High uncertainty in fish bioenergetics impedes precision of fish-mediated carbon transport estimates into the ocean ’ s twilight zone

Mesopelagic fishes may contribute substantially to marine carbon transport by consuming organic carbon near the surface at night and releasing it in the mesopelagic zone during the day. However, the magnitude and uncertainties associated with this transport are not well understood; fish-mediated carbon flux estimates range from less than 1 % to greater than 30 % of biologically-driven carbon export out of the epipelagic zone. While total mesopelagic fish biomass is an important source of uncertainty, information on fish bioenergetics and movement might also limit the precision of carbon transport estimates. Here, we ask how uncertainties in fish bioenergetics and behaviors affect carbon flux estimation, and which processes contribute most to uncertainty. We used sensitivity analyses to reveal that modeled carbon flux was most sensitive to respiration-related parameters, and per capita fish carbon flux estimates varied six-fold over the range of plausible parameter values. Biomass estimation can add at least ten-fold variation in ecosystem-scale carbon flux estimates. We conclude that it is not currently possible to estimate fish-mediated carbon flux precisely, but estimates may be constrained through future empirical work on the most influential parameters.


Introduction
Oceans play a key role in global carbon budgets and cycles by absorbing about a quarter of carbon dioxide emissions (Keeling & Garcia 2002;Friedlingstein et al., 2022).Understanding the mechanisms and drivers of carbon transport from the sea surface to the deep sea is essential for estimating the magnitude of net carbon flux into the oceans, which is a precursor for predicting how the oceans sequester carbon over time (Kwon et al., 2009;Burd et al., 2010;Boyd et al., 2019).
To gain this understanding, global carbon budgets need to account for distinct pathways of carbon transport from surface waters-where carbon fixation occurs-into deeper waters where carbon is sequestered from the atmosphere for decades to centuries (Siegel et al., 2021).These pathways include those within the biological carbon pump, which is responsible for roughly one third of the ocean's carbon dioxide uptake from the atmosphere, with the solubility pump being responsible for the remainder (Volk & Hoffert 1985).The biological carbon pump drives 60-70 % of the dissolved inorganic carbon gradient in the ocean (Toggweiler et al., 2003).The biological pump is often partitioned into passive and active carbon transport components.Passive carbon transport is driven by the gravitational sinking and physical mixing of particulate and dissolved organic carbon, has been more extensively documented (Ducklow et al., 2001;Boyd & Trull 2007;Boyd et al., 2019;Buesseler et al., 2020) and likely accounts for at least half of all carbon exported out of the epipelagic zone (0-200 m) via the biological carbon pump (Longhurst et al., 1990;Boyd et al., 2019).Much of the remaining biological carbon export out of surface waters is mediated by the more poorly understood process of active transport.Active carbon transport is largely driven by diel vertical migration (Boyd et al., 2019) whereby organisms swim up to the productive epipelagic zone to feed at night.During the day, they swim back down to the mesopelagic zone (~200-1000 m) (Klevjer et al., 2016), also known as the twilight zone, to hide from visual predators (Hays 2003).At depth, these migrating organisms release carbon through respiration and egestion, and some are consumed by deep-dwelling predators.These processes result in net downward flux of carbon (Longhurst et al., 1990;Archibald et al., 2019;Saba et al., 2021).
While the contribution of zooplankton to active transport is relatively well-studied (Ducklow et al., 2001;Turner 2015;Hernández-León et al., 2019b;Archibald et al., 2019;Boyd et al., 2019), the contribution of fish is often overlooked but has become a priority for biogeochemical cycling research (St. John et al. 2016;Drazen & Sutton 2017;Aumont et al. 2018;Pinti et al. 2023).Active carbon transport by fish has been estimated to range from less than 1 % to over 30 % of total biological carbon export (active plus passive transport) from the epipelagic zone (Hidaka et al., 2001;Davison et al., 2013;Hudson et al., 2014;Ariza et al., 2015;Belcher et al., 2019;Hernández-León et al., 2019a).Additionally, fish-mediated carbon transport has been proposed to explain why empirical observations often find more carbon in the mesopelagic zone than predicted from biogeochemical models (Burd et al., 2010;Davison et al., 2013).Better understanding of fish-mediated carbon transport is particularly important given that these fish are subject to a host of anthropogenic influences including fishing, warming temperatures, and deoxygenation.
Current estimates of mesopelagic fish-mediated carbon transport are highly imprecise owing to limited knowledge of the abundance, community composition, migratory behaviors and physiological rates of these fishes.Mesopelagic fishes are distributed nearly globally and include some of the most abundant vertebrates on the planet, especially fish of the families Gonostomatidae (bristlemouths) and Myctophidae (lanternfishes) (Gjøsaeter & Kawaguchi, 1980;Williams & Koslow, 1997;Hidaka et al., 2001;Davison et al., 2013).There is high uncertainty in their abundance, with estimates ranging from about 1-16 billion metric tons (Kaartvedt et al., 2012;Irigoien et al., 2014;Proud et al., 2019).If this range is accurate, they make up roughly 50 % to 94 % of total global fish biomass (Jennings et al., 2008;Wilson et al., 2009;Bianchi et al., 2021;Gjøsaeter & Kawaguchi, 1980).Meanwhile, global estimates find that fish may contribute a mean of ~16 % of total carbon transported via the biological carbon pump (Aumont et al., 2018;Saba et al., 2021).Acknowledging that flux boundaries vary by study, and that the above estimates are subject to high uncertainty, fish may contribute anywhere from 0.3 % to 40 % (Saba et al., 2021) of total, biologically-driven carbon flux out of the epipelagic zone.
While uncertainty in biomass and variation in migrating proportion is well-established (Klevjer et al. 2016;Proud et al. 2019), far less attention has been paid to the consequences of individual-level physiology and behavioral processes relevant for carbon transport.These lessstudied uncertainties include physiological vital rates (e.g., metabolism, digestion efficiency) and diel migratory behaviors that vary widely across taxa.The vertical extent of diel migration is important because it affects the depths (and therefore sequestration times) at which fishmediated carbon is transported.Physiological rates are poorly known for these taxa because these measurements are notoriously challenging to collect on such deep-dwelling, fragile organisms (Robison 1973;Torres et al., 1979;Smith & Laver 1981).Yet, digestion efficiency and metabolic demands directly influence the primary pathways by which mesopelagic fish transport carbon.
Here we explore the consequences of uncertainty in migratory behavior and physiological rates for estimating mesopelagic fishmediated carbon transport.We focus on these because the consequences of mesopelagic fish abundance are relatively well understood.Published models of fish-mediated carbon transport often calculate the carbon flux per unit of fish biomass, multiply that flux by the total estimated fish biomass, and consider several capture efficiencies to incorporate uncertainty in biomass.However, uncertainty associated with migration physiology are poorly understood and likely vary considerably across regions depending on biogeography and oceanographic conditions.
We address this problem by asking: what are the bioenergetic and behavioral parameter uncertainties in estimates of fish-mediated carbon transport, and which parameters contribute most to uncertainty in the magnitude of this transport?First, we develop a bioenergetics model using a framework similar to that used by Davison et al. (2013).Second, we use a literature review to determine the precision of the parameters used in the model.Third, we evaluate the consequences of imprecision on the modeled carbon transport using thorough sensitivity analyses.In this way, we can propagate the imprecision in parameter values through to the estimated carbon transport.Finally, we identify the most influential parameters beyond biomass of migrating and non-migrating fish, and discuss our model estimates in the context of other mechanisms of carbon flux.

