Arrow 1 in Figure 1 . Laboratory experiments represent the first step of the framework as our goal is to upscale results obtained at the sub-individual level to evaluate their implications at higher organizational levels. Experiment here is understood as a procedure done in a controlled environment in which the effect of one or multiple environmental factors (e.g., diets, pollution; thereafter named conditions), each with one or several treatment levels, are tested at the individual (e.g., growth in length) or sub-individual (e.g., mitochondrial performance) scales. An experiment is, in our definition, always a combination of a control condition and one or several stress* conditions. For example, an experiment testing the effects of a depletion of essential fatty acids on the European seabass Dicentrarchus labrax would follow the growth of individuals knowing their everyday length and weight and their swimming performances, e.g., aerobic scope, basal metabolism. This would be realized with a set of individuals fed with normal fatty acid concentration, i.e., the control condition, and another set of individuals with several levels of fatty acid-depleted food, e.g., −10%, −15%, and −20%, i.e., the stress* conditions. This would be done in controlled environments ideally at different stress* and temperature conditions to obtain functional responses.

A priori, any kind of organism can be used for the experiment, as long as enough knowledge and data are available on the species to further develop a DEB model, and if the size and other traits of the species allow to realize controlled experiments. The purpose of this step is to use the knowledge obtained by the experiments in the development of a DEB model with different conditions. In addition, this knowledge can be a source of data to contribute to the estimate of parameters in the DEB model in control conditions, along with data from literature (see arrow 2 and Section 2.1.2.). All along the experiment, it is essential to follow both the temperature conditions (if not already considered as a stressor condition) and the ingested food. In terms of organism variables, it is necessary to monitor variables that allow the estimation of growth, development, maintenance and reproduction when relevant. Measurements performed both in length (when relevant) and weight during the experiment are particularly useful for a successful integration of experimental results into DEB models. However, considering the initial question, other types of variables can become primordial, e.g., the number of eggs produced if the ecological question explores reproductive success in specific stress conditions. Longitudinal measurements (i.e., repeated in time for the same individual) are also particularly useful when individuals can be marked. Also, to gather other quantitative variables, such as age and food consumption rate, metabolic rate at different temperatures would provide more precision to the following steps. Variables measured both in control and stress* conditions are the most useful to better understand the mode of action (MoA) of the stressor, i.e., the metabolic processes potentially impacted by the stressor.

The choice of age/stage on which experiments are done can be restricted (e.g., availability of individuals, size of the tanks), but one of the powerful benefits of DEB is the possibility to extrapolate results to a larger spectrum of sizes, ages, and stages. Also, when relevant according to the organism of interest, if males and females are discriminated in experiments, this can also be included in the DEB model by using a zoom factor that will differentiate individuals by altering the value of the maximum food assimilation rate { p ˙ A m } in DEB, and the following different life stage thresholds: hatching, birth, metamorphosis, and puberty. Finally, results obtained for the different replicates/individuals will be paramount to tackle the uncertainty that will be considered in the upscaling process, even if an “average” individual can be used in the following modeling steps (see arrow 3 and Section 2.2.1.). Recent developments of the DEB model by, e.g., Boersch-Supan et al. (2017) and Koch and De Schamphelaere (2020) permit to catch the inter-individual variability and overcome the “average” individual approach.

At this step, it is also possible to use a pre-existing DEB model. In this case, if the experiment outputs were not included for the initial estimate of DEB parameters, the model still needs to fit well with the experimental data. Indeed, good fits with the new observations are a first requirement for the following comparison between the predicted individuals in control and stress* condition(s) that will permit deducing DEB parameters that can be influenced by the stress* and in which proportions.

Arrow 2 in Figure 1 . Even if a particular experiment focuses on a particular life stage, the full life cycle of an organism will be embedded in a DEB model. Predictions, e.g., of the egg energy content as well as ultimate weight could be made with the same set of parameters. It is therefore commonly done to complement the experimental data of interest with previous experiments and field observations reported in the literature to estimate most accurately the parameter set that best fits all the different sources of data.

Experiments, when possible, often provide information on the younger stages, while field observations will more often provide information on the juvenile and adult stages. If the ecological question at aim in the foreseen work is oriented on the trophic network, it is reasonable to think that for macroorganisms, adults’ stages can represent most of the biomass in the natural environment, and therefore particular attention is required on fitting these older stages in the DEB. Nonetheless, exceptions can occur for some species that can be present in certain habitats only during specific stages, e.g., juvenile fish in nursery grounds.

