The MVF model, a first-order advection-reaction partial differential equation, was first presented by for use in epidemiology, but was later more formally derived for use in the study of cellular systems by . Since it provides a natural framework for representing aspects of size dependency and fish life history, and generates biomass spectra that resemble those found in the field , the MVF model has seen a wide variety of applications in marine ecosystems and fisheries. Ecosystem models that have applied the MVF approach to large-scale fisheries studies generally make use of the classical size-structured equation, but differ in the formulations used to calculate growth, mortality, and reproduction, and differ in the structural organisation of fish groups.

Although the MVF model can be expressed by a variety of variables, it is usually presented in terms of the number of fish (the abundance) that evolve in time as a function of the fish age. As an alternative to age, size (measured as length or mass) is also used as an organizing variable, since it can be more descriptive than age for certain applications. Since fish growth , natural mortality , and harvest are generally size-dependent, we employ size in lieu of age. Moreover, we describe size in terms of mass as opposed to length, although there is a strong relationship between fish mass and length .

The MVF model uses a spectral framework to describe fish populations; that is, it describes the biomass of fish of mass m at time t by a continuous spectrum f(m,t) such that the fish biomass in the mass interval [m,m+dm] is f(m,t) dm. Although abundance is typically used in applications of the MVF model, and has been used in marine ecosystem applications, see for example , or , we use biomass to compare our results more directly with the harvest data that we use to evaluate BOATS. Regardless, since the abundance n and biomass f spectra are related by f(m,t)=n(m,t)m, in the continuous case, using one form over the other does not influence the model dynamics. We note that, in the numerical implementation of the model, there will be a small difference between the two since we use the geometric mean to represent a discretized range of masses (Sect. ). Hence, as fish grow they jump from one geometric mean to next, which may result in an accumulation of biomass.

Fish biomass evolves in time as ∂∂tfk(m,t)=-∂∂mγS,k(m,t)fk(m,t)︸1+γS,k(m,t)fk(m,t)m︸2-Λk(m,t)fk(m,t)︸3fk(m,t=0)=fk,m,0fk(m0,t)γS,k(m0,t)=Rk(m0,t), where fk(m,t) is the biomass spectrum in grams of wet fish biomass (gwB) per square metre of ocean surface per unit of the mass class (gwBm-2g-1), for an individual fish of mass m, at time t, belonging to group k. In Appendix , we derive the biomass form of the MVF model used in Eq. (). From the definition of the biomass spectrum above, we have that the cumulative biomass at time t of individuals of mass ranging from 0 to m is the integral Fk(m,t)=∫0mfk(m′,t)dm′. In this paper, spectral variables such as the biomass spectra fk(m,t) are written in lower case, whereas cumulative variables that are integrated over size are written in upper case.

Fish biomass is controlled by growth, mortality, reproduction, and recruitment (note that we present harvest in the companion paper, ). Term 1 on the right hand side of Eq. () represents the somatic growth in fish biomass that occurs at a rate γS,k(m,t) (gs-1). This term results from fish growing from one interval of mass, which in the discrete case is called a mass class, into the adjacent mass class (for example from a class of 1 to 2 kg to a class of 2 to 3 kg). Since the MVF model is founded on the conservation of numbers of fish (Appendix ), term 2 represents the biomass accumulation that occurs from fish growing in size. Term 3 of Eq. () represents the natural mortality Λk(m)fk(m,t) (gwBm-2g-1s-1), or all non-harvesting sources of fish mortality, which includes losses to predation as well as non-predation losses such as parasitism and disease, senescence, and starvation . Although we do not consider harvest mortality in this paper, in the full BOATS model (described by , in review) it is represented by another loss term on the right hand side of equation Eq. (). The growth rate γS,k(m,t) (Eq. ) and the mortality rate Λk(m) (Eq. ) depend on both mass and temperature.

Since the time evolution equation of the MVF model is a first-order partial differential equation, we specify an initial condition (Eq. ) and a boundary condition (Eq. ). The initial condition, or the fish biomass spectrum at the starting time fk,m,0, is discussed in Sect. . The boundary condition, which is defined at the lower mass boundary m0, determines the flux of biomass that is added to the biomass spectrum at the initial size class, and depends on the energy allocated to reproductive biomass, the recruitment, and the NPP. This term is detailed in Sect.  and summarized in Eq. (). A schematic of the ecological module of BOATS, with the main model components and processes, is presented in Fig. .