Methods
We developed a model of fish-mediated carbon flux that incorporated the bioenergetics and movement patterns of small, highly abundant mesopelagic prey fish to determine the fate of their bioenergetic output.The scope and aim of this model were to calculate carbon transported by individual migrating or non-migrating mesopelagic fish past a flux boundary of 150 m (though we allow this to vary in the sensitivity analysis).These calculations involved detailed accounting for carbon throughout a daily cycle.To inform this model with specific parameter values, we then conducted a literature review to determine minimum and maximum plausible values for each parameter.Sensitivity analyses revealed the total uncertainty in per capita estimates of fishmediated carbon flux.Results of this analysis are intended to serve as a relative, rather than absolute, calculation of per capita fish-mediated carbon flux.

Model scope and structure
We limited the extent of this model to calculate carbon flux for individual mesopelagic fish but not communities.In doing so we considered basic movement patterns and a commonly used flux boundary that approximated a feasible mixed layer depth and the boundary between the epipelagic and mesopelagic zones.We did not explicitly scale individual fish carbon flux to community-level fish carbon flux because uncertainties originating from unknown capture efficiencies (for netbased abundance estimates) and target strengths (for acoustic-based estimates) have already been established (Proud et al., 2019).Instead, we focused on individual level physiological processes, modeling rates that are relevant to carbon fluxes.We considered basic movement attributes such as whether individual fish undergo diel vertical migration, or whether they are full-time residents of the mesopelagic zone (hereafter referred to as either vertically migrating fish or non-migrating fish, respectively).We also considered how the vertical extent of migration affects total carbon flux below a pre-determined flux boundary.We did not explicitly consider the counterfactual of how much carbon would be transported in the absence of mesopelagic fish; in the absence of any individual fish, some of the carbon that would have passed through it may still be transported anyway by other mechanisms.Such a calculation would require a more sophisticated, region-specific model of zooplankton-mediated carbon flux, which is outside the scope of this study.
Both migrating and non-migrating adult mesopelagic fish were considered in the model.Vertically migrating fishes (e.g., myctophids) are defined here as those that migrate shallower than 150 m at night to feed.Non-migrating fishes (e.g., bristlemouths and other species that remain deeper than 150 m throughout the day) do not actively carry carbon across the flux boundary.However, they consume vertically migrating zooplankton or other migrating mesopelagic fish that may otherwise migrate back up above the flux boundary, and release carbon there in the absence of these non-migrating fish predators (Fig. 1).

Overview of carbon flux calculations
We adopted a bioenergetics modeling framework similar to that used by Davison et al. (2013).We modeled daily individual-level feeding, metabolism, egestion, and mortality for two distinct groups of mesopelagic fish to estimate their role in marine carbon transport.This model calculated carbon fate past a flux boundary at discrete, daily time steps throughout the daily activity of a migrating or non-migrating individual.Within each day, vertically migrating fish spend a portion of each day migrating, foraging, and resting, and non-migrating fish spend the day either foraging or resting.This allows for consideration of how migration depth and time spent at various activity levels (resting, foraging or migrating) affects carbon transport.Seasonal variation was not considered.

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The overarching bioenergetics equation in the model is based on a standard base energy budget (Kitchell et al., 1974, Kitchell et al., 1977).This equation relates growth to energy intake (ingestion) and energetic losses due to activity and standard metabolism, food assimilation (specific dynamic action), growth, egestion, and excretion.Energy was converted into units of carbon using weight specific energy content measurements in fish (mesopelagic fish, where possible) and their waste products.The model then calculated carbon egested, excreted, and respired below the chosen flux boundary, plus carbon exported via mortality (Fig. 2).

Carbon flux model parameterization
To parameterize the model, we derived nominal parameter values and plausible parameter ranges from the literature using data from mesopelagic fishes when possible, then from other teleost fish when not possible.These selected parameter ranges incorporated both natural variability in parameters where appropriate, plus uncertainty associated with error in the methods of measurement.These sources of uncertainty would be difficult to distinguish, and considering both captures the full uncertainty in estimating fish carbon flux from the best available knowledge.To estimate respiration rates, we used respirometry data (Ikeda 2016) to relate routine respiration rates of mesopelagic fish to fish size, water temperature, and migration behavior.We arbitrarily chose water temperatures that may be typical of a pelagic, temperate environment for the purposes of the sensitivity analyses, but temperature can be adjusted to suit any specific study site context.We used the same general equation for both types of fish (migrating or nonmigrating) and for each of their behaviors (foraging, resting, or migrating) (Appendix B, Eq. (B.1) and (B.2)).Respiration throughout a 24-hour cycle was summed across the time spent in each of these behaviors.We allowed assumed activity levels and time spent in each of these behaviors to vary in the sensitivity analysis due to limited empirical information about mesopelagic fish behavior, activity levels and associated respiration rates.Respiration rates in units of oxygen utilization were then converted to units of daily energy expenditure (Appendix B, Eq. (B.3) and (B.4)).We assumed that specific dynamic action, egestion, and excretion rates were proportional to ingestion rate (Appendix B, Eq. (B.5)).