Field observations useful for the calibration of DEB models are length- and weight-at-age of individuals, their ultimate length and weight, age-, length- and weight-at-maturity, and fecundity as a function of length (or weight) together with proxies for temperature and food conditions. When collecting such data from the literature, data from the spatial area of interest are, of course, more relevant but data from other populations can also be used considering some uncertainties in the estimation procedure (e.g., by using lower weight coefficients*, i.e., coefficients used during the estimation process depending on the confidence in the quality of each dataset and the priorities for an application for these datasets).

Arrow 3 in Figure 1 . In a steady-state situation, the consumption Q (often defined in grams of carbon per kilometer per year (gC.km⁻².year^-−1) in LIM, which will constitute the reference for the following) of a consumer trophic group i is defined as:

where U (gC.km⁻².year⁻¹) is the unassimilated consumption, P (gC.km⁻².year⁻¹) is the production, and R (gC.km⁻².year⁻¹) is the respiration of the group. P includes all mortalities (i.e., predation, fishing, and other mortalities such as senescence or diseases), net migration and biomass accumulation, while R represents all the losses not attributable to feces (and N-wastes when defined in biomass) constituting U (Christensen et al., 2005; Figure 3 ). All these flows are often constrained as ratios of Q or the biomass B of the trophic group. It is worth noting that in LIM, natural mortalities other than predation or harvesting M0 (senescence, disease, etc.) are often considered negligible in LIM for secondary consumers (e.g., Chaalali et al., 2015) and are difficult to distinguish from unassimilated consumption. With the use of DEB modeling as input in the LIM, it is now theoretically feasible to clarify this distinction if values of U and P are fixed or precisely narrowed, M0 being grossly the difference between U without DEB-defined constraints and U with them. Note that this would slightly modify the estimated assimilation and production flows.

It is possible to use different outputs from DEB models to develop ecological threshold values for the species of interest in a given food web ( Supplementary Table S4 ). However, special attention should be given to the definition of the different rates and the corresponding units. Output rates from DEB models are most of the time expressed on a daily basis, while food webs are often represented on a yearly basis in LIM, even if exceptions can occur (e.g., Saint-Béat et al., 2014; Haraldsson et al., 2018). For the following section concerning the dynamic representation of the food web (see arrows 5–7 and Sections 2.3.1–2.3.3), it is easier to represent annual flows. We will thus express DEB fluxes in an annual basis.

The transfer of individual’s values to a trophic group is not straightforward. LIM’s equations, similar to those of Ecopath, follow the ones of Winberg (1956) that were initially bioenergetic functions defined for individuals (Link, 2010). In a mass-balanced* ecosystem modeling framework like LIM, most inputs rely on observations made on individuals, e.g., Q/B ratio, R/B ratio, or diets. In this context, a trophic group could be considered as a single individual organism simplifying the diversity and demography of the group. This choice can be required for simplicity’s sake or due to the absence of precise data on the demography and ontogeny for several species. When comparing different parameterization of LIM models, we assume stable composition of size, age, and even species when the group is multi-specific. This issue was already treated by Christensen et al. (2005) in the “Capabilities and limitations” of the EwE User’s Guide with the proposition of using multiple life stanza by splitting the biomass pools when relevant, i.e., species or groups known to have a strong trophic ontogeny or size-dependent vulnerability to harvest. Moreover, recruitment is often difficult to predict in the aquatic environment and depends on many other parameters than the spawning stock biomass (Subbey et al., 2014). In consequence, population subdivision often might not be necessary because the less costly hypothesis for demography remains to keep it unchanged, except when valuable indices exist. However, changes in the production of a trophic group (for instance in the dynamic projection of the trophic network—see arrows 5–7 and Sections 2.3.1–2.3.3) due to environmental conditions and trophic interactions will be implicitly integrated in trophic flows as changes in the growth and reproduction of individuals. Inevitably, the processes represented by the model would imply changes in the recruitment and demography and even taxonomic diversity (when applicable) of the group in real life conditions, but are not here explicitly represented.

