Atlantis

Atlantis is a three dimensional spatially explicit deterministic end-to-end ecosystem model. It incorporates ocean physics, chemistry and biology [16–18]. It is coupled to a hydrodynamic model that provides currents, temperature and salinity. These data influence nutrient cycling, primary production and organism physiology and distribution. Flow of nitrogen is tracked through the food web as it passes between functional groups (groups of species aggregated by niche and life history). Each functional group is associated with sub-models describing consumption, production, respiration, reproduction and movement. Interaction rates are modified by a diet matrix, density-dependent feeding relationships, gape limitation, and predator-prey co-occurrence. Vertebrates are age structured and the nitrogen pool is divided into reserve weight (soft tissue and gonadal mass that can be reabsorbed) and structural weight (bone and hard tissues). Human impacts may be represented in several ways including fisheries catch and discards. Seasonal migration into and out of the spatial domain is represented, as well seasonal movement within the domain, vertical movement, density dependent foraging movement and random diffusion. The simulation operates at 12-hour time steps. In this article, we only review model processes most relevant to this application. A more comprehensive review of system equations is available here [16,17]. A summary of applications is available here [18].

The GOM implementation of Atlantis is described in Ainsworth et al. [19]. It incorporates the most recent available data on fish abundance from state and federal survey programs (e.g., from SEAMAP [20]). Biomass was distributed using a statistical methodology [21] and the diet matrix was determined using gut content analysis and statistical analysis [22,23]. Atlantis polygon geometry is based on bioregional features (e.g., physical processes, habitat, and climatology), exploitation patterns and management jurisdictions [19]. The model uses 91 functional groups: 61 are age-structured vertebrates (including fishes, birds, turtles and mammals), 19 are invertebrates, 6 are primary producers, 2 are bacteria and 3 are detritus. Model functional groups represent single species in the case of important fishery targets, or aggregated groups of species with similar life histories, distributions and feeding patterns. Vertebrates and some exploited invertebrates are tracked by numbers and body weight, while non-exploited invertebrates are handled as biomass pools. This allows us to represent processes and rates relevant to fisheries management (e.g., size-based gear selection, fecundity at age) and it minimizes problems arising from data quality in poorly studied species. Polygon geometry reflects circulation, habitat and management divisions (Fig 1). There are up to 6 water column layers per polygon, plus a sediment layer. Initial conditions represent the ecosystem in January 2010. Hydrodynamics are provided by the American Seas model (AMSEAS) based on the National Research Laboratory-developed NCOM model [24].

Growth and mortality effects

In Atlantis, the instantaneous rate of change in biomass (B, in units mg N) for each polygon, depth layer and age class of functional group i, is determined by Eq 1.

(1)

All rates are specific to each polygon and depth layer except for M1 and g_oil. T_IMM,i and T_EM,i are immigration and emigration, F_i is fisheries catch, and P_i,j is predation by predator j on group i. G_i,s and G_i,r are growth in structural and reserve nitrogen (these are functions of assimilation efficiency, growth rate and total consumption). R is entry of new individuals into the age class from recruitment or aging. In this application, M1 summarizes natural mortality from non-predation sources as in Eq 2.

(2)

Here, m_l and m_q are linear and quadratic mortality parameters set iteratively during model calibration and m_oil is mortality derived from the dose-response model in Eq 3. g_oil is the modifier on growth rate caused by oil exposure in Eq 4.

(3)

(4)

The m_oil term represents a weighted average of a pelagic mortality modifier (m_pelagic) and a benthic mortality modifier (m_benthic). The weighting reflects the proportion B of the affected group’s diet that comes from benthic prey. This serves as a proxy for how intimately a functional group associates with contaminated sediments. Similarly, g_oil is a weighted average of pelagic and benthic growth rate modifiers.

Modifiers are calculated via the dose-response models in Eqs 5 and 6 (note change in symbols: e.g., m_t = m_pelagic at time t when the amount of bioavailable oil φ_t is representative of the pelagic environment). The dose-response models were developed in a previous publication [25]. They are based on organism responses to petrogenic PAHs from exposure studies and field sampling of the DWH oil spill and elsewhere. Lesion and tumor frequencies were used as proxies for changes in mortality rate and otolith annuli measurements were used to infer growth rate impacts. Due to the limited available ecotoxicology data, it was not possible to develop mortality and growth responses specific to Atlantis functional groups. The mortality response was based on 106 species from 40 fish families and the growth response was based on nine species from six families [25]. These models were applied to all fish functional groups, assuming a similar physiological vulnerability to oil toxicity, although functional groups varied in their level of exposure according to the oil transport model described below.