Converting from bioenergetic outputs to carbon transport
We convert the routine metabolic rate to respiratory carbon flux of vertically migrating fishes, given by where C R,VM is the carbon flux due to respiration that is not related to specific dynamic action (mg C/d, per 1 g individual), R Q is the respiratory quotient that converts from moles of oxygen to moles of carbon dioxide, E M is the respiration (kJ/d -1 ) not related to specific dynamic action that can be considered exported by a vertically migrating fish, and Q ox is the oxycalorific coefficient (kJ mg O 2 -1 ) (Jobling, 1994).We assume that all migrating fish respiration during the daytime is exported, plus the fraction of the respiration during their migration that occurs below the flux boundary.None of the respiration that occurs during foraging is exported because this dissolved inorganic carbon, as carbon dioxide, is released shallower than the flux boundary.Given these assumptions, the carbon related to metabolism that is exported (E M ) equals where B is the flux boundary (m), D max is the maximum migration depth (m), and D min the minimum migration depth (m).Calculated in Eq. (B.3), M VM,r and M VM,m represent the energy expended on metabolism that is not related to specific dynamic action while resting and migrating (Eq.(B.3)).We estimate metabolism (and later, ingestion) assuming fish have a mass equal to 1 g as an example mesopelagic fish biomass (Davison et al. 2013, Belcher et al., 2019), and later we also consider a 10 g fish to compare results of the sensitivity analysis.
For non-migrating fish, all respiration occurs deeper than the flux boundary.So, respiratory flux equals Carbon passing through non-migrating fish that originated from zooplankton of non-detrital origin is the only portion of non-migrating fish-mediated carbon transport that contributes toward total fish-mediated carbon transport.
Zooplankton are implicitly represented in the model in ingestion rates but shown here for clarity.where C R,NM is the carbon flux of a non-migrating fish due to metabolic activity that is not related to specific dynamic action, M NM,t is the total energy expended on respiration for non-migrating fish (Eq.(B.3)), Z is the fraction of their prey of non-detrital origin, and the constants convert from grams to moles of oxygen and then from moles to grams of carbon.We assume non-migrating fishes eat zooplankton, which is true for the most abundant non-migrating fishes Cyclothone spp.(Sutton et al., 2010;Thompson & Kenchington 2017), and that they feed nonselectively on migrating and non-migrating zooplankton.Under these assumptions, some proportion of the carbon that they consume as zooplankton prey originated from detritus.We did not include the proportion of non-migrating fish prey of detrital origin in our calculations of fish carbon flux because that carbon may already have sunk deeper than the flux boundary via passive carbon transport.We only consider the prey of non-detrital origin, Z, that are ingested and later released by non-migrating fish.This carbon may otherwise have remained in the epipelagic zone through release by migrating zooplankton.Some non-migrating fishes are piscivorous and consume vertically migrating fish (Butler et al., 2001), and this pathway of carbon flux was included implicitly in the vertically migrating fish mortality term.
The fluxes in Eq. ( 1) for C R,VM and C R, NM calculate only the respiratory flux related to activity and maintenance.We calculate the respiratory flux due to specific dynamic action separately as where C SDA,i is the carbon flux due to respiration associated with specific dynamic action (mg C/d), ϕ SDA is the approximate fraction of total ingestion spent on specific dynamic action, and I is ingestion (kJ/ d).
For vertically migrating fish, E SDA is the export ratio used to include only the digestion-related respiration that occurs below the flux boundary after fish have migrated below it after feeding each night.This export ratio for specific dynamic action was derived based on assumptions about the timing of metabolic demands of digestion relative to feeding time (Appendix C).For non-migrating fish, the respiratory flux related to specific dynamic action is Egestion flux, consisting of sinking fecal pellets as particulate organic carbon (which could later leech out of pellets as dissolved organic carbon), is calculated for vertically migrating fish as where C F,VM is the fecal pellet egestion flux (mg C/d), ϕ F is the fraction of ingested energy lost in fecal pellets, I is ingestion (kJ/d), C p is the ratio of carbon to dry weight of fecal pellets, E F is the ratio of fecal pellets exported from surface waters, i.e., sinking deeper than the flux boundary before remineralization, and e p is the energy density of fecal pellet dry mass (kcal/g).The constant converts from kilojoules to kilocalories.If fecal pellets are released above the flux boundary, detritovores could consume them or bacteria could remineralize them before the pellets sink below the flux boundary.However, the sinking rates of fecal pellets of midwater fishes are some of the fastest known for organic detritus, averaging more than 1 km/d (Robison & Bailey 1981;Turner 2015).
Relative to their sinking rates, their dissolution and bacterial remineralization rates are low (Robison & Bailey 1981).Therefore, we do not attempt to monitor the timing of the release of fecal pellets.Instead, we assume in our sensitivity analyses that most egested carbon is released below or sinks below the flux boundary and we allow this ratio (E F ) to vary (Table B.1).
We use a similar general equation for non-migrating fishes where Z is the proportion of fish prey of non-detrital origin.We assume all egestion occurs below the flux boundary, but by multiplying by Z, the estimated flux is adjusted to exclude egested carbon of detrital origin.
Excretion is often excluded from bioenergetic models of fishmediated carbon transport because little carbon is excreted in the urine of teleost fishes (Davison et al., 2013).We assume that the ingested energy lost to excretion contains negligible carbon, given that excretion is a small portion (~7 %) of energy expenditure (Brett & Groves, 1979), 70-90 % of excretion is ammonia that does not contain carbon (Dosdat et al. 1996;Durbin & Durbin, 1983), and unlike for fecal waste, excreted waste largely occurs within one or two hours after feeding in small forage fish (Durbin & Durbin, 1983).We do consider the excretion of carbonates as a potential contribution to particulate inorganic carbon flux (Neukermans et al. 2023).Carbonate excretion is considered an understudied but relevant component of fish-mediated carbon flux related not to ingestion but to osmoregulation (Wilson et al., 2009).We calculate excretion flux to determine model sensitivity to this mechanism of fish carbon flux.Due to lack of published data for mesopelagic fishes, we use carbonate excretion rates from non-mesopelagic, teleost fishes (Wilson et al., 2009) to estimate this carbon flux as where C X, VM is the excretion flux of vertically migrating fish (mg C/d), X r is a carbon excretion rate for fish (mg C/d), and E X is the ratio of carbon excreted by vertical migrators that is released or sinks below the flux boundary.Carbonate excretion rates (X r ) from non-mesopelagic fish species range from about 18 to 40 nmol C hr -1 per gram of fish wet mass (Wilson et al., 2009 and references therein), so we use an intermediate value and allow this to vary widely in the sensitivity analysis (Table B.1).
Excretion flux for non-migrators is found as where the excretion flux for non-migrators C X, NM is modified by Z as for other fluxes to consider only carbon of non-detrital origin.Mortality from predation, disease and senescence results in carbon exported as particulate organic carbon in carcasses, and predation can result in carbon remaining in the mesopelagic zone if fish are consumed by mesopelagic residents.Using a regression model for annual mortality rate of fishes (Pauly 1980) and mesopelagic fish-specific values of L ∞ and K (Gjøsaeter, 1973;Wörner 1975;Childress et al., 1980), we estimate mortality rates as where μ is the instantaneous annual mortality rate, L ∞ is the asymptotic length of the fish in cm, K is the growth coefficient, and T is the water temperature in • C. We do not distinguish between migrating and nonmigrating fish due to limited K and L ∞ estimates.We used the previously defined mean water temperature over the water column (0-500 m) to calculate mortality rates of vertically migrating fish, and the previously used mean mesopelagic temperature (150-500 m) for mortality rates of non-migrating fish.In the sensitivity analysis, we calculate μ with literature values for a 0 , a 1 , a 2 , a 3 , L ∞ and K, such that only μ is allowed to vary.This determines the overall sensitivity of carbon flux estimates to mortality rate.
Using the mortality rate μ from Eq. (5a), we calculated mortality flux for a 1 g vertically migrating fish as where C μ,VM is the mortality flux (mg C/d), E μ is the proportion of carcasses exported deeper than the flux boundary, and C c the ratio of milligrams of carbon per gram of fish wet mass.The constants convert from annual to daily flux, and e is Euler's number.The right-hand expression is multiplied by W, the mass of the simulated fish (g), and 1000 converts the fish mass from g to mg.We do not assume that all mortality occurs below the flux boundary.Vertically migrating fish such as myctophids are predated on by some seabirds during deep dives (e.g., penguins) or shallow, nocturnal foraging trips (e.g., red-legged kittiwakes), and some of this carbon would presumably be released shallower than the flux boundary or even on land.Migrating fish are also predated on by diving mesopelagic predators such as seals, sharks and tunas (Braun et al., 2019;Rivière et al., 2019).After ascending to shallower waters, these predators may subsequently release carbon from their prey above the flux boundary.
We calculated mortality flux for non-migrating fish as where Z is the proportion of fish prey of non-detrital origin.
Finally, to find per capita fish-mediated carbon flux for vertically migrating fish and for non-migrating fish, we summed the respiration (Eq.( 1)), digestion (Eq.( 2)), egestion (Eq.( 3)), excretion (Eq.( 4)), and mortality (Eq.( 5)) fluxes.We assume that individual fish migrate every day, as assumed in some regional studies (Sobradillo et al., 2022), though this can easily be modified in the model for a region-specific study if empirical data show otherwise.

Informing the bioenergetics model with data
The model described above was informed using data from the literature such that the individual parameter perturbation and Monte Carlo simulation results represented plausible parameterizations of the carbon flux model.Specific parameter values and references are provided in Appendix B (Table B.1).