Outputs from DEB models do not solve the issue of linking individual scale outputs to populations, and assumptions must still be done. However, being available at all possible ages or lengths, these outputs can be produced for all sizes/ages of a population and weighted according to the known demography or a good indicator of it for each of the species present in the trophic group of interest to provide one value representative of each population. In the absence of knowledge on the demography, the easiest option is to keep the values obtained at a middle-adult stage with the DEB model representing most of the time for macroorganisms the main part of the biomass, e.g., the common length from the Fishbase database for fish (Froese and Pauly, 2000), if adult individuals are known to occur in the represented ecosystem. In addition, the vital rates (i.e., consumption, assimilation, egestion, production, respiration, excretion when relevant) only slightly evolve at the adult stage and thus limit the error due to uncertainty on demography. To obtain the annual DEB outputs, each flow is simply cumulated over the 365 days of every age or at the age of interest potentially divided by the average weight of the individual at that age to obtain its ratio per biomass (i.e., Q/B, P/B, R/B).

Cautions have to be taken if the trophic group is multi-specific. Similarly to the demography issue raised above for populations, assumptions have to be done to upscale from individual to trophic groups. In this case, the most straightforward one is to use the species for which experimental data are available as the reference and consider the results identical for all the species composing the group. This hypothesis seems reasonable because the species composing such group are supposed to be gathered according to their diet and vital rates’ similarities. Another solution would be to weight the values obtained from DEB modeling according to the biomass proportion of the species in the group to provide ecological thresholds* (i.e., minimum or maximum values of a constraint* defined by inequality) for the LIM. However, this would require having other constraints* (i.e., model input data to limit the solutions for the flows to possible values) value for the other species constituting the group, which is not always the case.

For a multi-age and multi-specific trophic group, a constraint* φ is defined as:

For each species s and age a composing the trophic group.

The better use of Q as constraint* for the LIM is to compute a value relatively to B of the trophic group, i.e., Q/B. To give the best flexibility to the LIM, we propose to use the thresholds* of P and R relatively to Q, i.e., the net growth efficiency P/Q and R/Q.

The conversion of DEB outputs to Q, P, and R is obtained from Nisbet et al. (2012), who worked on relating the DEB theory to other bioenergetic approaches to obtain, in our case, vital rates in gC.year⁻¹ and biomass in gC following the LIM.

The relation between the representations of the consumption rate in DEB noted Q D E B and LIM is the most straightforward. Q D E B (derived from the “food consumption rate” C of Nisbet et al., 2012) is computed as:

where p ˙ X is the ingestion rate (J.year⁻¹), 12 is the molecular weight of carbon (g.mol⁻¹), and μ X is the chemical potential of food (J.mol⁻¹).

The animal’s biomass in DEB B D E B is computed as the sum of the structure, reserve and (for adults) reproductive reserve:

where W V is the structural weight (g), W E is the reserve weight (g), W R is the weight of the reproduction buffer (g), d V is the density of the structural volume (g.cm⁻³), V the structural volume (cm³), w V is the molar weight of the reserve (g.mol⁻¹), E is the reserve energy (J), E R is the reproduction buffer energy (J), and μ E is the chemical potential of reserve (J.mol⁻¹).

The production rate in DEB P D E B (from the “growth” G of Nisbet et al., 2012) is computed as:

where p ˙ A is the assimilation rate (J.year⁻¹), κ G is the fraction of energy for growth that is fixed into structure, p ˙ G is the rate of energy allocated to growth (J.year⁻¹), and p ˙ D the dissipation rate (J.year⁻¹), which encompasses all processes not associated with the production of new reserve and structure ( Figure 2 ).

If the currency used in the LIM is the carbon, the respiration rate in DEB R D E B (from the sum of “specific dynamic actions” S and “standard metabolism and activity” MA of Nisbet et al., 2012) is computed as:

where κ X is the assimilation efficiency in the DEB, κ P is the fraction of the ingestion rate transformed into feces, and μ P is the chemical potential of feces (J.mol⁻¹). If the LIM is defined in biomass, excretion has also to be considered following Equation 8 from Nisbet et al. (2012). This case is not tackled in our methodology.