Organismal body temperature is a fundamental driver of physiological processes since it strongly controls rates of metabolic activity and therefore strongly influences growth, mortality, and reproduction rates . To model temperature dependence, which we represent by the function a(T), we apply the van't Hoff–Arrhenius equation a(T)=exp⁡ωakB1Tr-1T, where Tr (K) is the reference temperature of the process in question (growth or mortality, for example), kB (eVK-1) the Boltzmann constant, and ωa (eV) the activation energy of metabolism. Although there is at present no mechanistic derivation of the relationship between metabolic rate and temperature at the level of an entire organism, we interpret the exponential temperature dependence of Eq. () as an empirical parameterization of this complex relationship with strong observational constraints .

For all temperature-dependent rates, we use the average water temperature from the upper 75 m of the water column , since it is representative of an average mixed layer depth and so identifies the average temperature at which photosynthesis takes place , and since it is representative of the depths at which many fish live and are harvested . We further assume that fish adopt exactly the water temperature. Given that the greater majority of marine organisms are ectotherms, we feel that this is a reasonable assumption. Taking the average of the upper 75 m of the water column could create biases in regions with strong vertical temperature gradients, since different components of the ecosystem could live at substantially different temperatures, or in regions that are dominated by bottom dwellers in regions deeper than 75 m. However, given that many commercial fish spend significant time near the surface, but actively travel throughout the water column, we feel that this depth is an appropriate first approximation of the average temperature felt by the community. Note that the temperature we apply is generally not accurate for mesopelagic ecosystems, which could make up a large part of marine biomass , but since the majority of these ecosystems have not been commercially exploited, they are not included in our modelled community.

Fish growth rates are key mass-dependent quantities that characterize each fish group and are limited by the energy available to consumers, and, ultimately, by the photosynthetic primary production. We assume that there is a constant energetic content of biomass , and so treat biomass and energy as equivalents. We envision that energy is supplied to a fish of mass m by the transfer of biomass through the food web by means of predation. Following macroecological theory, this complex process is parameterized by assuming that a fraction of the energy from NPP is transferred up through the food web to become fish biomass production, depending on the average trophic efficiency, the average predator to prey mass ratio, and the phytoplankton size (Eq. ). Individual fish then allocate this energy input to either somatic growth (that is, the formation of additional biomass, which we from here on refer to simply as growth γS,k(m,t), gs-1) or to formation of reproductive biomass γR,k(m,t) (gs-1), and so we have that ξI,k=γS,k+γR,k, where ξI,k(m,t) is the input of energy to a fish at time t in group k. We rearrange to write the growth rate as γS,k=ξI,k-γR,k.

It is important to recognize that the individual fish growth rate cannot exceed a biologically determined maximum rate, no matter how much food is available. This aspect of fish growth is based on empirical observations and allometric arguments, and founded on the work of and expanded upon by many others including , , and . To take this growth rate limitation into account, we assume that the realized input energy ξI,k(m,t) cannot exceed that supplied by NPP through the trophic scaling, or that determined by empirical growth limits, and so have that the energy input is ξI,k=min⁡[ξP,k,ξVB,k], where ξP,k(m,t) is the energy input to fish from NPP as transferred through the food web, and ξVB,k(m,t) is that input from a purely empirical allometric framework following . Essentially, ξVB,k(m,t) describes the maximum growth rate of fish in the case that food is extremely abundant.

We define ϕΠΨ,C as the fraction of NPP that is potentially available to the sum of all commercial fish groups. In the present work, we assume that ϕΠΨ,C is equal to 1, and therefore omit it from the equations. This simplifying assumption implies that the entire global ecosystem of animals larger than 10g would have consisted of potentially commercial species prior to fish harvesting (including bycatch). Obviously this is incorrect, in that the existence of non-commercial animals larger than 10g requires that ϕΠΨ,C<1 in the real world. However, given the weak observational constraints on biomasses of non-commercial animal species at the large scale, and the fact that the species composition of all marine ecosystems has been heavily altered by human activity, it is very difficult to estimate the true value of ϕΠΨ,C. Despite this difficulty, sensitivity tests revealed that biomass and harvest in the model are approximately linear with ϕΠΨ,C (not shown). Since we constrain the parameters in BOATS by comparing linear correlations of modelled and observed harvest (see Sect. , Table , and ), and given the linearity of modelled harvest vs. ϕΠΨ,C, the value of ϕΠΨ,C would have a negligible effect on the spatial correlation criterion used for the optimized parameter choices. Nevertheless, it should be pointed out that using alternate values of ϕΠΨ,C would change the predicted biomass and harvest, all else being equal.

