Modeling Food Web Interactions in Benthic Deep-Sea Ecosystems

Deep-sea benthic systems are notoriously difficult to sample. Even more than for other benthic systems, many flows among biological groups cannot be directly measured, and data sets remain incomplete and uncertain. In such cases, mathematical models are often used to quantify unmeasured biological interactions. Here, we show how to use so-called linear inverse models (LIMs) to reconstruct material and energy flows through food webs in which the number of measurements is a fraction of the total number of flows. These models add mass balance, physiological and behavioral constraints, and diet information to the scarce measurements. We explain how these information sources can be included in LIMs, and how the resulting models can be subsequently solved. This method is demonstrated by two examples-a very simple three-compartment food web model, and a simplified benthic carbon food web for Porcupine Abyssal Plain. We conclude by elaborating on recent developments and prospects.

ABSTR ACT. Deep-sea benthic systems are notoriously difficult to sample. Even more than for other benthic systems, many flows among biological groups cannot be directly measured, and data sets remain incomplete and uncertain. In such cases, mathematical models are often used to quantify unmeasured biological interactions. Here, we show how to use so-called linear inverse models (LIMs) to reconstruct material and energy flows through food webs in which the number of measurements is a fraction of the total number of flows. These models add mass balance, physiological and behavioral constraints, and diet information to the scarce measurements. We explain how these information sources can be included in LIMs, and how the resulting models can be subsequently solved. This method is demonstrated by two examples-a very simple three-compartment food web model, and a simplified benthic carbon food web for Porcupine Abyssal Plain. We conclude by elaborating on recent developments and prospects.

INTRoDUCTIoN
Deep-water sediments are among the largest and most elusive of Earth's ecosystems. Compared to other ecosystems, we know little about the taxonomy, natural history, and trophic linkages among the organisms that inhabit deep-sea sediments. This limitation makes it difficult to predict the impact of human activities on deep ecosystems, but such predictions are greatly needed because human pressures on this ecosystem are increasing rapidly (Glover and Smith, 2003).  (Soetaert et al., 2002).
Diagenetic models focus on the role sediments play in the global cycles of essential elements (C, N, P…) and consider sediments to be shaped by physical processes and biochemical reactions. In contrast, benthic food web models study the flows of energy or matter between biological functional groups.
There are several good reviews and recent books that deal with diagenetic modeling and its applications (e.g., Burdige, 2006). In contrast, texts on food web modeling are typically case studies with only short introductions to the methodology employed. In this paper, we focus on the latter types of models and give a practical account of their applications and results. (Sarmiento and Gruber, 2006) and longterm removal of CO 2 (Middelburg and Meysman, 2007). On the other hand, processing of organic matter through the food web and resulting secondary production ultimately fuels commercially interesting demersal fish communities (Graf, 1992).  Figure 1). Before detritus becomes incorporated into the sediment, part of the food is consumed by suspension feeders that filter organic matter from the water mass overlying the bottom (Gage and Tyler, 1991) or by sedimentdwelling deposit-feeders (Blair et al., 1996;Drazen et al., 1998). The remainder of the food is ingested by sedimentinhabiting detritivores of various sizes (e.g. Graf, 1992) and by bacteria (Lochte and Turley, 1988) that respond rapidly to the supply of particulate organic matter in terms of increased metabolic activity Moodley et al., 2005) or growth and reproduction (Tyler et al., 1982;C.R. Smith et al., 2008;K.L. Smith et al., 2008). The detritivores are consumed by predators, which may themselves be preyed upon by larger animals such as fish. The waste products of all consumers become food for detritivores and bacteria (detritus) or are exchanged with the water column (CO 2 , nutrients).
The flows of food to the primary benthic consumers and the recycling of matter from one biotic component to another comprise the benthic food web.
How benthic communities process the primary material and convert organic matter as it passes through each trophic link has significant consequences for ecosystem properties, such as food web stability (e.g., Rooney et al., 2006), and for linking benthic secondary production with higher trophic levels that are eventually harvestable by humans (Pauly et al., 1998). In addition, food web structure and functioning affects biogeochemical properties such as carbon sequestration (Middelburg and Meysman, 2007), carbon turnover (Meysman and Bruers, 2007), and nutrient regeneration (Vanni, 2002). Thus, the identification and quantification of energy pathways through the major ecosystem components is a basic element of food web studies. The ultimate goal is to achieve a quantitative understanding of the functional interactions between biological components in order to eventually predict the response of deep-water systems to global change phenomena.
Quantification of biological interactions in terms of carbon or organic matter flows is strongly hampered by the lack of sufficient high-quality empirical data (Brown, 2003). This is because the elucidation of food web flows from direct measurement or experimentation is notoriously difficult, even for  comparatively well-studied shallowwater benthic systems (e.g., van Oevelen et al., 2006a,b