The assimilation efficiency as defined in LIM obtained from the DEB and noted A D E B is computed as:

As explained above, the use of Q D E B for the Q/B input in the LIM requires its division by the average weight of the individual during a year. The use of P D E B and R D E B as P/Q and R/Q ratio only requires their division by Q D E B .

An equality equation* (defined as x = φ, with x as unknown flow and φ as empirical data) for the LIM can then be defined using DEB outputs for a given constraint* φ. A DEB model classically gives only one output and not the uncertainty around it. However, using only these values as equations in the LIM is giving too much credit on the DEB model predictions, while it is possible to consider its variability and uncertainty by building LIM inequations* (defined as x ≥ φ or x ≤ φ).

As previously evoked, there are ways to obtain more complex outputs than a single value from the DEB, i.e., the Bayesian method developed by Boersch-Supan et al (2017); Boersch-Supan and Johnson (2019) or the confidence interval definition proposed by Marques et al. (2019). If available, the outputs from these methods can provide much better estimates of the thresholds* provided for the LIM, even if we still have to set their limits, at least when using the Bayesian method (e.g., 5th and 95th percentiles of predictions). Once the thresholds* are defined by the modeler, a set of inequations* can thus be defined for a constraint* φ knowing the minimum and maximum values that x_i can reach.

If the last option is not feasible, another one is to use an equivalent of the Ecopath pedigree* (i.e., a coded statement categorizing the origin of a given input and by the same token specifying the likely uncertainty associated with the input; Christensen et al., 2005; Supplementary Table S5 ), as previously done by Nogues et al. (2021), i.e., giving a fixed percentage of uncertainty ζ around the output considered as an average value. A typical set of inequalities* for φ can thus be obtained:

ζ has to be given according to the confidence the user has on DEB’s outputs to describe the species in the area of interest.

Arrow 4 in Figure 1 . Despite the contribution of inputs from the previous steps, literature still constitutes a valuable source of information to evaluate the flow intensity within biomass-based trophic networks. These can belong to different sources, e.g., experimental studies, in situ observations, empirical relationships, other calculations from other values or even from other models when judged credible, which inform on the diet and vital rates of the groups. If there is no pre-existing LIM of the concerned ecosystem, all information necessary on how to build such model can be found in the work of Niquil et al. (2011).

The trophic network being static in LIM, the biomass is fixed, and a median or mean value is therefore to be given when having a time-series of biomass for the group observed. It is equivalent to consider that the variation of biomass is negligible throughout the time step of the model (e.g., the year). It is constituted by the different stages/ages of the species, except when the choice was made to use a multi-stanza group (see arrow 3 and Section 2.2.1).

Information on the species of interest, in addition to the DEB, is useful, at least concerning its/their diet. Concerning vital rates, the use of good quality thresholds* can be valuable in addition to outputs from the DEB because despite the potential of adding an equivalent of Ecopath pedigree* uncertainty around the predictions (see arrow 3 and Section 2.2.1.), DEB modeling still mostly relies on a single output when not using a methodology that considers uncertainty (see Section 2.2.1). By using additional literature-based thresholds*, the plausibility coverage by the thresholds* obtained from the DEB can be reduced.

Arrow 5 in Figure 1 . A comparison of two LIM models could be done if biomass observations were available in the control and stress* conditions. However, as it is rarely the case, the purpose of the LIM step was to obtain a reliable balanced* version of the trophic network to simulate the control state that will serve as a basis for comparisons with the stress* condition(s) with CaN modeling. We do not consider that LIM is well adapted for projection in the future of scenarios as, except in situations of extreme biomass variation like during phytoplankton blooms (e.g., Haraldsson et al., 2018), LIM works with a static biomass, neglecting biomass variations, compared to flow values. By modifying vital rates of the species of interest directly from the DEB to the LIM, one would assume that all the biomasses of the different trophic groups would remain stable and that flows would adapt with this situation, which seems unlikely in reality. Here, to build projections of food webs under stress* condition(s), the variation of biomass needs to be estimated by a model due to the important effects of biomass variations on trophic network’s structure and functioning. The LIM step can even be eventually by-passed to use directly the CaN modeling, which is, in our context, the best tool to answer the question of trophic relationships in the transition from control to stress* conditions, but this would generate a minimized exploration of uncertainty on CaN input parameters.