The ‘hockey stick’ dose-response models in Eqs 5 and 6 were selected from among linear, exponential and step-function models using an Akaike information criterion. The hockey stick model implies that there is a PAH concentration threshold below which there are no adverse effects and above which there is a log-linear decrease or increase in growth or mortality rates, respectively. The effect size for mortality (m_t) and growth (g_t), vary with time step t.

(5)

(6)

Parameters α and γ are slopes and β and δ are the PAH concentration thresholds where an effect manifests. These were fit by maximum likelihood estimation [25] (α = 0.2885, β = 907.4306, γ = 0.0531, δ = 28.422). Since experiments exposed animals to oil over the course of one or more weeks, a mean exposure time of ω = 15 days is assumed so that we may approximate the daily effect. Note that we do not necessarily assume direct mortality from oil, but rather an increase in the likelihood of mortality from any source due to reduced health and behavioral changes [26]. Although certain components of the oil are known to be more toxic than other components, there are too few response data available to discern toxicity based on oil composition. Dose response models for mortality and growth are assumed applicable to all fish functional groups and age classes (although total effect size is modulated by sediment association as in Eqs 3 and 4 and by co-occurrence of oil and life stage distribution). We assumed that no avoidance behavior occurred.

Organisms used in lab and field exposure studies to develop the dose-response models were typically from chronically oil-exposed areas [25]. Since resistance has been documented in fish populations to a range of organic pollutants, either through genetic adaptation or physiological acclimation [27], we acknowledge that fish populations exposed to DHW oil could react more severely to oiling than those examined in exposure studies. Therefore, we conduct sensitivity analysis on the threshold parameter of the hockey stick model, reducing the threshold parameter β (by -0%, -20%, -40%, and -60%) for simulations involving the mortality modifier. These simulations are referred to as β907, β726, β544, and β363, respectively (the numerical value corresponds to the PAH threshold in ppb). The threshold for growth effects is already low (δ = 28.4 ppb) so a sensitivity analysis is of less value on this parameter.

Exposure studies used to build the dose-response relationships consistently reported PAH concentrations in the sediment [25]. Unfortunately, we did not have enough sediment oil data from the field to use directly as the oil driver in the simulations. The few sediment oil data that we had existed for only small and non-contiguous areas of the GOM (I.R. and D.H., unpublished data). However, we had water column data for the entire GOM from oil transport modeling. We therefore inferred what the sediment concentrations must have been based on the water column data. To do this, we multiplied time- and depth-integrated water-column oil concentration values by a sediment-to-water column ratio (K). This ratio was informed by a comparison of oil transport modeling results with field-collected sediment cores sampled by C-IMAGE (I.R. and D.H., unpublished data). There is potentially a concentration factor up to 1000 times (S1 Fig). In reality, sedimentation varies by suspended particulate load and microbial activity over the study area, as well as by variations in oil density in the water column and by the sediment composition. These factors are not taken into account by our simple empirical ratio. However, Atlantis polygons are more than 25,000 km² on average and include areas of both below- and above-average deposition rates, thus small-scale spatial variations can be ignored. Persistence of the sedimentary oil is managed by a depuration model, explained below.

We vary the concentration factor (K) in sensitivity analysis (by -0%, -20%, -40%, and -60%) for simulations involving mortality and growth modifiers. Hereafter, these simulations are referred to as K1000, K800, K600, and K400 respectively. Therefore, our sensitivity analysis consists of 16 oil spill simulations varying K and β. Where results from a single oil spill simulation are shown, we indicate the worst-case scenario, [K1000 β363]. This offers an upper bound for potential impacts under our assumed conditions and it should make qualitative effects on ecosystem structure more apparent.

Benthic and pelagic modifiers in Eqs 3 and 4 are calculated in the same way, but they assume different amounts of bioavailable oil φ_t at time t. This amount is determined by an uptake/depuration model that implicitly represents accumulated body burden (Eq 7).