We further assume that each of the three fish groups has access to an equal fraction of the available primary production, ϕΠΨ,C/3. By assuming that constant portions of the available photosynthetic energy are available to each of the commercial fish groups, all groups are assured to coexist stably. Ecologically, our assumption implies equal resource partitioning to each group, both when they are at the larval stage (through recruitment) and as juveniles and adults (through growth) . This can be thought of as reflecting a separate ecological niche for each group that remains stable over time, and implies that excess NPP, which would result from growth-rate limitation of one group, is not available to other potentially commercial groups, but rather supplied to non-commercial species. Non-commercial species could include, among others, unharvested mesopelagic fish, planktonic invertebrates such as cnidarians, and benthic invertebrates such as amphipods and nematodes. Although this and the previous assumption are not strictly accurate representations of the marine ecosystem, we feel that they are commensurate with the simple three-group representation of the ecosystem and the scarcity of appropriate data constraints, and could be improved in future work.

Each individual fish receives an equal part of the fish production that is input to its mass class, which we here identify as an infinitesimal mass interval of width dm. Where ϕC,k is the fraction of ϕΠΨ,C that is available to group k, and π(m,t) the fish production distribution, the individual fish production is therefore the fish production in the mass interval ϕC,kπ(m,t)dm divided by the number of individuals in the mass class nk(m,t)dm (Eq. ). Since the abundance spectrum nk(m,t) is equal by definition to fk(m,t)/m, the primary-production-based input of energy to each individual fish is ξP,k=ϕC,kπdmnkdm=ϕC,kπmfk.

Since we assume that the NPP that is transferred up through the trophic web is uniformly input to all individuals in a given mass class, if the biomass in a mass class falls due to a removal (such as harvesting, for example) then this is equivalent to a decrease in the number of individuals in that mass class. This implies that more fish production would input to each individual, and so in such a scenario ξP,k would increase. This input of energy depends on the biomass (also referred to as density dependence) and the fish production. The fish production term depends on temperature through the representative mass of phytoplankton mψ(t) (Eq. ), which is a function of the temperature-dependent large fraction of phytoplankton ΦL(t) .

In conditions that are not limited by food availability, the standard von Bertalanffy (somatic) growth rate equation is γVB,k=Amb-kam-krm, where the Amb term represents the energy input from food intake after assimilation and standard metabolism, and kam and krm represent the energy used in activity and reproduction, respectively . The allometric growth rate (not to be confused with the somatic growth rate γS,k), which we write as A=A0aA(T), is the growth constant A0 (g1-b s-1) modulated by the van't Hoff–Arrhenius temperature dependence for growth aA(T) (Eq. ).

The energy input we wish to resolve is that for both growth and reproduction, and so we add the reproduction term krm to both sides of Eq. () to find the energy input to be ξVB,k=Amb-kam.

Although the interpretation of the terms in Eq. () do not exactly correspond to von Bertalanffy's original interpretation of a balance between anabolic growth and catabolic decay, we refer to this equation as the von Bertalanffy energy input ξVB,k. We consider different values of the activation energy of metabolism for growth ωa,A and mortality ωa,λ (Eq. ), which result in different temperature dependence curves aA(T) and aλ(T). The parameter b (unitless) is the allometric scaling constant, and ka (s-1) is the mass specific investment in activity. We follow and define a new constant ϵa=ka/(ka+kr), which when combined with the idea that there is zero growth at the asymptotic mass m∞,k , allows us to express the mass specific investment in activity as ka=Aϵam∞,kb-1. At each group's asymptotic mass, we therefore have that ξVB,k(m∞,k)=A(1-ϵa)m∞,kb.

Equation () for the input of energy to growth and reproduction is therefore ξI,k=min⁡ϕC,kπmfk,Amb-kam, the minimum of a term that depends on biomass and one that does not. Applying the definition of the fish production spectra that we introduce in the next section (Eq. ), we have a change in growth regime when fk is such that fk<ϕC,kΠψmψmτAmb-kam.

When biomass is low enough that this equation holds, NPP no longer influences the input energy, fish will grow at their maximum physiological rate, and any unused energy available to fish production is assumed to be transferred to unresolved parts of the ecosystem. For low productivity systems, the model could overestimate biomass since a larger fraction of primary production will be transferred to commercial species. However, in high productivity systems, the allometric limit is more likely to set growth rates; therefore a larger fraction will be transferred to the non-commercial groups. That said, the potential for, and the magnitude of, such a feature will depend on the particular values of the growth rates at the site in question (Eq. ).

We assume that the energy allocated to reproduction γR,k(m,t) is proportional to the total input energy ξI,k(m,t) such that γR,k=ΦkξI,k, where Φk(m) is the mass-dependent fraction of input energy that is allocated to reproduction. From Eq. (), we write the growth rate as γS,k=1-ΦkξI,k.

