1+Z<%42'$4/),&+M,2'/4+T'-&-0*+,/6+Z(-&-0*B+YLL:Y!Y@YW7"+

U./)B+G"Z"+,/6+]4//46*B+X"T"B+7VVW"+U-%.&,)'-/+$-64&&'/0+-1+;,5$,/',/+2-(I+&-95)42B+9.+/+0

&1:.)1+""B+245-.2(45"+M,2'/4+,/6+P2453?,)42+X454,2(3B+!R:VaW@VRL"+

U./)B+G"Z"B+]4//46*B+X"T"+,/6+P2.5342B+H"S"B+7VVW"+Z5)'$,)'/0+)34+5'>4@)2,/5')'-/+$,)2'<+1-2+

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U*)3-/+H-1)?,24+P-./6,)'-/B+FLLR"+U*)3-/+F"a"7"+J,$%)-/B+A4?+J,$%53'24B+`HG"+

X+S4=4&-%$4/)+O-24+;4,$B+FL7L"+X:+G+&,/0.,04+,/6+4/='2-/$4/)+1-2+5),)'5)'(,&+(-$%.)'/0"+X+

P-./6,)'-/+1-2+H),)'5)'(,&+O-$%.)'/0B+e'4//,B+G.5)2',"+


37

X466B+]"H"B+[,2$,/B+H"A"B+P2.5342B+H"S"+,/6+[-3/5-/B+O"X"B+FLLR"+G+$-&4(.&,2+,%%2-,(3+)-+

'64/)'1*+%24*+-1+)34+5-.)342/+2-(I+&-95)42"+T.&&"+Z/)-$-&"+X45"B+VR:FYY@FYR"+

X446B+S"O"B+X,55?4'&42B+G"B+O,22B+M"J"B+O,=,/,.03B+]"O"B+M,&-/4B+S"U"+,/6+H'404&B+S"G"B+FL77"+

^,=4+6'5).29,/(4+-=42?34&$5+)-%@6-?/+,/6+9-))-$@.%+(-/)2-&+-1+%2'$,2*+

%2-6.()'-/+'/+O,&'1-2/',+I4&%+1-245)5"+Z(-&-0*B+VF:F7LR@F77a"+

H,&)4&&'B+G"B+O,$%-&-/0-B+P"+,/6+O,2'9-/'B+["B+FLLV"+H(244/'/0+'$%-2),/)+'/%.)5+'/+$-64&5+?')3+

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H,&)4&&'B+G"B+;,2,/)-&,B+H"+,/6+O,$%-&-/0-B+P"B+FLLL"+H4/5')'=')*+,/,&*5'5+,5+,/+'/0246'4/)+-1+

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H,&)4&&'B+G"B+;,2,/)-&,B+H"+,/6+O3,/B+]"U"H"B+7VVV"+G+_.,/)'),)'=4+$-64&@'/64%4/64/)+$4)3-6+

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H(341142B+M"B+O,2%4/)42B+H"B+P-&4*B+["G"B+P-&I4B+O"+,/6+^,&I42B+T"B+FLL7"+O,),5)2-%3'(+53'1)5+'/+

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H(341142B+M"+,/6+O,2%4/)42B+H"X"B+FLLY"+O,),5)2-%3'(+240'$4+53'1)5+'/+4(-5*5)4$5:+&'/I'/0+

)34-2*+)-+-9542=,)'-/"+;24/65+'/+Z(-&-0*+h+Z=-&.)'-/B+7R:a!R@a\a"+

H(341142B+M"B+=,/+A45B+Z"J"B+J-&$024/B+M"+,/6+J.0345B+;"B+FLLR"+U.&54@62'=4/+&-55+-1+)-%@

6-?/+(-/)2-&:+;34+(2')'(,&@2,)4+3*%-)345'5"+Z(-5*5)4$5B+77:FFa@FYW"+

H4')>B+X"S"B+Q'%('.5B+X"A"B+J'/45B+G"J"+,/6+Z00&45)-/B+S"T"B+FLL7"+S4/5')*@64%4/64/)+%246,)'-/B+

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HI,&5I'B+c";"+,/6+c'&&',$B+["P"B+FLL7"+P./()'-/,&+245%-/545+?')3+%246,)-2+'/)421424/(4:+=',9&4+

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H$')3B+G"S"M"B+T2-?/B+O"["B+T.&$,/B+O"M"B+P.&)-/B+Z"G"B+[-3/5-/B+U"B+],%&,/B+#"O"B+Q->,/-@

M-/)45B+J"B+M,(I'/5-/B+H"B+M,2>&-11B+M"B+H3,//-/B+Q"["B+H3'/B+f"["+,/6+;,$B+["B+FL77"+

#$%,()5+-1+P'53'/0+Q-?@;2-%3'(+Q4=4&+H%4('45+-/+M,2'/4+Z(-5*5)4$5"+H('4/(4B+

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38

H)2,'/B+Z"M"G"+,/6+[-3/5-/B+O"X"B+FL7F"+#/)4/5'=4+1'53'/0+-1+$,2'/4+(-/5.$425+(,.545+,+

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Corrigendum to Sensitivity analysis and pattern-oriented validation of TRITON, a model with...

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Transcript of Corrigendum to Sensitivity analysis and pattern-oriented validation of TRITON, a model with...