Sensitivity analyses
We performed individual parameter perturbation to examine how each parameter governs carbon transport across its plausible range, and to determine which parameters contribute most to uncertainty in estimates of fish-mediated carbon transport.We then used Monte Carlo simulation to examine the global sensitivity of each model parameter, identify interactive effects between parameters, and propagate error.For the purposes of these uncertainty analyses, we considered a 1 g vertically migrating fish and a 1 g non-migrating fish, and then also considered a 10 g fish to examine how fish size affected the results.
After describing probability distribution functions for each parameter based on existing literature (Table B.1), we used 5,000 Monte Carlo simulations to randomly sample the parameter space and calculate carbon flux for each sampled vector of model parameters.The regression Fig. 3. Individual parameter perturbation plots for a 1 g vertically migrating fish, showing the full allowed range for each parameter value on the x-axis and carbon flux on the y-axis.A greater slope indicates higher sensitivity of the carbon flux estimate to that parameter.The x-axis units corresponding to each parameter can be found using Table B.1, and percent change in carbon flux across the allowable parameter range can be found in models that result from random draws of respiration rate coefficients fit the data from Ikeda 2016 well (Appendix A, Fig. A.1).Each of these regression models represented a separate Monte Carlo simulation, which pulled regression coefficients from a multivariate normal distribution based on the estimated coefficients and their variance-covariance matrix.For all other parameters in the carbon flux model, we assumed a uniform distribution and allowed them to vary within their feasible range (Table B.1) for each simulation.Where migrators and nonmigrators had common parameter values, we used the same parameter values for both fish categories within each simulation.

Results
The individual parameter perturbation revealed that carbon transported by mesopelagic fish was generally most sensitive to parameters related to oxygen utilization and respiratory flux (Figs. 3 and 4) as respiration is the main component of an individual's energy and carbon flux.Egestion was also a relatively important process for migrating fish but not for non-migrating fish.This difference was in part due to lower ingestion rates by non-migrators, caused by their assumed lower activity levels, colder environments, and thus slower metabolisms as residents of the mesopelagic zone.Furthermore, the importance of egestion flux was magnified for migrators because, unlike for other migrator flux pathways, we assumed most egested carbon would sink below the flux boundary even if released shallower.Carbon flux from non-migrators was dependent in part on diet composition, specifically the proportion of prey of non-detrital origin.This sensitivity is a consequence of our assumption that only a portion carbon passing through non-migrators contributes to carbon transport (consumption of detrital-origin prey does not move carbon from surface to mesopelagic depths).In contrast, we assumed that a large proportion of carbon ingested by migrators is actively transported across the flux boundary, transporting ingested carbon deeper than it may have been transported otherwise.Relationships between parameter values and carbon flux were monotonically increasing or decreasing, and either linear or log-linear.
Our migrating fish carbon transport model was highly sensitive to several parameters (Fig. 3 and Table E.3).For a 1 g migrating fish, the baseline respiration rate (α 0 ), the effect of habitat depth on respiration rate (α 3 ), and the energy density of fecal pellets (e p ) had the largest influence on carbon transport.Estimated carbon flux varied by roughly two-fold (by 120% for α 0 and by 98% for e p and α 3 ; Table E.3) over the ranges we considered for these parameters.In other words, higher respiration rates in migrating fish foster higher rates of carbon flux as migratory fish consume prey in the epipelagic and respire in the mesopelagic zone.This is partly due to the higher consumptive demands that a high metabolic rate impose.Carbon transport declined with increasing energy density of fecal pellets (e p ). Carbon transport increased as the respiratory quotient (R Q ) increased because a higher ratio of carbon dioxide released to oxygen utilized results in higher calculated release of carbon.Carbon flux also increased as the fraction of oxygen utilization attributed to non-digestion related respiration increased (θ).This occurred because a higher θ led to higher overall respiration rates as predicted by the respiration rate regression (modified from Ikeda 2016).The effect of mass on respiration rate (α 1 ) had no effect on carbon transport because our calculations assumed a 1 g fish.However, the sensitivity analysis results were generally robust to fish size, except that the parameter α 1 became more influential when we instead considered a 10 g fish (Appendix D, Fig. D.1).Generally, parameters related to excretion (E X and X r ) had little effect on carbon flux (Fig. 8 and Table E.3).
The non-migrating fish model shared some but not all sensitivities with the migrating fish model (Fig. 4 and Table E.4).The estimated Fig. 4. Individual parameter perturbation plots for a 1 g non-migrating fish, showing the full allowed range for each parameter value on the x-axis and carbon flux on the y-axis.A greater slope indicates higher sensitivity of the carbon flux estimate to the parameter.The x-axis units corresponding to each parameter can be found using Table B.1, and percent change in carbon flux across the allowable parameter range can be found in Table E.4.Parameters are grouped and color-coded by carbon flux pathway.

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carbon flux of non-migrators was most sensitive to the respiratory quotient (R Q ), the activity scaling factor while foraging (A f ) and the baseline respiration rate (α 0 ).Over the ranges we considered for RQ, A f and α 0 , estimated carbon flux varied over two-fold (by 127 % for RQ, 123% for A f ; and 117 % for α 0 , Table E.4).These parameters dictated consumptive demand and the rate at which carbon dioxide was produced, thus affecting carbon flux.Carbon transport increased with the temperature in the mesopelagic zone (T meso ) and the ratio of prey of nondetrital origin (Z).As was true for migrating fish, carbon flux for nonmigrating fish was insensitive to the effect of mass on respiration rate (α 1 ).We removed the effect of habitat depth on respiration rate (α 3 ) from the sensitivity analysis because it did not influence carbon transport of non-migrators; by definition, this coefficient was multiplied by zero when the indicator variable represented a non-migrator.As with migrating fish, excretion-related parameters did not substantively alter carbon flux estimates for non-migrating fish.
We used Monte Carlo simulation to identify interactive effects of parameters on carbon transport, to identify primary pathways by which mesopelagic fish transport carbon, and to reveal ranges of carbon transport rates given plausible parameter values.Generally, interactive effects of parameters mean that the sensitivity of the model to one parameter depends on the value of one or more other parameters.These can be diagnosed using visual plots of Monte Carlo output versus parameter values, where the range of model outcomes varies depending on the parameter value (Essington 2021).Overall, the Monte Carlo analysis did not reveal evidence of substantial interactive effects among parameters (Figs. 5 and 6).In general, the estimated inner 80 % quantile had similar widths, and the lower and upper quantile bounds tended to increase or decrease in the same direction as the median values.Moreover, the relationships between mean carbon flux and parameter values in the Monte Carlo simulations were nearly identical to those revealed in the individual parameter perturbation.
The distribution of carbon flux estimates from the Monte Carlo simulation suggests that there was high uncertainty in carbon export for both migrators and non-migrators, and that migrators of the same mass exported more carbon per unit time than non-migrators (Fig. 7).For both migrating and non-migrating fish, there was over a six-fold difference between the minimum and maximum estimates (inner 90th percentile range, i.e., from 5th to 95th percentile) of per capita carbon export.A 1 g vertically migrating fish transported 0.16 % to 0.76 % of its body weight in carbon per day (inner 90th percentile range), or 51 % to 277 % of its weight in carbon per year.In contrast, a 1 g non-migrating fish transported 0.050 % to 0.32 % of its body weight in carbon per day (inner 90th percentile range), or 18 % to 120 % per year.In generating all Monte Carlo results, we selected the same parameter values for migrators and non-migrators within each simulation.To directly compare magnitudes between migrators and non-migrators, we calculated the difference in carbon transport for each random draw of parameter values.This revealed that a 1 g migrating fish transported 0.19 to 5.1 g of carbon more per year than a non-migrating fish (inner 90th percentile range of differences).
We used the Monte Carlo analysis to examine the relative importance of carbon flux pathways-respiration, specific dynamic action, egestion, mortality and excretion-and how that varied between migrating and non-migrating fish (Fig. 8, Table E.2).For both categories of fish, respiration was a major contributor to total carbon flux and excretion was a negligible contributor.However, the relative importance of respiration and egestion fluxes varied between migrating and nonmigrating fish.Respiration and egestion had similar importance for migrating fish, and respiration was the dominant pathway for nonmigrating fish.Additionally, mortality was proportionally more important for non-migrating than migrating fish.
Although the basic bioenergetics equation was the same for migrators and non-migrators, differences in carbon export accounting drove the above differences in the relative contribution of pathways.For nonmigrators, all carbon released via respiration and mortality (save for the small fraction attributed to detrital prey) was considered exported below the flux boundary.For migrators, we excluded the respiration that occurred while actively foraging and the mortality that occurred due to predation in more productive surface waters above the flux boundary.As a result, respiration and mortality contributed a lower percentage of total carbon flux for migrators than for non-migrators.With specific dynamic action and excretion contributing similarly, egestion necessarily became a more important flux for migrators than for nonmigrators.Furthermore, most of a migrator's egested carbon was counted toward export because we assumed that most of their dense fecal pellets sink below the flux boundary even if released near the sea surface.Therefore, due to differences in carbon export accounting, egestion was relatively more important, and respiration and mortality relatively less important, for migrators than for non-migrators.

