Food Webs and Biodiversity: Foundations, Models, Data

Acknowledgments

I am deeply in dept to my colleagues Ken H. Andersen, Louis-Félix Bersier, Tak Fung, Alan J. McKane, Géza Meszéna, Matthew Spencer, Rudolf P. Rohr and several anonymous reviewers for invaluable advice and comments that helped improve this book. Preparation of the manuscript would not have been possible without the trust and support it received from Keith D. Farnsworth and the friendly and patient encouragements from the team at Wiley-Blackwell, especially Nicky McGirr and Fiona Seymour. Support for research and manuscript preparation came from a Beaufort Marine Research Award by the Republic of Ireland, from the Centre for Environment, Fisheries & Aquaculture Science (Cefas) in Lowestoft, U.K., from the U.K. Department of Environment, Food and Rural Affairs (M1228), from the European Commission (agreement no. 308392), and from my wife Nana Sato-Rossberg during work after hours.

A. G. R.

List of Symbols

Part I

Preliminaries

1

Introduction

Food webs are the networks formed by the trophic (feeding) interactions between species in ecological communities. It is widely acknowledged that food webs are complex in some sense, both in their structural and their dynamical properties. What is causing this complexity? Does it play any important role for the functioning of ecosystems? If yes, which? And how do food webs depend on or control the diversity of the species they harbour? Considering that it is not easy to observe feeding interactions even between a single predator and a single prey species in the wild, observing entire food webs is a daunting task. Our knowledge of food webs is the result of inumerous days of concentrated field work, and yet the picture we have of their structural and dynamical properties resembles more a collage of ragged sketches than a photograph (de Ruiter et al., 2005; Duffy et al., 2007; Thompson et al., 2012).

It is the role of theory to bring these pieces together in a more orderly form. This, however, turned out not to be easy, either. While theorists over the last century or so were able to connect many pieces of this puzzle (Bersier, 2007), the big picture has not emerged, yet. A major hindrance is that food-web theory has no clear foundations, no obvious point to start from, which would allow bringing successively more pieces into place, step by step. This book is a bold attempt to outline the body of a coherent theory of food webs built on solid foundations.

My background is in the science of complex systems. This science has no strong foundations either. Rather, it is characterized by a folklore of methods that have proven to help understand a variety of complex systems. Common to these methods is, however, that they all make use of the language of mathematics. For complex systems, one equation is very simple:

(1.1) equation

Now, I am fully aware that mathematics is not particularly popular among ecologists, but I also know that the desire to understand food webs is strong. 5 All efforts have therefore been made to present the heavy diet contained in this volume in as small and tasty bites as possible. Some more spicy bites have been locked away in text boxes that need not be opened. Besides, refreshers on some basic mathematical notation and tools are provided in an appendix. Cross referencing of equations and relevant text sections is used lavishly to make it easier to start reading in-between and to trace complex arguments backwards.

The book was written to be suitable for an ecological graduate seminar in which students and teacher work together to read and understand either parts of it or the entire text. It may be of interest to empirically and theoretically minded community ecologists, and also to mathematicians and complex-system scientists looking for inspiration in ecology. It is hoped that study of the text gives readers a better understanding of what causes what and how in relation to the interplay between food webs and biodiversity. Incidentally, answers to five out of May's (1999) nine Unanswered questions in ecology will be offered (questions of spatial ecology are not covered here). Readers less familiar with the language of mathematics will find ample examples of how this language can be used to develop complex arguments without doing proper mathematics, i.e. to talk about real things rather than proving theorems.