We now derive an expression for Φk(m). Following , we assume that the allocation to reproduction is proportional to mass , and that it also scales with a size-dependent rate sk(m) that defines the size-structure of the transition to maturity (Eq. ). This gives us γR,k=krmaxskm, where krmax is a normalizing constant. Combined with Eq. (), we have that γR,k=Φkξ‾I,k=krmaxskm, where ξ‾I,k is a representative input energy that we employ to guarantee that the allocation to reproduction does not change with biomass. For the representative input energy, we take the maximum possible value; that is, the von Bertalanffy input energy described in Eq. (), and so have that ξ‾I,k=ξVB,k. We therefore determine Φk(m) for the energy input regime that is not limited by fish production, and find that Φk=krmaxskmξVB,k.

We determine krmax by applying the definition of the asymptotic mass, namely that it is the mass at which energy is only allocated to reproduction and so Φk(m∞,k)=1. This gives Φk(m∞,k)=krmaxsk(m∞,k)m∞,kξVB,k(m∞,k,t)=1, and so we have that krmax=ξVB,k(m∞,k,t)sk(m∞,k)m∞,k.

We replace this value of krmax into Eq. () to find that Φk=sksk(m∞,k)mm∞,kξVB,k(m∞,k,t)ξVB,k.

Applying Eq. () for ξVB,k, and noting that sk(m∞,k) is essentially equal to 1, we find that Φk=sk1-ϵam/m∞,kb-1-ϵa.

Bringing this development together with Eq. (), the individual fish growth rate is γS,k=1-sk1-ϵam/m∞,kb-1-ϵamin⁡ϕC,kπmfk,Amb-kam.

As in , we assume that the mass structure of the allocation of energy to reproduction sk(m) is a sharply transitioning function that shifts from near zero to near one around the mass of maturity mα,k. Based on and , we further assume that the mass of maturity is proportional to the asymptotic mass m∞,k such that mα,k=ηm∞,k (Table ). Although other functional forms are plausible, sk(m) must have a transition in mass that is proportional to m∞,k (or to the maturity mass) , and so we use the functional form used by , sk=1+mmα,k-cs-1, where the parameter cs determines how quickly the transition from zero to one takes place (Fig. ). For reference, we calculate the reproduction allocation mass scale, the range over which the majority of the change in reproduction allocation takes place, as the inverse of the derivative evaluated at the maturity mass, (dskdm|m=mα,k)-1, which we find to be 4mα,kcs.

We model the biomass production of fish by assuming that both phytoplankton and fish production are part of the same energetic production spectrum . Unlike in the approaches of and , among others, we do not model the growth and decay dynamics of phytoplankton biomass. Instead, we represent fish production over a spectrum of individual fish masses, π(m,t) (mmolCm-2g-1s-1). Following and , we base this formulation on (1) the NPP Πψ(t) (mmolCm-2s-1) (Sect. ), (2) the representative size at which NPP takes place mψ(t) (g) , and (3) the trophic scaling exponent τ that indicates how efficiently energy is transferred through the trophic web, where τ depends on the trophic efficiency α and the predator to prey mass ratio β, and is equal to log⁡(α)/log⁡(β) . The fish production spectrum follows π=Πψmψmmψτ-1.

As in , we assume that α and β, and hence τ, are constant. From the expression for fish production detailed in Eq. (), we determine the individual fish growth rate using Eq. ().

Although variability in the trophic scaling τ, that could depend on environmental or ecosystem characteristics, is potentially of significant importance, we take here the simple assumption that the trophic scaling is globally constant, as other authors have . We note that, using a large database of individual prey eaten by individual predators, found that the predator to prey mass ratio increases with predator mass. Given that we apply an average value of β, and assuming that all else remains equal, the work of implies that we would underestimate β for large m and overestimate β for small m, and so (by Eq. ) we underestimate πk for large m and overestimate πk for small m. Essentially, a mass-dependent β would tend to decrease the steepness of biomass spectra relative to what is shown here. It is also commonly assumed that the trophic efficiency α is constant . Based on acoustic biomass estimates and modelling work, suggests that trophic efficiency can instead be significantly different in low and high productivity regions, at different levels in the food web (from phytoplankton to mesozooplankton and from mesozooplankton to fish) and that it can also depend on environmental parameters such as temperature (through its influence on organismal metabolic rates) and water clarity (which affects visual predation). Quantifying variability in τ is an important target for future work.