Discussion
This modeling exercise revealed that our understanding of carbon transport by mesopelagic fish is highly uncertain, even putting aside the inherent uncertainty in mesopelagic fish abundance.Estimated carbon flux from surface to deep (greater than 150 m) depths varied by at least six-fold when considering parameter uncertainties over the range of plausible parameter values.Some of the most influential parameters that governed the rate of carbon flux are difficult to precisely measure because of the deep depths at which these fish live and the subsequent challenges in measuring respiration and other metabolic expenditures.These findings, compounded with uncertainty in the biomass of mesopelagic fish (which itself can span up to three orders of magnitude,    Davison et al., 2015), imply that total uncertainty causes fish-mediated carbon flux estimates at an ecosystem scale to range over three orders of magnitude.Consequently, it is not possible at this time to precisely estimate mesopelagic fish-mediated carbon flux, although further empirical research could reduce this uncertainty.Reducing uncertainty in estimates of carbon export via the biological carbon pump could have enormous economic and decision-making value in the realm of climate policy and management (Jin et al., 2020).
For both vertically migrating and non-migrating fish, the model was highly sensitive to parameters that are challenging to further constrain through empirical work.Carbon transport by both categories of fish in our model was highly sensitive to respiration rate and activity scaling factors used to account for variable respiration rate throughout a 24hour cycle.Respirometry measurements of mesopelagic fish are challenging for several reasons.These fish generally live in offshore habitat (except in some deep, inland fjords) so sampling is costly.Migrators can be captured near the sea surface at night, but non-migrators only exist in high pressure, deep environments, making them challenging to capture and keep alive in a shipboard lab at one atmosphere of pressure.Many migrators and non-migrators are fragile and not able to endure encountering hard surfaces as in a respirometer, and often survive for less than an hour to a few days (Robison 1973;Torres et al., 1979;Smith & Laver 1981;Donnelly & Torres 1988).In situ measurements may circumvent some of the trauma caused by capture of mesopelagic fish in nets for respirometry measurements on a shipboard laboratory, but these will require considerable advances in undersea technology to be effective.
Some of the remaining terms in the migrator and non-migrator models could be estimated better through dedicated study using available tools and methods.For example, future empirical studies of mesopelagic fish bioenergetics and fecal composition could further constrain e p , the energy density (kcal/g) of fecal pellets.The proportion of pellets sinking below the flux boundary (E F ) could be estimated using sediment traps measurements, and morphological or genetic identification of mesopelagic fish fecal pellets.Parameter uncertainty for time spent resting versus foraging could potentially be reduced using highresolution acoustic work (Kaartvedt et al., 2020), or using underwater robots like Mesobot that can optically track individuals and observe their behavior (Yoerger et al., 2021).The proportion of prey of non-detrital origin could potentially be constrained with comprehensive stable isotope analyses of region-specific mesopelagic food webs.By obtaining more empirical measurements, future empirical research could build a better statistical understanding of these parameters and their probability distribution functions in addition to their appropriate ranges.We assume basic, uniform distributions for most parameters purely due to data limitation.
Two considerations are important to extrapolate individual to ecosystem-level carbon flux.One, there is tremendous uncertainty in abundance and biomass.Acoustic-based biomass are sensitive to depths, fish size distributions, species composition, and assumed target strengths of these fishes, so that biomass estimates can span three orders of magnitude (Davison et al., 2015).Net-based biomass estimates are subject to high uncertainty as well, in part because of unknown capture efficiency (i.e., the proportion of fish in the area swept or volume filtered that are caught and retained in nets) that can cause biomass estimates to vary by up to an order of magnitude (Davison et al., 2015).Two, scaling up this fish carbon flux model to the ecosystem level required information on the size structure and species composition of the fish community.The results presented here are based on a model that assumes a 1 g fish (though we also consider a 10 g migrating and non-migrating fish, Figs.D.1 and D.2).However, respiration rates per gram of fish biomass vary by the individual size of the fish due to allometry, so the size structure of the fish community would need to be incorporated in calculations of community-level fish carbon flux.
We caution against using these results to make inferences at the ecosystem level about carbon sequestration because this model only estimates carbon flux, and flux is not interchangeable with sequestration.Sequestration times are not quantitatively addressed here, but inorganic carbon exported to 150-200 m (e.g., from respired carbon dioxide) has mean sequestration times of 30-50 years (Boyd et al., 2019).In some cases physical mixing can extend deeper than 500 m within a single year (de Boyer Montégut et al., 2004), which could bring up carbon transported below 150 m to the sea surface on relatively short time scales.Biogeochemical models and policy decisions relating to mesopelagic fish resources ought to consider the high uncertainty in current estimates of carbon flux and fish biomass, while also understanding that carbon flux is not interchangeable with carbon sequestration on climate relevant time scales.
To contextualize the implications of uncertainty in fish carbon flux in a back-of-the-envelope calculation, we compare estimates of fishmediated carbon flux at the ecosystem level in one location to estimates of total flux (active plus passive flux) in the same location.Based loosely on recently published data from the Scotia Sea, we assume here that passive flux (such as measured by sediment traps) is 0.6 to 3.2 mg C m − 2 d -1 in the autumn (Manno et al., 2015), migrating fish density is 500-5200 mg wet mass m − 2 , and all fish weigh 1 g (Belcher et al., 2019).Under these assumptions, migrating fish carbon flux in this region could be anywhere between 0.37 % and 82 % of total carbon flux (90th inner quantile derived from Monte Carlo analysis of per capita carbon flux rates, Table E.1).At other times of the year when particulate organic carbon flux for this region in the winter is 7.1 to 13.1 mg C m − 2 d -1 (Manno et al., 2015), fish carbon flux would be 0.09 to 28 % of total carbon flux on an annual basis.These simple calculations do not consider the uncertainty in the relative contribution of zooplanktonmediated active carbon transport to total carbon transport, which is also poorly constrained but could contribute a mean estimate of 15-20 % of total carbon flux (Archibald et al., 2019).This thought experiment also does not consider seasonal variation in fish movement, metabolism or biomass, but illustrates the potential variation of the contribution of fish to the biological carbon pump in various environmental contexts.The implication of these compounding uncertainties is that fish are either an enormous or a relatively unimportant component of total biological carbon export out of the epipelagic zone.
Our sensitivity results align to some extent with previous findings in terms of which parameters are most influential and which fish category contributes most to carbon flux, though our methods were distinct.Migrating fish almost certainly contribute more to carbon flux than nonmigrators on a per capita basis (Davison et al., 2013).However, if nonmigrating fish were much more abundant than migrators, this would not be true on an ecosystem level.There is also structural uncertainty in our model that we did not consider; we assume non-migrating fish increase carbon export by consuming zooplankton that would otherwise migrate back to sea surface, but they may also increase remineralization of carbon (and thus decrease further export) by consuming particulate organic carbon and releasing dissolved inorganic carbon that does not sink (Sarmiento-Lezcano et al., 2022).Components of our model that are distinct from others include an updated respiration rate regression model (Ikeda 2016) and comprehensive consideration of all fish carbon flux pathways (Saba et al., 2021), including the specific dynamic action carbon flux.Yet, one fundamental finding that aligns with previous work is that carbon flux is highly sensitive to respiration rate parameters (Davison et al., 2013).The non-migrating fish model was also sensitive to Z, the proportion of prey that is non-detrital, and thus the fraction of carbon passing through non-migrating fish that we count toward active transport.In a food web model that used a different, bottom-up modeling approach (Anderson et al., 2019), the extent to which zooplankton were detritivores was an important parameter in estimates of fish-mediated carbon flux.Fish longevity was also an important parameter (Anderson et al., 2019), which we incorporated into our model via the mortality terms, but we found mortality parameters to be only moderately influential.This was likely because we do not assume all mortality results in carbon export past the flux boundary.We estimated mortality based on a regression (Pauly 1980) using asymptotic fish length, growth rate and water temperature.Longevity and growth rates are not well known for most mesopelagic fish, with most age studies done on myctophids and very few on bristlemouths or other abundant taxa (Caiger et al., 2021).Future models of fish carbon flux could incorporate a more sophisticated treatment of fish longevity or age-structured populations, and a more detailed model of fish carbon flux by fish taxon in the population being modeled (e.g., Woodstock et al., 2022).Here, due to lack of data, we do not track growth of individual fish through time; rather, we use a single growth rate that is an average from the literature of either migrating or non-migrating mesopelagic fishes (Childress et al., 1980).
We constrained this model based on the assumption that vertically migrating mesopelagic fishes forage near the sea surface.This is based on the dominant hypothesis about vertical migration behavior, which is that migrators expend energy to migrate to the surface for the reward of being able to forage on more abundant, epipelagic prey (Hays 2003).However migrating mesopelagic fish likely consume prey that they encounter in the mesopelagic zone (Moku et al. 2000;Sobradillo et al. 2022;Kaartvedt et al. 2023).If fish were found to meet a high proportion of their daily energy needs by consuming non-migrating, mesopelagic zooplankton, then the downward carbon transport would be less than that estimated in this study.Therefore, if the model were applied in a specific study site where substantial daytime feeding were observed, then the model could be modified to allow for daytime, mesopelagic feeding by vertically migrating fishes.
Standardizing how to quantify zooplankton and fish-mediated carbon flux would allow future research on the topic to advance more efficiently and facilitate synthesis of existing findings for science communication.Some studies on fish-mediated carbon flux only consider respiratory flux (Ariza et al., 2015;Belcher et al., 2019Belcher et al., , 2020;;Hernández-León et al., 2019a), while others only consider fecal flux (Bray et al., 1981;Staresinic et al., 1983;Angel 1985).Most studies exclude excretion and specific dynamic action fluxes.Furthermore, when authors refer to the relative contribution of fish to carbon flux, sometimes this is the percentage of fish flux divided by total active plus total passive flux.In other cases, this is the percentage of fish flux divided by passive flux only (as presented in T able 1 of Saba et al., 2021).Variation in definitions for terms like "relative contribution" and even "biological carbon pump" makes inter-comparison between studies difficult.Flux boundary depths and net capture efficiency values also vary in the literature, leading to incomparable carbon transport quantities and expected storage times across studies (Boyd et al., 2019).Finally, some studies consider vertically migrating fish only, while others consider both vertically migrating and non-migrating fish in calculations of mesopelagic fish-mediated carbon flux calculations.Standardization or making all components of the analysis openly available and reproducible could aid in inter-comparison of results among studies.
This study provides an open-source model of fish-mediated carbon flux that quantifies uncertainty in per capita fish carbon transport rates.We highlight parameters to prioritize for future research to reduce uncertainty related to this understudied component of the biological carbon pump, while realistically characterizing challenges to constraining fish carbon flux estimates.These caveats of uncertainty, relatively low storage times, and counterfactuals should be considered in claims related to mesopelagic fish-mediated carbon transport, as well as in decision-making regarding nature-based climate solutions or harvest of mesopelagic fish resources.The current state of knowledge is simply not sufficient to make precise estimates of carbon transported by these animals at an individual scale, much less at an ecosystem level.As this field is relatively new, our sensitivity analyses reveal which parameters contribute most to fish-mediated carbon flux uncertainty at the scale of the individual fish, which can help guide future empirical work to tackle this uncertainty incrementally.(Ikeda 2016) showing the effect of fish body mass as a predictor (effects of temperature and habitat depth not shown).The data used to estimate the regression model coefficients were modified by including only data from mesopelagic fish, setting the habitat depth predictor to be categorical (for either vertically migrating or non-migrating fish), and centering the mass and temperature predictors around 1 g and 15 • C to reduce covariance between estimated regression coefficients.The estimated coefficient values (and their associated uncertainty ranges, Table B.1) were used in subsequent sensitivity analyses.