Biomasses have to be set in our methodology for the initial year of the simulation. Otherwise, these would not be constrained and generate dynamics far from the goal of having a representation of the control condition. The simplest way is to use the B values used for the LIM. After this first year, no value is given for the biomasses for the rest of the simulation (see arrow 6 and Section 2.3.2).

Every iteration of the LIM is a potential version of the trophic network under the control conditions. If we aim to limit the number of simulations within the CaN modeling, it is safer to initialize the CaN parameters from the mean value of the flows obtained from the LIM (Saint-Béat et al., 2013). However, a way to take advantage of the uncertainty of the LIM is to use a given number of randomly chosen solutions of the vector of flows. By this operation, the uncertainty of the LIM would be spread on the main parameters of the CaN model. From each of these versions of the trophic network, multiple simulations would finally be done with the CaN modeling. This technical solution increases drastically the number of simulations to run and analyze but remains the most rigorous for uncertainty evaluation.

Multiple outputs from LIM can be used in the CaN modeling. Within this latter, the main parameters driving the simulations are the refuge biomass* (i.e., the absolute minimum biomass that a trophic group can reach), the prey digestibility* (i.e., a correction factor that accounts for variations in energy content between prey), the assimilation efficiency*, the satiation* (i.e., the maximum consumption rate per unit biomass of the predator), the inertia* (i.e., a parameter that expresses the variations in biomass from 1 year to the next are bounded between a maximum growth rate and a maximum mortality rate), and the other losses* (Lindstrøm et al., 2017).

Several constraints* can be included to represent the trophic network. Imports, often parameterized in LIM, can be added as a constant or bounded value, proportional or not to the biomass of the group. This is particularly useful if the user wants to represent the primary production. It is, however, possible to do like Lindstrøm et al. (2017) and represent the primary producers as constant net imports (see Drouineau et al. (2021) for the different ways to implement the constraints*). According to the type of ecosystem represented, the consumption of detritus can be significant and represents a substantial part of the energy source. In these conditions, the simplest way to represent detritus is to include them as trophic imports for the groups concerned, for example, as a ratio of their biomasses. In the following, Q for secondary consumers will thus include the trophic imports when existing.

Starting from a LIM, it is likely that the currency of the trophic network will be organic carbon. Therefore, there are two ways to operate the CaN modeling, either keeping carbon as a currency, which is not originally the case in CaN modeling (i.e., originally in biomass), or transferring the model into wet weight or any other desired currency. If the choice is the latter, all biomasses and flows have thus to be converted using conversion factors (e.g., Brey et al., 2010).

The refuge biomass* can be set to 1% of the initial biomass, as proposed by Planque et al. (2014).

If the model is kept in carbon, which seems the easiest way to proceed, prey digestibility* is set to one for all the trophic groups to mimic the LIM results. In this case, the assimilation efficiency* γ has a similar definition than for LIM and is for each group equal to:

based on the outputs of the LIM. Otherwise, both must be defined from literature in accordance with each other (Lindstrøm et al., 2017).

The satiation* σ is based on the Q/B ratio from the LIM and includes the trophic imports. Keeping this value for a trophic group is equivalent to assuming that this group has the same feeding rate than the one it could achieve under “safe” laboratory conditions. According to Christensen et al. (2005), such an assumption is reasonable at least for marine mammals that will always try to feed to satiation*, while it is rarely the case for fish (Ahrens et al., 2012). Depending on the knowledge on the satiation* of the different groups in a particular ecological context, the Q/B value can be increased by a certain percentage to obtain the satiation* parameter. The satiation* is computed as follows:

where λ is a ratio equal to or greater than 1 defined for each trophic group using literature or expert knowledge.

The inertia* ρ results from the P/B ratio at satiation* obtained from the LIM. Following the implementation of the inertia* in CaN modeling, the inertia* is computed as follows:

where σ is the previously computed satiation* for each trophic group.

The other losses* µ are obtained from the following computation:

Unless the user has insights about the amount of other natural mortalities, the other losses* are considered equal to the respiration flow from LIM. For primary producers, other mortalities are defined as:

where γ is the previously computed assimilation efficiency* (that considers losses to sediment organic matter in LIM), and I refers to the import flow.

The exports of trophic groups are also a common output from the LIM. They correspond to emigration but also to fish/harvest when occurring in a given ecosystem. Like imports, exports can be added as explicit constraints* for the CaN projections according to the user’s knowledge on the process.