The idea underlying the structure of the book is as follows. In Part II the foundations of the theory are laid out by introducing a limited number of concepts and their mathematical representations. These are energy and biomass budgets, allometric scaling laws, population dynamics and trophic interactions, trophic niche space, and community turnover and evolution. Particular care was taken to assure a rooting of each of these concepts in reality by linking its mathematical representation to measurements. In the case of trophic niche space (Chapter 8), this requires some effort. Part II closes in Chapter 10 with a brief illustration of a theoretical food web built using a model that combines all these concepts. In Part III, the mathematical representations of the basic concepts are then used as building blocks to construct a variety of other models, which are evaluated mathematically and/or in simulations. The mathematical analyses will occasionally lead to the emergence of new concepts. Part III also contains several reality-checks, in which structures predicted to arise by the models are compared with those found empirically. It comes back to models combining all basic elements in Section 22.2, now offering a good understanding of the mechanisms at work. Logical dependencies among the chapters of Parts II and III are illustrated in Figure 1.1 Part IV looks back at these theoretical considerations, and asks what their implications for ecology as a science and for conservation are. Among others, it contains a collection of assumptions and predictions made by the theory that merit testing in the field (Section 23.2).

Figure 1.1 Flow of reasoning in this book. Boxes above the dashed line correspond to chapters introducing concepts, boxes below to chapters analyzing mechanisms. Arrows indicate logical dependencies.

2

Models and Theories

Some basic remarks about the role of models in ecology in general and this book in particular help set the stage for what follows.

2.1 The Usefulness of Models

Most human thinking makes use of models, that is, simplified descriptions of reality. The very fact that we need to orient ourselves and make decisions in a world of which we are just small parts makes this a logical necessity. We cannot have exact representations of reality in our brains or computers because these are smaller than reality as a whole.

When I see a glass of water on a table in front of me, I involuntarily invoke a mental model of a glass of water. I will expect the glass to feel cold and hard, and the water to form waves when I move the glass. I expect the water to remain in the glass if I and everybody else leave it where it stands, and expect the glass to break into pieces if I drop it on the floor. I will usually rely on these expectations, even though, for any of these, I can think of conditions where they are wrong. Even for the most conventional of glasses of water, my mental model is wrong. For example, the water and the air above it permanently exchange molecules, and the water will eventually evaporate. In an even more precise, though tedious, description in terms of quantum fields the conceptual distinctions between glass, water and air fully disappear. As George E. P. Box (1979) famously wrote: All models are wrong but some are useful.

When is a model useful? There are at least four criteria that are relevant to the usefulness of models as tools for orientation and decision making in a complex world. (1) We want models to be easily specified and to build on well-known concepts, so that they are easily remembered and communicated. (2) We want models to be easily applied to make predictions, without requiring tedious computations. (3) We want models to be general, valid over a wide range of situations or system parameters. Finally, (4) we want models to be accurate –if only in the general sense discussed below.

Usefulness of a model by one of these criteria often comes at the expense of usefulness by another. Because of this, there can be several models of one and the same thing that are all useful in different ways. The most popular kind of models used in ecology, for example, linear regression models and their 9 variants, are easily formulated and easily applied, but are not particularly accurate and generally found to be valid only on a case-by-case basis. An ecosystem model describing different functional groups in an ecosystem that increase or decrease each other's abundances through non-linear interactions (e.g., Kishi et al., 2007) is likely to be more accurate and to be valid for a wider range of situations than linear models for the relationships between these compartments; however, its description will also be more complex, encompassing several interrelated equations, and a computer will probably be needed to evaluate it.

Similarly, the logistic equation (Section 14.3.4), a simple non-linear model relating the rate of change of population size to population size, will generally be more accurate than a linear model for the time-dependence of population size, but perhaps not as accurate as a model taking interactions with other populations into account. The logistic equation has the advantage over most other non-linear models of population dynamics to be analytically solvable. That is, there is a simple formula to compute the population size that the model predicts at any time in the future (or the past), given the current population size and two model parameters. We do not need a computer for this, a pocket calculator is fully sufficient. With some experience in reading formulae, we can even estimate the size of future populations using the analytic solution without actually evaluating it. Such estimates will be sufficient for many practical purposes –especially because, anyway, the simplicity of the logistic equation imposes limits to the numerical accuracy at which it can describe real systems.

Among the models used in this book to describe food webs, biodiversity, and interactions between the two, some will neither be particularly simple nor easy to evaluate. While these models may be capable of describing reality over a wide range of circumstances, they are unlikely to be accurate in the sense of correctly predicting the sizes of populations in the field. To capture the particular strengths of these models, a more general notion of accuracy than numerical accuracy is required.