Appendix B. Bioenergetics equations, parameterization, and data used to inform carbon flux model
Using a modified regression that is specific to mesopelagic fishes but based on a literature synthesis of fish respiration rates more generally, we calculate respiration rates using where R O is the respiration rate in units of μL oxygen utilization per hour, i indicates fish category (vertically migrating or non-migrating), j indicates behavior, α 0 is the intercept of the respiration rate regression, and f(W), g(T i,j ), and h(H i ) are functions for fish wet mass (mg), temperature ( • C), and fish category, respectively.Vertically migrating fish have three possible behaviors in the model (foraging, migrating or resting), while non-migrating fish have two possible behaviors (foraging or resting).Temperature values used in g(T i,j ) vary depending on fish category and type of activity.Temperatures used here are theoretical and generic (roughly representing offshore, temperate regions), but study site-specific temperatures can be used for region-specific studies.For vertically migrating fish, we use a theoretical mean temperature over the water column (0-500 m) to calculate R O when migrating, mean epipelagic (0-150 m) water temperature when foraging, and mean mesopelagic temperature (150-500 m) when resting at depth.For non-migrating fish, we use the mean mesopelagic temperature while resting and foraging.For the purposes of this analysis, we assume a maximum migration depth of 500 m, as this is a relatively common daytime deep scattering depth in many regions of the global ocean.We then allow maximum migration depth to vary in our sensitivity analysis as it does in nature (Klevjer et al., 2016).Later (in Eq. ( 3)), different activity scaling factors are used for each behavior category to calculate routine metabolic rates from respiration rates.
More specifically, using the appropriate fish mass and temperatures as described above (modified from the Ikeda 2016 regression), the functions are used with Eq. (B.1) to calculate respiration rates, where α 1 is the effect of fish mass on respiration rate, α 2 is the effect of water temperature on respiration rate, α 3 is the effect of fish category on respiration rate, and H is a binary, indicator variable indicating whether the fish is vertically migrating (H = 1) or non-migrating (H = 0).The data used to create the modified respiration rate regression for mesopelagic fish only are available with all other code on Github (https://github.com/hmcmonagle/Uncertainty-in-fish-mediated-carbon-transport).
We then convert respiration rate from units of oxygen utilization to units of energy utilization and find daily rates of energy expenditure across all daily activity states (kJ/d).We multiply the metabolic rates for each daily activity by the proportion of a day spent in each of those activities, and then sum to find total energy expenditure per day throughout a cycle of foraging, migrating and resting.This is only modified slightly for non-migrators, where we set the proportion of a day spent vertically migrating equal to zero.This can be found using where M is the total, daily energy expenditure in units of kJ/d due to metabolism (maintenance of basic cell function and locomotion), i indicates fish category, and j indicates behavior, in that j equals 1 for foraging, 2 for resting or 3 for migrating.Q ox is the oxycalorific coefficient in units of kJ/mg O 2 (Nelson, 2016;Brett & Groves 1979), P i,j is the proportion of each day spent in a given behavior, and A i,j is the activity scaling factor to adjust routine metabolic rates by differential activity levels during foraging, resting or migrating.Activity scaling factors are largely unknown for mesopelagic fishes, so they are based on scaling factors for other fishes (Brett & Groves, 1979) and allowed to vary widely at, above or below the routine metabolic rate (Table B.1).These activity scaling factors could be modified as appropriate based on empirical observations of fish activity levels at various depths.
The constants convert from an hourly to daily rate, from micrograms of oxygen to grams of oxygen, and from microliters of oxygen (Ikeda 2016) to micrograms of oxygen (Rounds, 2011).We convert into mass units of oxygen because volumes of respired oxygen would be affected by depth and thus pressure.The effect of pressure on respiration rate may not be accounted for in the respiration rate regression (Ikeda 2016) due to limited measurements to date of mesopelagic fish respiration at pressures greater than 1 atmosphere (the approximate pressure of measurements taken in shipboard laboratories).The activity scaling factors adjust the previously calculated respiration rates to appropriate levels based on in situ activity level during various behaviors.Thus, the routine metabolic rates throughout the day represent realistic rates based on activity that are between the maximum metabolic rate (active metabolic rate) and the minimum metabolic rate (standard metabolic rate).Unless a fish is starved, some measured oxygen utilization can be attributed to metabolism related to specific dynamic action, so we include θ as a correction factor.Mesopelagic fish are challenging to collect and maintain in a laboratory setting; therefore, oxygen consumption rates used to build the respiration rate regression model (Ikeda 2016) were often measured within about an hour of capture of each fish (Donnelly & Torres 1988), rather than on starved fish.The best practice in respirometry is that measurements are conducted on fish after they are fasted in the laboratory to remove the uncertainty in how much oxygen utilization is attributable to specific dynamic action, but this is often not possible for mesopelagic fish.The implication of this is that some of the fish were likely to have still been digesting food when the measurements were taken.If this is the case, then oxygen consumption could be attributed both to metabolism and to specific dynamic action.We allow θ to vary in our sensitivity analysis from 0.54 to 1 to allow for the possibility that both metabolism and specific dynamic action were contributing to oxygen utilization in the data we used (Ikeda 2016).The minimum correction factor of 0.54 is the fraction of the daily energy spent on non-digestion related metabolism divided by the fraction spent on digestion (specific dynamic action) in a general fish energy budget (Brett & Groves 1979).
We assume that the respiration rate data used in the Ikeda 2016 regression were measured when the fish was at an activity state between resting and actively swimming routine metabolic rates.In the sensitivity analysis, we allow the activity scaling factor during the resting behavior to vary between 50 % and 100 % of the metabolic rate measured in Ikeda 2016 (Table B.1).This allows for a potential decrease in the routine metabolic rate during periods of torpor.A resting phase in daily activity cycles has been observed by submersible observation during the day for some migrating fish (Backus et al., 1968) and by direct respirometry measurements of non-migrating fish that also exhibited a circadian rhythm with lower resting, nocturnal respiration rates (Smith & Laver 1981).We allow the activity scaling factor for foraging and migrating behaviors to vary between 100 % and 400 % of the metabolic rate measured in Ikeda 2016 (Table B.1).This increases the routine metabolic rate, as predicted by the regression, during periods of elevated activity while foraging or migrating.
The proportions of a day spent foraging (P f ), migrating (P m ) or resting (P r ), used to adjust the energy expenditures by activity in Eq. (B.3), were calculated as P r = 0.5 (B4a) where P r is the fraction of a day spent resting, P m is the fraction of a day spent migrating (which by definition is zero for non-migrating fish), P f is the fraction of a day spent foraging, D max (m) is the maximum depth of migration, D min (m) is the minimum depth of migration, and S is the average migration swimming speed of vertically migrating fishes across studies (m s − 1 ).Swimming speeds of vertical migrations are often approximately 0.05 m s − 1 (Davison et al., 2013), although these fish swimming speeds can vary by several times higher or lower (Christiansen et al., 2021;Kaartvedt et al., 2022).We convert units from seconds to hours and then divide by 24 to find the fraction of a day spent on a one-way migration.The additional factor of two (Eq.( 4b)) accounts for both upward and downward migrations each day.
We assume that for both vertically migrating and non-migrating fishes, about one half of the day is spent resting (P r ), although we allow this to vary in the sensitivity analysis.For studies on seasonal differences in fish-mediated carbon flux, these P r nominal values could be adjusted, but seasonal variability in carbon flux is beyond the scope of this paper.Though the time spent resting at their daytime depth will be greater for vertically migrating fishes in the summer when daylight hours are greater (at non-zero latitudes), we assume this will be cancelled out by lower time spent resting in winter when daylight hours are less.Lastly, P f is the fraction of a day spent foraging, which is based on the remaining fraction of a day after migrating and resting.Since time spent migrating for non-migrating fishes is zero, the nominal value of P f is equal to the nominal value of P r for non-migrating fishes.
The assumption that all mesopelagic fishes spend half the day resting is allowed to vary in the sensitivity analysis to follow.
Energy budget accounting is based on the overarching bioenergetic equation G = I -(SDA + F + X + M).In this equation, M was found previously (Eq.(B.3)) based on mesopelagic fish respirometry data (Ikeda 2016).The energy spent on growth was previously estimated for four migrating and four non-migrating mesopelagic fish (Childress et al., 1980).We averaged these values and converted to units of kJ/d to obtain an estimated daily energy expenditure on growth for migrating and non-migrating mesopelagic fish (G VM and G NM in Table B.1).Given the paucity of data for the energy mesopelagic fish devote to growth, and the values originated from fish of various ages, we assume growth scales linearly with biomass.We allow these values to vary by 50 % in the sensitivity analysis.
We solved for ingestion rate as where I, G and M (kJ/d) represent energy ingested or spent on growth and metabolism not attributed to specific dynamic action, respectively, and ϕ SDA , ϕ F and ϕ X are the approximate fractions of total ingestion lost to specific dynamic action, fecal waste, and excretion (Table B energy spent on both somatic and reproductive growth (grouped due to a lack of empirical information for these different energetic demands in mesopelagic fish).As a check, we independently calculated ingestion rates by dividing consumption-to-biomass (QB) ratios for mesopelagic fish (Palomares & Pauly 1989;Koehn et al., 2016) by the energy density of mesopelagic fish prey (Q z , Table B.1).Both methods produced comparable ingestion rates, so we based our sensitivity analysis on Eq. (B.5).
A literature review guided the selection of nominal, minimum and maximum values for each parameter (Table B.1).Where a range of values was not available in the literature, the parameter value was allowed to vary by 10 %, 20 %, 50 % or 80 % depending on data quality to reflect the level of uncertainty for these parameters (Table B.1).We note that some parameter ranges were selected based on values from the literature for epipelagic fish due to a lack of published data for mesopelagic fish specifically (e.g., Φ SDA , Φ F , Φ X , θ, A f , A m , A r , R Q , Q ox , e p , X r and C p ).For the same reason of limited data availability, other parameters were based on zooplankton literature (e.g., Z, E F ), while others could only be estimated roughly based on supporting information in the fish literature (e.g., E μ ).In each of these cases, parameters were allowed to vary substantially to reflect these uncertainties.
The component of this general bioenergetics equation with the most data available from previous studies for mesopelagic fish specifically was the respiration rate.A recent review of fish respiration rates (Ikeda 2016) included 38 mesopelagic fish species and fit a regression model of respiration rate based on in situ temperature, body mass of the fish, and the habitat depth.We modified this regression model by fitting the regression to data for mesopelagic fish only (defined as fish that inhabit depths between 150 and 1000 m for at least part of the day).We also modified the habitat depth predictor from a continuous variable of minimum habitat depth to a categorical variable based on whether the fish was vertically migrating or nonmigrating.