DEVELoPMENT oF A FooD WEB MoDEL
The various data sources and equations that comprise a food web model are represented in Figure  Step 1. Establishing the Topological Food Web Food web modeling starts with choosing relevant abiotic and biotic components and specifying the links between them ("who eats whom"). In general, the biota are subdivided into size categories, in accordance with the observation of distinct abundance peaks in certain size ranges (Schwinghamer, 1981;Duplisea, 2000), and in line with the existing fields of expertise of separate researchers. If appropriate, size categories are subsequently divided into feeding types.
This stage of the modeling process depends on observational detail (e.g., to guide the choice of major components) and also on intuition and common sense.
Generally, a larger organism feeds on a smaller organism but not vice versa.
Thus, even if the feeding process has never been witnessed, this assumption is used to draw a feeding link from small to large organisms.
A universally valid physical constraint is that, for each chemical element, mass is conserved. Application of this conservation principle allows writing elemental mass balances of the general form indicating that the temporal change (dX/dt) of the mass of a compartment (X) equals the difference between incoming (f in ) and outgoing (f out ) flows.
Thus, when f in is larger than f out , the mass will increase in time (as its rate of change is positive). This mass balance principle forms the backbone of a food web model. To understand how the mass balance principle is applied, it is useful to have a closer look at how an organism processes its food.
When organisms ingest food, only part of the food is assimilated (i.e., transported across the gut wall), and the rest is expelled as feces ( Figure 3). Some of the assimilated mass conservation bound on and relations between flows site-specific data equations "who eats whom" primary food sources export terms maximum rates growth efficiency assimilation efficiency biomass flow measurements individual respiration Step 1: Topological food web Step 2: Physiological constraints Step 3: Site-specific measurements Consider a very simple food web model comprised of three components (in the figure at right): Blue boxes represent (1) detritus, (2) bacteria growing on detritus, and (3) fauna grazing on bacteria. The system is driven by an external import of detritus (f 0 ). There are two consumption flows (f 1 and f 2 ), one feces production flow (f 3 ), and two respiration flows (f 4 and f 5 ); demersal fish graze on the fauna (f 6 ). Neither Co 2 (the respiration product) nor the fish are modeled explicitly as food web compartments; rather, they are considered external compartments. The three mass balance equations relate the rate of each compartment's change to the seven source and sink flows. If we assume that the compartments are invariant in time, they can be written as These equations can be written in a more general way as: to relate the rate of change (left-hand side, assumed 0) to a sum of products, where each product is composed of the flow times a coefficient. It is convenient to collect these coefficients in a matrix, leading to the following notation: The positivity of the flows is written as Physiological considerations (Step 2 in Figure 2) are implemented through the inequality constraints. For example, bacterial carbon is high quality food for benthic fauna; therefore, a reasonable assumption is that feces production (f 3 ) is small, between 10% and 30% of faunal ingestion (f 2 ). This gives the following two inequalities: Growth respiration is assumed to be between 20% and 40% of assimilated detritus (bacteria) or assimilated bacteria (fauna): Site-specific flow measurements can be directly included. Suppose that detritus deposition has been measured with sediment traps at f 0 = 1 g C m -2 yr -1 . This measurement gives the following equation:

Fauna Bacteria
Detritus fraction of the food is used as building blocks for growth and reproduction (secondary production), and some of it is oxidized to provide the energy required to maintain basal metabolism, form new biomass, reproduce, and move. For heterotrophic organisms, the energy needed for growth and for maintenance is paid by respiration, that is, the oxidation of simple organic compounds, while other, so-called chemo-autotrophic organisms produce biomass from chemical energy and inorganic compounds. From the modeler's point of view, the functioning of an organism is analogous to a chemical "factory" that uses raw materials (food) to produce valuable goods (biomass), while consuming energy (respiration) and producing waste (feces) in the production process ( Figure 3).
The principle of mass conservation states that, if the organism is not predated upon, then the ingested food is either respired, defecated, or will serve biomass increase because of growth ( Figure 3). We write this as where C is the biomass of the organism, and dC/dt is its growth (i.e., the rate at which this biomass changes in time).
This so-called mass-balance equation equates biomass changes to feeding flows minus loss terms (respiration, feces production, mortality).
ingestion defection predation basal respiration growth respiration respiration Figure 3. The mass balance of one organism from the food web (left) and the factory analogue (right).
functional group j is denoted as , then the mass balance for a biotic food web component (i = 1,..n) is given by This equation can be made more compact by denoting detritus as component "0" and carbon dioxide as component -1, so that we obtain In food web models, the flows are the unknowns (x) to be quantified. The mass-balance equations are just sums and subtractions of these unknown quantities. These linear equations are conveniently cast into matrix notation as in which x is a vector with the unknown flows, and b is a vector with the rates of change of the components.
Because flows have a direction (i.e., they are non-negative quantities), the following inequalities also hold: x ≥ 0. (2) Step 2. Physiological and Behavioral Constraints The physiology and behavior of organisms imposes lower and upper limits on their feeding and growth rates. When organisms search for food, the encounter rate and external handling time determines maximal foraging capacity (Holling, 1966). The ingested food is hydrolyzed and assimilated, but these processes are limited by physiological and digestive constraints (Jumars, 2000). Consequently, animals can only process a finite amount of food per unit of time (i.e., there are upper bounds on weight-specific ingestion rates). Often, these maximal weight-specific rates scale inversely with organism size (Peters, 1983 Figure 3, where respiration delivers the energy to produce a certain amount of goods). Classically, this is represented by growth efficiency-the ratio of secondary production (growth) to assimilated food, which is generally on the order of 60-70% (Calow, 1977) and at most 80% (Schroeder, 1981).
In addition, there is a similar relationship between feeding and defecation: organisms cannot produce more feces than the amount of food they ingest, but they have to assimilate a certain fraction to balance the loss terms. Depending on the quality of the food, a small or large fraction of it will be expelled as feces (Calow, 1977;Schroeder, 1981). Most often, this dependency is expressed by the so-called assimilation efficiencythe ratio of assimilated food (the food that is not defecated) to ingested food (Conover, 1966), which is roughly on the order of 20%, 60%, and 80% for detritivores, herbivores, and carnivores, respectively (Hendriks, 1999). Rather than assuming that growth and assimilation efficiencies are exactly known, it is more realistic to impose upper and lower bounds on these efficiencies.
The constraints on ingestion rates mentioned above as well as growth and assimilation efficiency put bounds " " on various flows and on relationships between flows. These constraints can also be cast in a matrix equation, comprising inequality conditions: Step 3. In Situ Measurements The data types mentioned above make are coupled to data in another currency (e.g., N, P, or O 2 ) (e.g., Vézina and Platt, 1988;Jackson and Eldridge, 1992;Gaedke et al., 2002).
Because of the valuable information contained in site-specific measurements, the available in situ measurements are generally adopted as they are (i.e., without uncertainty) and implemented as equality equations, which can be written in a matrix form that is identical to the mass balance equation Step 4 A straight line is characterized by two unknown parameters: the slope and intercept. These parameters are quantified by fitting to a data set so that the data are optimally reproduced.
With one observation and two unknown parameters, the model is under-determined, and there are infinite numbers of different straight lines that all exactly pass through that single data point (A in the figure at left). Similarly, under-determined food web models will have an infinite number of solutions.
With two data points, the model is even-determined, as there are also two parameter values to derive. only one straight line passes exactly through two points. Similarly, even-determined food webs will have one unique solution (B in the figure at left).
The over-determined state is encountered when there are more data points than unknowns. Not all data points can be exactly reproduced due to unavoidable measurement error, but a unique parameter combination reproduces the data optimally (C in the figure at left).  (Niquil et al., 1998;Kones et al., 2006). Moreover, the parsimonious web often takes extreme values (i.e., it lies at the boundaries of the solution space) (Diffendorfer et al., 2001;Kones et al., 2006). x 1