(7)

Hydrocarbons may accumulate through direct uptake from the water by gills or skin, uptake of suspended particles or through ingestion through food. In all cases, there is potential for bio-magnification of toxins. We have combined all processes into a single uptake function, where the rate of uptake is the product of an uptake constant (μ = 1 for benthic environments and μ = 0.1 for pelagic environments) and pollutant load O_i,t as determined from oil transport modeling. In the depuration constant ρ, we have summarized into a single term the effects of gill elimination, metabolic transformation, fecal egestion, growth dilution and elimination via egg deposition and sperm ejection [28]. A rate of depuration is assumed that achieves 99% clearance in 20 days (ρ = 0.2424). Note that ecotoxicology experiments will improve this estimate (D. Wetzel, Mote Marine Laboratories).

Oil transport and fate modeling

Oil concentrations were taken from a probabilistic framework for oil droplet-tracking based on the Connectivity Modeling System (CMS) [29], an open-source Lagrangian stochastic model. Examples of oil applications of the CMS (oil-CMS) are provided by Paris et al. [30]. The geographical distribution and depth of the so-called deep plume is similar to observations by Kessler et al. [31]. Surface oil is informed by Le Hénaff et al. [32] and validated against remote-sensing observations. However, the configuration and oil concentration calculations of the model applied here have improved over Paris et al. [30] since we have used published values of the actual oil spilled [33,34]. Degradation dynamics now consider new results from high-pressure experiment data [35,36] and hydrocarbon fractionation is implemented more realistically with all fractions in a single droplet, allowing dissolution (C.B.P. and N.P. unpublished).

The oil-CMS computes the 3-D oil particle trajectories and their evolution. Its fourth order Runge-Kutta advection scheme utilizes 3-D momentum, temperature and salinity data from the high-resolution ocean model with data assimilation. The oil-CMS estimates droplets’ terminal velocity depending on their size and density and on ambient fluid properties (temperature, salinity, density and viscosity). Additional considerations account for oil dissolution, biodegradation rates and surface evaporation [35]. The initial conditions of the model are specified a priori. The most recent simulations of the Deepwater Horizon incident are summarized in an article in preparation (C.B.P. and N.P., unpublished manuscript). Briefly, we used an initial oil droplet size distribution of 1–500 microns [33]. Post-processing analysis of modeled oil droplet location and properties through time translated the model output data into oil concentrations. This used estimates of the actual oil spilled [34,36] and the assumed droplet size distribution at the time of the release [37].

CMS operates at 1/25^th degree horizontal resolution over 20 vertical layers. The oil droplets are released bi-hourly for 87 days and tracked for 167 days in the area bounded by 25^oN and 30.75^oN latitude, 93^oW and 84^oW longitude. Oil concentrations are integrated over the vertical dimension to match Atlantis’ depth layers (partitioned at 10 m, 20 m, 50 m, 200 m and 2000 m) and provided to Atlantis at 24-hour intervals. Dose-response calculations are made at each horizontal grid point and averaged over Atlantis’ polygons, adjusting for the proportion of grid points per polygon that do not contain oil. We use a 100-day ‘spin-up’ time before introducing oil in the Atlantis simulation to allow transient biomass dynamics to settle. Oil forcing lasts for 167 days and depuration continues thereafter according to Eq 7.

Fisheries closures

We conduct model runs that simulate DWH oil spill emergency fishery closures as spatiotemporally dynamic fishing closures. The schedule is reported by the National Centers for Environmental Information [38]. ArcGIS shapefiles [39] for closures were processed relative to the Atlantis polygon map using an Intersect tool and the proportion of each polygon overlapped a closure was closed to the appropriate fishing fleet(s) in Atlantis. This reduces local fishing mortality proportionately without reallocation of spatial effort. We simulate closures from April 20, 2010 to April 19, 2011, the duration of the emergency closures, updating spatial configuration at daily time steps.

Recruitment

Additional simulations test potential impact on recruitment. Taxon-specific impacts on fish larvae were calculated in a previous study based on the overlap of larval fish distributions and observed surface oil from the DWH oil spill [40]. Distributions of larval fish were created using 27 years of samples collected by the Southeast Area Monitoring and Assessment Program (SEAMAP). Recorded counts were standardized to create monthly average abundance distributions. Surface oil distributions were provided by a study conducted by NOAA, which binned the surface oil features on a daily basis using the presence/absence of oil and a semi-quantitative estimate of oil thickness (density). See Murawski et al. [1] for further explanation. The estimated proportions of fish larvae potentially exposed to DWH oil were calculated as the abundance of fish larvae located within the oiled area during the months of April through July divided by the total abundance of fish larvae throughout the entire year in the northern GOM study area. It is assumed that any larva exposed to oil was killed [40].