Table B.1
Fish-mediated carbon flux parameters.The variables used in each equation are given with their mean (nominal) value, range, units, and citation where applicable.Blank mean values indicate that the value was calculated based on other input parameters, blank ranges indicate that the value entered is a constant and not allowed to vary in sensitivity analyses, and blank units indicate that the parameter is unitless.Parameters apply to both vertically migrating (VM) fish and non-migrating (NM) fish unless otherwise noted.The abbreviation W is used for wet weight, DW for wet weight, and SDA for specific dynamic action.Sutton and Hopkins 1996;Ochoa et al., 2013;Olson et al., 2014;O'Dwyer et al., 2015;Watanuki and Thiebot 2018;Stewart et al., 2018;Goetsch et al., 2018;Braun et al., 2019;McBride et al., 2022 where 24P m finds the time spent migrating, the fraction that modifies it finds only the proportion of total migration time at which the fish is above the flux boundary, and this is divided in half to include only the downward migration not also the earlier upward migration.For later meals, the integration period must be adjusted to account for the amount of time after the meal that the individual is above the flux boundary, which equals: The fraction of specific dynamic action occurring below the flux boundary equals 1 -f(t meal ).Because there are multiple meals throughout the foraging period, we must calculate the product of f (t meal ) for each meal, and the proportion of total feeding that occurs at each time t meal , g(t meal ).For simplicity, we assume that feeding occurs continuously between time 0 and 24P f , and follows a uniform probability function g(t meal ).Under those assumptions, proportion of total specific dynamic action that occurs when individuals are below the flux boundary (E SDA , the fraction exported) equals: Given that g(t meal ) is constant for all t meal , and its integral over 0 to P f equals 1, this simplifies to: Here we assumed that p(t) could be represented by a gamma probability density function.We re-parameterized this function with respect to the mean, or peak, of the distribution (t peak ) and the time elapsed until 90 % of the specific dynamic action from a meal had occurred (t 90 ).In the sensitivity analysis, this allowed for simulation of alternative parameterization that had intuitive, biological meanings.We approximated the integral using Simpsons rule to find E SDA for vertically migrating fish.