When considering complex systems such as ecological communities, accuracy is sometimes usefully understood as meaning the ability to reproduce some narrowly defined properties of reality or, in the world of science, of empirical data. Consider, for example, a model for food-web topology, that is, a model for how species are interlinked through feeding interactions. The model might output random samples of food-web topologies, each a set of nodes and a set of directed links that point from one species to one of its consumers. Clearly, there are no numerical data here, and thus there is no obvious way in which the model could be accurate in the numerical sense. Instead, one can establish a correspondence between model and data by showing that certain statistical properties of the data, e.g., the distribution of the number of links pointing to a consumer (Camacho et al., 2002b), are well reproduced by the model output (Camacho et al., 2002a). Depending on the kinds of properties considered, different models will be the most general, most simple, or most easily applied models to reproduce these properties, which is another reason why there are many different useful models of a single complex system such as a food web.

2.2 What Models should Model

Criteria for choosing the properties used to compare between empirical and model data can also be derived from utility considerations. For similar reasons as explained in Section 2.1 for models, we want the properties (1) to be easily defined and (2) to be (computationally) easy to verify given sufficient data. In addition, we want (3) the amount of data (samples) required to verify these properties to be small. Apart from the obvious practical reasons for concentrating on properties satisfying these three criteria, there is another reason for considering such properties: They are also likely to be those most relevant for the effects that the system studied has on the rest of the world. Any other system, by being notably affected by specific properties of the system studied, implicitly detects these properties. But properties of a system are unlikely to have an effect on the rest of the world if they have complex definitions, are difficult to compute, or require large amounts of data to be detectable. Ecologists would say that such properties do not contribute to the functioning of the system. Utility of properties, in the sense above, is therefore closely related to their relevance.

A property of ecological communities for which many models have been developed is the distribution of the numerical abundances of species in a community (species abundance distribution, SAD). Often one finds that, although a community is dominated by just a few species, there are several less common species, and many more species with low abundance. SAD have moderate demands with respect to the criteria for properties listed above: They are quite easily defined and computed from data. The data requirements are low if one considers the empirical distributions themselves as the input data, but high if each observed individual is considered a data point, because often hundreds or thousands of individuals of the most abundant species are counted before observing one of a rare species. McGill et al. (2007) reviewed 27 models that were all built to reproduce and explain empirical SAD and concluded that several of these models reproduced the empirical data equally well, even though the models and the ecological mechanisms they invoked were quite different.

Thus, there clearly is a fourth criterion for choosing the properties used to compare between models and data. We want (4) the properties of empirical data that a model reproduces to be characteristic, in the sense that there are only a few models capable of reproducing them. This criterion is related to the use of models as predictive tools: If there are only a few simple and easily applied models that reproduce a given property over a wide range of parameters, then chances are good that such models also reproduce other properties that we did not test for, and that these models work for parameter values where they have not been verified, yet. At the least, this is more likely for these models than for models only reproducing less specific properties of data.¹ This fourth criterion also excludes many trivially reproduced properties of data, e.g., the nearly universally but trivially valid observation that a food web contains at least one species.

One way of making the properties one seeks to model more specific is to define new properties as combinations of other properties. McGill et al. (2007), for example, suggested developing models that make many predictions simultaneously, e.g., by not only reproducing SAD, but also their relations to the distributions of species over trophic levels, or the times since species invaded communities. Models capable of reproducing such specific, composite properties (and being accurate in this sense) can be useful even if this comes at the expense of model complexity, higher computational demands, and restricted ranges of validity.

The more complex a system is, the larger the variety of models tends to be by which it can be usefully described. Indeed, one could define complexity of a system as the characteristic of being usefully describable by a large variety of different models. Food webs, for example, are complex in this sense.

It is not unusual that different models describe the same complex system in terms of different kinds of interacting objects. Coming back to the initial example: What is described as interacting quantum fields in one useful model of a system can be a glass interacting with the water it contains in another useful model. This phenomenon can be interpreted as the emergence of novel concepts (a glass, water) in simplified, approximate descriptions of complex systems. The usefulness of these concepts derives from the usefulness of models making use of them and the roles the concepts play in expressing the properties that useful models reproduce.