Box 2. Determining the Determinacy State of a Model
x 2 x 3 x 1 x 2 x 1 x 2 x 3

Parsimonious solution
Range estimation Bayesian sampling selects one solution estimate of flow range flow distribution in ensemble The Porcupine Abyssal Plain (Northeast Atlantic), located at ~ 4850 m depth, is one of the best-studied deep-sea sites. Here we give a simplified model of this ecosystem (Figures 5 and 6). A more elaborate model is under construction. Detritus from the water column (det_w) adds to the sedimentary detritus compartment (det), where it is taken up directly by nematodes (nem) and macrobenthos (mac), and dissolves to become dissolved organic carbon (doc). Part of the dissolved organic carbon is taken up by bacteria (bac), and the other part effluxes to the water column (doc_w). Bacteria in turn are grazed upon by nematodes and macrobenthos or may lyse (e.g., by viruses). Respiration by the biotic compartments induces a flux to the dissolved inorganic carbon pool in the water column (dic_w). Nematodes and macrobenthos produce feces that add to the detritus compartment. Nematodes are preyed upon by macrobenthos. Finally, nematodes and macrobenthos are preyed upon by megabenthic predators, but because they are not considered in this model, these grazing fluxes are described as export fluxes from the food web. Site-specific data on stock sizes of the biotic and abiotic compartments and process rates from the literature (Tables 1 and 2) are combined with generic physiological constraints (Van oevelen et al., 2006c) (Table 3) and added to the model. The implemented data are internally consistent, and the model is solved (Figure 6) by estimating the parsimonious (simplest) solution, the Monte Carlo solution, and the associated flow ranges ( Figure 6A) (see Figure 4 for conceptual visualization). overall, the flow ranges are relatively small, indicating that, notwithstanding the limited amount of data, the flows in the food web are well constrained. However, some flows (e.g., det → nem and nem → det) are highly uncertain ( Figure 5), and strongly positively correlated ( Figure 6). This result indicates that it is possible to quantify the net flux from detritus to nematodes, but not the separate flows.

Process Expression in Flows Value Reference
Efflux of dissolved organic carbon doc → doc_w 0.83 Lahajnar et al., 2005 Sediment community respiration bac → dic + nem → dic + mac → dic 5.43 Witbaard et al., 2000 Box 3. The Benthic Food Web at Porcupine Abyssal Plain pool, such that sediment detritus constitutes a complex mixture of organic matter with different origins, composition, age, and reactivity (Middelburg, 1989;Middelburg and Meysman, 2007 Blair et al., 1996;Moodley et al., 2002;Aberle and Witte, 2003;Witte et al., 2003;Moodley et al., 2005;Nomaki et al., 2005). In principle, it is also possible to trace the bacterial pathway by isotope enrichment of the dissolved organic carbon pool in the sediment or amending enriched cultured bacteria into sediment cores. Although this method has so far only been applied in intertidal mudflats (Carman, 1990;Van Oevelen et al., 2006a,b;Pascal et al., 2008), only technical difficulties ham-  spatial dynamics of the food web structure. For example, Donali et al. (1999) (Gaedke, 1995).

ACKNoWLEDGEMENTS
The polychaete drawing in