The production spectrum is the product of two terms. The first is the initial value determined at the representative phytoplankton mass mψ(t), which corresponds to the NPP normalized by the representative phytoplankton size. The fish production spectrum then follows a power law dependence in m with a scaling exponent of τ-1. This mass scaling represents larger phytoplankton (larger mψ(t)) being trophically closer to fish than smaller phytoplankton, thereby permitting more energy to be transferred from phytoplankton to fish . The power law dependence that we use is based on and . The model is forced with observations of NPP, and so we run the model in units of mmolC. For analysis and presentation, we convert to grams of wet biomass (gwB) by assuming that there are 12 gC per molC, and that there are 10 gwB for every g of dry carbon .

Phytoplankton mass ranges over several orders of magnitude . We take a simple approach and express the spectrum of phytoplankton as a single representative mass at which NPP takes place. Due to the wide range of phytoplankton mass, we calculate the representative mass as mψ=mLΦL(t)mS1-ΦL(t), and so take the geometric mean of the mass of a typical large, mL, and a typical small, mS, phytoplankton, weighted by the fraction of production due to large or small phytoplankton, ΦL(t) and 1-ΦL(t), respectively. We calculate this fraction using the phytoplankton size structure model of , which resolves small and large phytoplankton and assumes that small zooplankton are able to successfully prey upon increasing production of small phytoplankton, but that large zooplankton are unable to do so as effectively for large phytoplankton production. propose an empirical relationship for the large fraction of NPP ΦL(t) in terms of temperature TC(t) (∘C) and the NPP, the Eppley factor ekETC(t) where kE (∘C-1) is the Eppley temperature constant for phytoplankton growth, and Π* (mmolCm-3d-1) the productivity normalized to a temperature of 0 ∘C. The model resolves a high fraction of the variability in phytoplankton community structure , and provides a mechanism to explain how the fraction of large phytoplankton biomass increases with increasing phytoplankton biomass. Although we use this particular formulation for the large fraction in Eq. (), future work could examine alternatives .

The natural mortality term represents all forms of natural (non-fishing) mortality. It mainly consists of predation, but also includes non-predatory sources of mortality such as parasitism, disease, and senescence . This term is of first-order importance in determining energy flows in marine food webs, and so also in determining biomass. In pursuing our principle of using empirical parameterizations to represent complex processes that are incompletely understood, we follow the work of and and take the mortality rate to be Λk=λm-hm∞,kh+b-1, where λ=eζ1(A0/3)aλ(T) (see Appendix  for a full derivation of this form). ζ1 is a parameter estimated from mortality data , A0 (g1-bs-1) is the growth constant from Eq. (), and aλ(T) is the van't Hoff–Arrhenius exponential for mortality as described in Eq. (). provided a mechanistic underpinning for Eq. () by calculating the optimal number of daughters per reproducing female over that female's lifetime. Unlike other empirical mortality rate frameworks, such as that of , the mass dependence m-h does not depend on the allometric growth scaling b, and so the mass dependence of the mortality rate is not determined by internal biological parameters, but by predation and competition . The losses due to natural mortality, term 3 in Eq. (), are linearly proportional to biomass as in , and in keeping with the classical MVF model.

It is important to highlight the fact that unlike some other models, we do not adopt an explicit representation of predation-dependent mortality . The mortality rate only depends on the organism mass, asymptotic mass, and temperature, and is linear in biomass. This choice is motivated by the wide range of predator–prey mass ratios in marine ecosystems , and the complexity and non-stationarity of food web relationships. In applying this parameterization, we avoid the complication of choosing a difficult-to-constrain prey selectivity function, and benefit from applying mortality rates that are directly founded in observed rates. Without necessarily losing realism, this parameterization simplifies the complicated dynamics that result from more sophisticated prey selectivity formulations .

Since the prey mortality rate does not depend on the predator biomass, we do not resolve top-down trophic cascades . At present, a scarcity of data hinders a formal verification of generalized trophic cascades in the open ocean, which would be desirable for the formulation of their impact within the BOATS framework. However, we do represent bottom-up effects through the growth formulation described in Eq. (), since a change in biomass in one size class is carried upward through the trophic web as fish grow to larger mass classes.

Fish reproduction and recruitment comprise a set of complex ecological processes that result in new fish biomass entering a fishery . This first involves fish allocating energy to reproduction and releasing eggs and sperm during spawning. Fertilized eggs must then survive predation until they hatch to become larvae, when they must again survive predation until they grow into juveniles . The end of the juvenile stage is generally defined as when fish reach sexual maturity or when they begin interacting with other adult members of the fishery . The definition of a recruit is more nuanced since it generally depends on the fishery in question and can be based on a particular size or age, the size or age of sexual maturity, or the size or age at which fish can be caught . For the model, we refer to recruitment as the flux of new biomass into the lower boundary mass (m0) of each group.