Fig. 1 .
Fig. 1.Conceptual model of fish-mediated carbon transport.Black arrows indicate metabolic and ecological processes used to calculate carbon transport.Red and grey arrows represent vertical movement patterns.Boxes indicate carbon stores.Parameter descriptions are provided in TableB.1.Excretion and specific dynamic action fluxes are not shown for simplicity.Ingestion at mesopelagic depth is not represented for vertical migrators in this model.Carbon passing through non-migrating fish that originated from zooplankton of non-detrital origin is the only portion of non-migrating fish-mediated carbon transport that contributes toward total fish-mediated carbon transport.Zooplankton are implicitly represented in the model in ingestion rates but shown here for clarity.

Fig. 2 .
Fig. 2. Flow chart of model calculations and unit conversions used to estimate fish-mediated carbon transport.Red boxes indicate inputs into the model.Blue boxes represent components of the model.Purple boxes indicate model outputs in terms of carbon transport pathways.

Fig. 5 .
Fig. 5. Vertically migrating fish results of the Monte Carlo simulation (n = 5000).Only parameters that vary in the sensitivity analysis are plotted.Parameter values are converted to z-scores on the x-axis.The resulting carbon flux estimate for each randomly assigned vector of parameter values is on the y-axis.Parameter symbol descriptions and units are provided in Table B.1.Parameters are grouped and color-coded according to their use for calculating carbon flux.The inner 80th percentile is shaded.

Fig. 6 .
Fig.6.Non-migrating fish results of the Monte Carlo simulation (n = 5000).Only parameters that are allowed to vary in the sensitivity analysis are plotted.Parameter values are converted to z-scores on the x-axis.The resulting carbon flux estimate for each randomly assigned vector of parameter values is on the y-axis.Parameter symbol descriptions and units are provided in TableB.1.Parameters are grouped and color-coded according to their use for calculating carbon flux.The inner 80th percentile is shaded.

Fig. 7 .
Fig. 7. Total carbon flux associated with vertically migrating and nonmigrating fishes in milligrams of carbon per 1 g individual per year.Estimates were generated from Monte Carlo simulations (n = 1000) and plotted with a kernel density estimator.Dashed lines show the inner 90th percentile range for each distribution (5th to 95th percentile).

Fig. 8 .
Fig. 8.Total carbon flux associated with each fish carbon flux pathway, by percent of total, for vertically migrating and non-migrating fishes.Ranges within each carbon pathway are based on Monte Carlo simulations (n = 1000).

Fig. A1 .
Fig. A1.Regression model fit to previously published fish respiration rate data(Ikeda 2016) showing the effect of fish body mass as a predictor (effects of temperature and habitat depth not shown).The data used to estimate the regression model coefficients were modified by including only data from mesopelagic fish, setting the habitat depth predictor to be categorical (for either vertically migrating or non-migrating fish), and centering the mass and temperature predictors around 1 g and 15 • C to reduce covariance between estimated regression coefficients.The estimated coefficient values (and their associated uncertainty ranges, TableB.1)were used in subsequent sensitivity analyses.

Fig. D2 .
Fig. D2.Individual parameter perturbation plots for a 10 g non-migrating fish, showing the full allowed range for each parameter value on the x-axis and carbon flux on the y-axis.A greater slope indicates higher sensitivity of the carbon flux estimate to the parameter.The x-axis units corresponding to each parameter can be found using Table B.1.Parameters are grouped and color-coded by carbon flux pathway.Appendix E. Summary statistics from figures in tabular format Table E.1 Summary statistics (inner 90 % percentile range of model estimates) of fish carbon flux (mg C per individual per day) from the Monte Carlo simulation for a 1 g fish.These values correspond to those in Fig. 7.

Table E
.3.Parameters are grouped and color-coded by carbon flux pathway.H. McMonagle et al.

Table B
Percent of fish-mediated carbon flux by bioenergetic pathway.We assume carbon is transported either through respiration of carbon dioxide that is not associated with specific dynamic action, respiration that is associated with specific dynamic action, egestion of particulate organic carbon, excretion of inorganic carbon, or mortality via predation, senescence or disease below the flux boundary.Calculations are based on nominal parameter values (TableB.1),andvaluesbelow are shown as percentages of total carbon transported by each category of fish (vertical migrator or non-migrator).These values correspond to Fig.8.Percent change in vertically migrating fish-mediated carbon flux (mg C per individual per day) from the minimum to the maximum parameter value, based on individual parameter perturbation.Minimum and maximum parameter values are based on the literature where published ranges were available.
(continued on next page) H.McMonagle et al.

Table E
Percent change in non-migrating fish-mediated carbon flux (mg C per individual per day) from the minimum to the maximum parameter value, based on individual parameter perturbation.Minimum and maximum parameter values are based on the literature where published ranges were available.