Analysis

Unless otherwise stated, all results shown are averaged over the shaded polygons in Fig 1. Out of the 64 dynamic polygons in the model, these 12 showed the greatest proportional change in biomass (averaged across fish functional groups) due to the oil spill (i.e., relating to the net effects of processes such as direct toxicological impacts, seasonal or ontogenetic movement of impacted populations, or movement of impacted prey resources). These areas also roughly correspond to areas of injury assessment [14] efforts. For clarity, we have aggregated results from Atlantis fish functional groups into eight ‘guilds’: snappers (Family: Lutjanidae), groupers (Family: Serranidae), Sciaenidae, elasmobranchs, large pelagic fish, small pelagic fish, small demersal and reef fish, and large demersal fish. Atlantis functional groups that comprise these guilds are presented in S1 Table. Species that comprise Atlantis functional groups are presented in Ainsworth et al. [19]. Simulations are 50 years’ duration, from 2010 to 2060.

We compare Atlantis’ predictions on age structure changes against tagging data from Southeast Data, Assessment and Review (SEDAR) reports. We also compare Atlantis’ predictions on species composition and body size changes against ROV fish count data and laser-scaled fish size estimates. Comparisons against ROV data are done at the level of functional groups. There were sufficient ROV abundance data to compare against 16 model functional groups. There were sufficient ROV size data to compare against 9 model functional groups. The ROV data are from 16 natural reefs sites on the northern Gulf shelf located from Mobile Bay to Choctawhatchee Bay (W.P., unpublished data) (Fig 1). The sites range from 17 to 75 m depth. Observations were made between August 2009 and Sept 2015 and so they represent pre- and post-spill conditions. Since the fish numbers from Atlantis represent an average over oiled polygons, they encompass a full range of age classes, habitats and depths. It is not meaningful to compare densities with ROV surveys in absolute terms since model data represents the average over polygons. Instead, we concentrate on trends, scaling median values to match the ROV data. Laser-scaled fish size estimates were converted to body weight based on length-weight relationships in Fishbase [41]. This comparison requires a caveat that only a fraction of species and age classes constituting these functional groups are present in surveys.

Fisheries closures and recruitment effects

The model suggests that fisheries closures and loss of larvae due to oil exposure (hereafter called recruitment effects) have little impact on ecosystem biomass (S2 Fig). Closed areas are responsible for a minimal biomass increase; no more than 1/3 of 1% in any species guild relative to a no-oil scenario (see S1 Table for guild compositions). Recruitment effects similarly have a negligible impact. When averaged across Atlantis functional groups, the recruitment impact equated to a 5.8% (σ = 4.5%) loss of the larval population in the year of the oiling. This was sufficient to reduce biomass only about 1/10^th of 1% for any guild. Across species guilds, the MPA effects were responsible for no more than 3.6% of the variance of biomass, and the recruitment effects were responsible for no more than 0.2% (both measured monthly from spill up until to one year post-spill). As modeled, these simple treatments result in short-lived effects that do not impact long-term recovery. Therefore, we omit fisheries closures and recruitment effects from subsequent analysis to concentrate on direct and indirect effects of oil toxicity on juvenile and adult life stages.

Impacts on biomass

Fig 2 shows biomass in the oil scenarios relative to the no-oil scenario for 8 species guilds including the mean and range of variation predicted under different settings of the assumed water column-to-sediment concentration factor K and oil-effects threshold β. The largest decreases in biomass relative to a no oil scenario occur within 7–16 months after the spill (median 10 months). Guilds comprised of large reef fish (i.e., snappers, groupers and sciaenids) as well as elasmobranchs and small pelagic fish exhibit a major impact, decreasing by 25–50% biomass relative to the no oil scenario in the most heavily oiled areas. This closely matches observed declines from ROV surveys conducted before and after the spill (see section Comparison against ROV survey data). Large impacts also occur in large pelagic fish and large demersal fish. The model suggests decreases in those guilds by 40–70%. A larger effect occurs in small demersal and reef fish. This guild, which constitutes the benthic and reef forage base, decreases by 50–75%. S2 Table provides biomass minima (i.e., minimum biomass reached by guilds throughout the simulation) for all variations of K and β. Note that small differences between the guilds are not meaningful in light of the error range of the model, but it can be said that small demersal and reef fish exhibit the largest effect followed by large pelagic fish and large demersal fish.