Finally, constraints* can be described on the diet of the trophic groups. These constraints* can be added on trophic links as presented by Drouineau et al. (2021) for explicit structural constraints*, e.g., when a prey is always more abundant than another one in the diet of a predator. However, the use of constraints* on the diet has to be done carefully, because too many constraints* could lead to not consider the potential variation of diets among trophic groups in the projections, especially in new configurations such as the stress* state(s). On the other side, no constraints* could represent an underuse of the knowledge on the system and make the CaN model explore implausible situations. The choices of the thresholds to use thus have to be made by the user. One possibility is to add a constraint* only when the proportions of two preys in the diet of a predator differ by a fixed magnitude, e.g., when a prey is always 10 times more abundant than another in a diet. In addition, it can also be relevant to define a constraint* on the minimum predation undergone by a defined group.

Arrow 6 in Figure 1 . According to the data availability on the biomass-based trophic network and the ecological question raised for the experimental study, partial or full time series, e.g., biomass from surveys, catches, primary production, etc. (see Drouineau et al., 2021 for examples of additional explicit constraints*), can be added to the CaN model to represent a given period. However, for the moment, CaN modeling does not allow users to change the values of inertia*, satiation*, and other losses* during a simulation. Therefore, it is not currently possible to design a simulation plan considering a time series for the stress* condition(s), except if these conditions already happened on a different time series than the one used for control condition, which is rarely the case. A solution is to start both control and stress* condition(s) with the same initial biomasses (see arrow 5 and Section 2.3.1). The model is then run for a given number of years free of time series constraints, i.e., close to the Non-Deterministic Network Dynamics (NDND) framework if considering only the biomasses (Planque et al., 2014; Lindstrøm et al., 2017), using potential additional permanent constraints* on the diets and/or import and export flows. In this case, time series can also be a way to calibrate* (i.e., adjust the parameters of a model by integrating the uncertainty of the parameters and/or the model to obtain a representation of the modeled system that satisfies a predefined criterion) the model with the control condition if the stability of the system is targeted, i.e., by modifying the ratios used to link consumption and satiation* and/or the thresholds of diet constraints*.

If the user does not use the values obtained from the LIM, it is still possible to use the empirical relationships given by Lindstrøm et al. (2017) to derive the values of inertia*, satiation*, and other losses* for every trophic group.

Arrow 7 in Figure 1 . As previously demonstrated, LIM outputs can be used to obtain a range of values of parameters (assimilation efficiency*, satiation*, inertia*, other losses*) for all the trophic groups in the control condition. The projection of stress* condition(s) for the focus species or trophic group is then input with perturbations of these same parameters quantified from DEB outputs.

The relative variations of Q, P, and R in the stress* state(s) compared to the control state are used to proportionally modify the assimilation efficiency*, satiation*, inertia*, and other losses*. The other values remain the same, including potential constraints* on diets. Like previously exposed in Section 2.2.1, the modification is justified by the analogy between the individual variations and the ones undergone by the group. Therefore, any modification of Q, P, and R are expected to affect all individuals of the group with the same magnitude. Like described in Section 2.2.1, this magnitude is obtained by weighting the values predicted by the DEB model depending on the demography of the species and/or the species composing the trophic group. The assimilation efficiency*, satiation*, inertia*, and other losses* for a stress* condition ψ are defined as:

where Q D E B ψ , P D E B ψ , R D E B ψ and A D E B ψ are obtained from the DEB model representing the stress* condition with the computations described in Section 2.2.1 (Equations 3–7) and 2.3.1 (Equations 11–15). If a secondary consumer is subject to trophic import, the constraint* can be modified accordingly.

The change in these three constraints* is supposed to modify the trajectories observed on projections performed with the CaN model. Depending on the trophic level of the group(s) modified following the methodology, cascading effects can be observed on the other compartments of the food web.

Other meta-parameters must be defined in the “sampleCaN” function of the RCaN package. For more details, see the work of Drouineau et al. (2021). It is noteworthy that the choice of the settings will greatly influence the time of sampling of the polytope* (i.e., a finite region of n-dimensional space bounded by hyperplanes), which could be limiting if multiple outputs of the LIM are used as input for the CaN modeling.