Recruitment is driven by biomass-dependent (density-dependent) processes, such as predation and disease, as well as by biomass-independent (density-independent) processes such as environmental change. These processes strongly and nonlinearly affect mortality throughout the egg, larval, and juvenile stages . To model the number of recruits that result from a given spawning stock of biomass, one must make assumptions on the nature of these processes. The widely used stock-recruitment models of , , and , and the generalization of these models by and , make such assumptions and operate in terms of the spawning stock biomass; that is, the biomass that is of reproductive age.

We model recruitment by considering both the NPP and the production and survival of eggs by adult fish. Our formulation is based on the Beverton–Holt stock recruitment relationship which employs a Holling Type 2 functional form,, as used by and , with NPP setting the upper limit and the half-saturation constant (Eq. ). This form allows for an approximately linear decrease to zero recruitment as the spawning stock biomass goes to zero, but sets an upper limit that depends on the NPP when the spawning stock biomass is large, in order to represent the role of food availability in determining larval survival.

The flux of biomass out of a mass class is the growth rate multiplied by the biomass in that mass class (Eq. ). Since the recruitment is also a flux of biomass (one that occurs at the lower mass boundary), to define it in terms of NPP RP,k(m0,t) (gwBm-2s-1), we apply Eq. () and find that RP,k(m0,t)=γP,k(m0,t)fk(m0,t)=ϕC,kπ(m0,t)m0fk(m0,t)fk(m0,t)=ϕC,kπ(m0,t)m0, where m0 is the lower bound of the smallest mass class, and π is the fish production spectrum from Eq. (). Alternatively, the recruitment from the production and survival of eggs to recruits, Re,k(m0,t) (gwBm-2s-1), depends on the energy allocated to reproduction, γR,k(t) (Eq. ), by all nk individuals over all mass classes, which we write as Re,k(m0,t)=ϕfsem0me∫m0m∞,kγR,knkdm.

The model biomass includes both males and females, which are assumed to mature at the same mass . As in other model studies , males and females of reproductive age continually reproduce, yet only the female contribution is counted in the flux into the smallest mass class, since the male contribution to a fertilized egg is negligible compared to that of the female. Hence, when the integral part of Eq. () is multiplied by the fraction of females, ϕf, we have the biomass of eggs produced. Dividing by the mass of an egg me therefore gives the number of eggs produced, which when multiplied by the survival fraction se, expressing the probability that an egg becomes a recruit, gives the number of recruits. From the number of recruits produced per unit time, we multiply by the mass of a recruit, m0, to determine the biomass flux of recruits.

Applying the same form as the stock-recruitment model developed by see we take the overall recruitment Rk(m0,t) (gwBm-2s-1) to be Rk(m0,t)=RP,k(m0,t)Re,k(m0,t)RP,k(m0,t)+Re,k(m0,t).

Following , we take the half-saturation constant (the value of Re,k(m0,t) at which the overall recruitment is one half of the maximum recruitment allowed by productivity) to be RP,k(m0,t). Figure  shows how the overall recruitment Rk(m0,t) changes as a function of RP,k(m0,t) and Re,k(m0,t). As is the case for a Holling Type 2 functional form, as biomass and therefore also the egg- and survival-based recruitment Re,k(m0,t) increases, the overall recruitment saturates toward the primary production-based limit RP,k(m0,t). This indicates that for sites with high biomass, NPP limits recruitment. At the other extreme, when Re,k(m0,t) is small relative to RP,k(m0,t), the recruitment is approximately linear in Re,k(m0,t) and so has a weak dependence on RP,k(m0,t) such that at low biomass the egg production and survival limits recruitment.

Tables  and  detail the fish model parameters and variables, respectively. The group and mass class structure, and the numerical discretization of the continuous biomass spectra, are presented in Sect.  and Sect. , respectively.

The ecological model requires temperature and NPP information as forcing input to calculate the time evolution of biomass (Eq. ). These variables can be provided by an ocean general circulation model that includes a lower trophic level model. Here, we instead use observational estimates, which would be expected to provide a more realistic simulation. For temperature, we use the World Ocean Atlas 2005 , which brings together multiple sources of in situ quality-controlled temperature interpolated to monthly climatologies on a 1∘×1∘ grid. We discuss our usage of temperature in Sect. , and as discussed above, use the average water temperature from the upper 75 m of the water column to force temperature-dependent rates. For NPP, we take the average of three satellite-based estimates to capture some of the variability that exists in different NPP models . We note that satellite-based estimates suffer from a range of shortcomings, including lack of productivity sources other than phytoplankton (e.g. seagrass and corals), and biases in coastal regions and estuaries . Although overall minor (see ), these uncertainties will carry through to the modelled biomass and harvest.