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Fig 2. Biomass trajectories for species guilds.

Biomasses are summed across all functional groups within these guilds. Shaded area shows range of outcomes observed in sensitivity analysis on concentration factor K and threshold β. Black line shows the mean of the 16 sensitivity runs.

https://doi.org/10.1371/journal.pone.0190840.g002

Recovery

There is no consistent relationship between biomass minima and recovery time (Fig 3) but small-bodied fish recover the fastest. For example, the small demersal and reef fish guild and most of the groups constituting the small pelagic fish guild return to pre-spill conditions within 10 years. One group each from the elasmobranchs, snappers, and groupers guilds takes longer than 20 years to recover. Three functional groups did not recover within the 50-year time horizon: two large pelagic fish and a grouper (S1 Table). More than half of the Atlantis functional groups comprising the large pelagic fish guild take longer than 30 years to recover. The large pelagic fish guild was also among the most heavily impacted guilds in terms of biomass lost (Fig 2). However, this finding is not borne out by commercial catch statistics (S3 Fig). This discrepancy is informative and we will return to it later.

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Fig 3. Biomass minima versus years to recovery.

Data are relative to no-oil scenario. Criterion for recovery: achieving 99% of biomass of the no-oil scenario. Functional groups > 50 years to recovery did not recover within simulation period. Points show Atlantis functional groups arranged by guild. Represents oil simulation [K1000 β363].

https://doi.org/10.1371/journal.pone.0190840.g003

Age structure

Reef-associated and demersal fish guilds experienced a loss of young individuals with the oiling event and therefore an immediate shift in age structure towards older individuals (S4 Fig). This is typified by the large demersal fish guild (Fig 4). This trend is supported by aging studies for almost every species assessed in the GOM since 2012 (S5 Fig, S3 Table). Only Gulf menhaden age composition fails to follow this trend. However, within a few years after the loss of these young fish, the model predicts a depressed number of reproductive aged individuals relative to the no-oil scenario, and therefore a shift towards a relatively younger age structure that persists for 5 or 10 years for most guilds (S6 Fig).

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Fig 4. Differences in age composition for the large demersal fish guild.

No oil (dark gray); oiled (light gray). Data represent October 2010 for heavily impacted polygons. Represents oil simulation [K1000 β363]. A) Relative proportion, B) absolute biomass.

https://doi.org/10.1371/journal.pone.0190840.g004

Spatial patterns

Ecological impacts from the DWH oil spill manifest at great distances away from the slick and sub-surface plume. The spatial pattern of impact for groupers (Fig 5) is typical for reef and demersal species (S7 Fig). Reduced biomass is predicted on West Florida Shelf populations, particularly in the south, and as far away as the Texas shelf and Campeche Bay. This can be attributed to impacts on the mobile prey base and/or on prey populations interconnected by larval transport. Such indirect trophic interactions have been shown to influence recovery dynamics with previous oil spills [42]. A decrease in condition factor (Fig 6) verifies that groupers are unable to consume enough prey for their needs despite an increase in the per capita rate of consumption (S8 Fig). Condition factor is represented as the ratio of reserve (soft tissue) to structural Nitrogen (hard tissue). The decrease in grouper condition factor is relatively large. During the period when condition factor is lowest in the oiled scenario (black solid line in Fig 6, around 2011), the grouper guild’s condition factor is 2.01, which is equivalent to the 32^nd percentile of all functional groups’ pre-spill values. However, without the oil effect, the grouper guild’s condition factor would be 2.32 at the same point in time, equivalent to the 60^th percentile of functional group pre-spill values. Reduced condition factor is predicted in all guilds to some degree (S9 Fig).

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Fig 5. Absolute biomass reduction for no oil versus oil scenario for grouper guild.

Biomass minima is shown occurring at 10 months after the oil spill. Oil simulation [K1000 β363].

https://doi.org/10.1371/journal.pone.0190840.g005

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Fig 6. Condition factor of grouper guild.