Fish span several orders of magnitude in mass, and we therefore discretize the mass spectra into logarithmic mass classes. In order to directly compare our results with the Sea Around Us Project (SAUP) harvest database , we consider three fish groups each with a different asymptotic mass. We first convert the maximum lengths used in the SAUP (30 cm for the small group, 90 cm for medium group, and up to our maximum resolved length for the large group) to asymptotic length assuming that the maximum length is 95 % of the asymptotic length , and then apply a length–weight relationship of the form m=δ1lδ2 to calculate the asymptotic mass. This results in asymptotic masses of 0.3, 8.5, and 100 kg for the small, medium, and large groups, respectively.

Although the asymptotic masses differ, all three groups have the same mass class structure, with lower and upper bounds of m0 = 10 g and mu = 100 kg, respectively. Since the groups have different asymptotic masses mk,∞, there are therefore fewer resolved mass classes for groups with smaller asymptotic mass. We define the mass classes by dividing the mass spectrum into NM classes with lower bounds mi,L such that mi,L=m0mum0i-1NM, where i is the index of the mass class that ranges from 1 to NM. Based on this definition, we describe a mass class as an interval Ii=[mi,L,mi+1,L] of length Δmi = mi+1,L-mi,L (i=1,…,NM). We divide the spectrum into 50 mass classes (NM=50). Although we use fewer mass classes than some other studies , we have tested higher temporal and spatial resolutions and find that our interpretations would not be influenced by our choice of temporal or spatial resolution.

When we calculate and present mass-dependent quantities, we consider a mass mi that represents the average or central value of its class. For this, we apply the geometric mean of the lower and upper bounds of a mass class, which we calculate as mi=mi,Lmi+1,L1/2, since the upper bound of a mass class is the same as the lower bound of the adjacent class.

The biological part of our model is a system of three nonlinear first-order (in mass) partial differential equations that describe the evolution of the biomass spectra of three fish groups. Each equation is forced with the same net primary production and temperature information, and the equations do not interact with one another. Here, we use the standard notation of a subscript i to describe a mass cell, and a superscript n to describe a temporal cell. The notation k, as in the main text, refers to a fish group. For example, fk,in+1 represents the biomass spectral value f of group k, at mass class i at time n+1.

Since the McKendrick von-Foerster model is an advective equation in biomass, as is true of advective equations, transport errors are a concern . To limit such errors, and because growth is always defined to be positive (or zero), we apply an upwind scheme . This numerical scheme uses only biomass information that is upwind of the cell of interest; that is, it only uses biomass information at cells i and i-1 to integrate and determine the biomass at cell i at the next timestep. We use a forward difference scheme for the temporal rate of change, and explicitly calculate the growth (γ) and mortality (Λ) rates; that is, we use the current temporal state of biomass fk,in to update the biomass, as opposed to using the future biomass state fk,in+1 as in an implicit scheme, and integrate biomass as fk,in+1=fk,in+-γk,infk,in-γk,i-1nfk,i-1nΔmi+γk,infk,inmi-Λk,infk,inΔt.

The model is stable and converges as we decrease Δt.

To begin our results and analysis section, we make a series of simplifying assumptions in order to derive an analytical biomass spectrum fk,m,0, which we use as a reference point for evaluating aspects of the full model. Since this analytical biomass state is a reasonable approximation of the full model, we also use it as an initial biomass condition for our simulations.

Beginning with the evolution of biomass in Eq. (), we assume that the input energy expressed in Eq. () is solely controlled by NPP, so that ξI,k(m,t)=ξP,k(m,t)=ϕC,kπ(m,t)m/fk(m,t), and that there is no allocation of energy to reproduction, so that Φk(m)=0. These two assumptions result in a growth rate of γS,k(m,t)=ϕC,kπ(m,t)m/fk(m,t), which allows us to calculate the equilibrium biomass spectrum (∂∂tfk(m,t)=0) in terms of the fish production spectrum (Eq. ) and the mortality rate (Eq. ). We consider constant forcing and so apply the annual average NPP Π‾ψ and temperature (which are contained in the mortality rate λ and representative phytoplankton mass mψ terms), and find that the equilibrium biomass spectrum of each each group is fk,m,0=ϕC,kΠ‾ψ(1-τ)λmψτm∞,kh+b-1mτ+h-1.