Condition factor is represented as reserve:structural Nitrogen ratio. High rN/sN indicates good body condition. Dotted line: no oil scenario, solid line: oiled [K1000 β363].

https://doi.org/10.1371/journal.pone.0190840.g006

Changes in the food web

Since the demersal and reef forage base was impacted more severely than the pelagic forage base, the model predicts a shift towards pelagic prey relative to demersal prey for all guilds (S8 Fig). The guilds are broad enough taxonomically to include some opportunistic feeders. At the same time, the model predicts an overall increase in the per capita consumption rate since so many predators have been eliminated (Fig 2). In some cases, the increase in per capita consumption rate may be partly due to the shift in age structure in predators towards (more voracious) young individuals, which happens subsequently to the initial loss of young individuals due to oiling (S6 Fig). The increased abundance of pelagic forage relative to demersal forage is indicated in the ecosystem’s increased pelagic-to-demersal ratio (S10 Fig). Other ecosystem indicators reveal structural changes in the food web. Major impacts on the forage base results in an increase in the ecosystem’s piscivory to planktivory ratio, while the loss of dominant predator functional groups results in system-wide decreases in mean trophic level and Shannon’s biodiversity index.

Impacts on fisheries catch

The model predicts reduced catches the year after the oil spill (2011) in fleets targeting pelagic and reef-associated fish. Note that we have assumed no spatial reapportioning of effort or other adaptive changes by fishers. Generally, the model shows a 20–40% reduction in catch relative to the no oil scenario (Fig 7). Substantive losses are predicted in estuarine gillnet fisheries, mackerel fisheries and pelagic longline fisheries. Impacts on pelagic and some reef fisheries last into the 2020s according to the model, but there is a modest improvement in menhaden and royal red fisheries over that time. Fig 7 shows the predicted effect of the oil spill alone for comparability with the other results presented. S11 Fig indicates predicted losses of catch due to fishery closures.

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Fig 7. Projected annual catch for whole GOM.

Catch presented relative to no-oil scenario. Error bars show range of sensitivity analysis; bars show mean.

https://doi.org/10.1371/journal.pone.0190840.g007

Comparison against ROV survey data

S12 Fig compares fish densities from ROV data with Atlantis estimates. The ROV data (W. Patterson, unpublished data) are influenced not only by changes in population size but also by fish movement between reefs. Nevertheless, there is qualitative agreement between the ROV data and model predictions in many cases. The model agrees with survey data that there was an initial decrease in fish numbers following the spill for all relevant functional groups except scamp and sciaenidae. The survey data shows post-spill fish density increases for six groups: large reef fish, shallow serranidae, small reef fish, jacks, other demersal fish and vermilion snapper. In all cases, the model agrees. The model also correctly predicts a lack of post-spill fish density increases in five groups: deep serranidae, red grouper, red snapper, greater amberjack and lutjanidae. However, in three groups, the model predicts post-spill fish density increases where none are seen in the observational data (gag grouper, scamp and sciaenidae). This suggests that unmodeled dynamics thwart recovery; notably, there are no recoveries in the ROV data that the model fails to predict. There are too few data available to assess predictions regarding large sharks or skates and rays. Thus, 11/16 functional groups show good agreement with data, capturing the initial decrease and subsequent increase in fish density (or lack thereof), 3/16 show marginal agreement (agreeing with the initial decrease or subsequent increase but not both), 2/16 cannot be assessed.

S13 Fig shows changes in body size predicted by the model relative to the ROV data. The model predicts realistic body sizes for jacks, lutjanidae, other demersal fish, red grouper, small reef fish and vermilion snapper indicating that the emergent body growth rates, which are a product mainly of consumption rates and mortality rates, are realistic. The model underestimates body sizes for gag grouper, red snapper, and scamp, though this result would be consistent with cases where fish present on the reefs are younger than the population average. The model is generally less variable than the observational data but this is expected since the model does not capture fish movement and averages body size data spatially and temporally. The model fails to predict an increase in the body size in red grouper and lutjanidae and a decrease in small reef fish. However, the quick rate of change in those data suggests this may partly be due to fish movement and not population-level change in weight at age. The model successfully predicts the increase in scamp body weight. Other groups, which do not show a clear directional change in the observations, are adequately represented by the model’s constant body size.