As expected from the MVF model, biomass follows a power law spectrum with respect to mass. Given that the power law scaling exponent is τ+h-1, biomass scales as a function of the trophic and mortality scalings, which we assume are constant. On the other hand, the intercept of the spectrum (in logarithmic space, when m=m0=10 g) depends on a variety of parameters such as the NPP and trophic efficiency, as well as the natural mortality rate and the representative phytoplankton mass. Unlike the mass scaling, the intercept is also group dependent through the fraction of primary production allocated to each group and the asymptotic mass.

As in other studies, we use features of the modelled biomass spectra, shown in Fig. , to interpret the model results. Work on marine ecosystems indicates that biomass spectra, when plotted in log-log space, are approximately linear over most of the size range and have slopes that range from -1.0 to -1.2 . Ignoring harvest, group biomass spectra generally decrease with size, except at the maturity mass at which energy begins to be allocated to reproduction (Fig. ), where there is a decrease in the growth rate and thereby an accumulation of biomass that may result in a local maximum or a local decrease of the spectrum slope . As expected from Eq. (), the group intercepts differ, but only by little since in our formulation the only difference arises from the weak asymptotic mass dependence m∞,kh+b-1 in the mortality term. Biomass is larger at the cold-water site, despite it having a lower NPP (Fig. ). In particular, large group biomass is larger at the cold-water site, which is consistent with the findings of .

There is a nonlinear decrease in biomass at larger mass classes (Fig. ). The shape of the biomass spectra are determined from the growth and mortality rates. Since the growth rate consists of NPP and allometric regimes (Eq. ), and the mortality rate of a single regime (Eq. ), any changes in the shape of the biomass spectra are determined by the growth rate. We generally find that the NPP regime (Eq. ) limits energy input in smaller mass classes, whereas the allometric regime (Eq. ) plays the limiting role in the largest mass classes.

Total biomass (Fig. a) increases monotonically for increasing NPP, yet decreases monotonically for increasing temperature. Increasing temperature not only reduces the primary-production-based growth rate γP by reducing the representative phytoplankton size (Eq. ), it also significantly drives up the mortality rate, generating a clear pattern of reduced biomass. Under the allometric regime of growth (Eq. ), higher temperature implies a greater growth rate, which on its own results in an increase in biomass (not shown). However, this feature is more than counterbalanced by the mortality rate increase, which results in an overall lower biomass for higher temperature.

We calculate the total biomass spectrum as the sum of the biomass of each mass class over all groups. We use the biomass value at the first mass class to define the intercept, and calculate the slopes based on the non-reproducing parts of the spectra (the mass classes that are smaller than the maturity mass mα,k) since this is generally the linear part of the spectra , using linear regression on the log-transformed data . The spectral intercept (Fig. b) depends on both NPP and temperature, monotonically increasing with increasing NPP, but nonlinearly changing in temperature due to the multiple sources of temperature dependence in the intercept (Eq. ). The biomass slope does not depend on NPP (Fig. c), as indicated in Eq. (), and the resulting total slope values (grey curve in Fig. c), given the parameters used in this single realization of the model, are consistent with published values from marine ecosystems that range from -1.0 to -1.2 . However, we find flatter slopes for lower temperatures, to values as low as -0.9. This implies that our model would result in generally higher biomass than if the slope of the spectra fell between -1 and -1.2. Equation () also indicates that the slope is not a function of temperature. That equation applies for the small group (blue curve in Fig. c) over all temperatures, and for the medium group at low temperatures. However, when the input energy is determined by the von Bertalanffy limit, as is the case for high temperatures in the medium group and all temperatures for the large group, a rise in temperature steepens the biomass slope. Overall, NPP only influences spectra by shifting the intercept, whereas temperature both shifts the intercept and changes the slopes of biomass spectra when the input energy is set by the von Bertalanffy limit.

The model illustrates hypothetical inferences, based on the macroecological theory it uses, that need to be compared to suitable observations. Further validation of the model at specific locations and at the size-class level of detail remains a challenge because of the scarcity of suitable data sets. To further validate BOATS and comparable models, we require size-class-resolved observations at the ecosystem level, at a high enough resolution to detect variations in spectral properties, and at a sufficient number of sites so as to detect bulk variations due to different temperature and NPP. This type of detail at the ecosystem level is not available even in current stock assessment databases, and it should be considered an important target for future data syntheses.

BOATS was written in MATLAB version R2012a , and was also tested in version R2010b. The zero-dimensional version of BOATS (for a single patch of ocean, that is, a single site), which includes the model run script, required functions, and forcing data, is available for download at doi:10.5281/zenodo.27700.