2.3 The Possibility of Ecological Theory

Elsewhere I have argued that, by formalizing the above sets of criteria for choosing models and properties, one can define sets of optimal models and properties –optimal in the sense that further improvements with respect to any single criterion can only be made at a cost by other criteria (Rossberg, 2007). These sets of optimal models and properties contain quite a few but by no means all conceivable models and properties. Yet, for any kind of empirical data there is at least one optimal model reproducing it. Even though the models we seek in science need not necessarily be optimal in any sense, knowing that optimal models exist makes it a meaningful endeavour to search for good models and to strive to improve them.

One might conclude from this alone that any area of science should invest in theoretical research aimed at constructing useful models and successively improving them. However, some study systems might be extremely complicated, in the sense that even optimal models defined as suggested above would be either too complex, too difficult to evaluate, or too inaccurate to be of any practical use. Ecologists sometimes appear to see their study systems as being of this kind. These sceptical ecologists could be right. Then, indeed, there was no place for theory in ecology. However, this view that ecology is theory resistant might just as well be due to a misunderstanding: The expectation that ecological models should, just as Newton's mechanics, be simple, universally applicable, incredibly precise, and for simple cases analytically solvable –all at the same time! This, we learned by now, one cannot expect.

Rather, any good ecological model will represent a compromise between the competing criteria listed above. The set of possible models one has to take into consideration when looking for those that are useful is therefore tremendous. Depending on the system properties that a model is meant to reproduce, the verification of models can be arduous as well. Finding good models despite these difficulties is a task for theoretical ecology.

Fortunately, methods exist that can be used to overcome these challenges. Most of them make use of the language of mathematics. For example, several mathematical techniques are available for extracting from complex models simpler models (which might then be less accurate, either numerically or in terms of the set of properties they reproduce). Other mathematical techniques are available for isolating the mechanisms by which certain model elements generate given properties, and which then sometimes allows us to combine model elements in such a way that more complex, composite model properties are reproduced by more complex models. Despite this prevalence of mathematical methods, the practice of theoretical ecology will only occasionally boil down to doing mathematics in the sense of proving theorems. One important reason is that many mathematical techniques used to analyse and manipulate models are approximation techniques for which the full range of validity is difficult to predict a priori.

Thus, while being guided by mathematical techniques and, of course, knowledge and data of empirical systems, much theoretical work will remain a matter of trial and error. Hence, there is always a risk that the fact that a model reproduces certain, even composite, properties of empirical systems might just be the fortuitous outcome of many trials. Nevertheless, a theorist confident about the validity of a model with respect to the system properties it was meant to describe will elevate it to the status of a theory. But this is not the end of the story.

2.4 Theory-Driven Ecological Research

Only by being confirmed in independent empirical tests, i.e. withstanding attempts at falsification, a theory becomes scientific knowledge. This theory-driven empirical research has in common with traditional empirical research in ecology that both investigate patterns in empirical data (i.e., properties of data sets and their parameter dependencies). A major difference is that the choice of patterns studied is now motivated by theory. Since, as explained in Section 2.2, good models and, hence, good theories describe patterns that are relatively easily accessible empirically, chances are they have been known to empiricists for a long time already. Advanced theories, on the other hand, might predict more subtle patterns that are more specific to the systems studied. It is then the role of empiricists to advance ecology by scrutinizing such theories.

Because, as explained, ecological models will generally address only specific patterns and be wrong about other aspects of observations, misunderstandings regarding the nature of predicted patterns can easily render empirical tests mute. Since mathematical methods are used to derive theories and the patterns they predict, the predictions will often be formulated in a mathematical language: Even moderately complex mathematical formulae would become unwieldy when spelled out in words. It is therefore unlikely that ecology will much advance as a science through construction and testing of theories if those involved are entirely unfamiliar with the language of mathematics.

Recent developments towards more quantitative perspectives in empirical ecology are steps in the right direction, theory-driven ecology, however, goes one step further. It designs experiments and sampling campaigns specifically to scrutinize theoretical predictions. Theory will often be useful also in optimizing study designs. Theory-driven ecology means that theorists and empiricists strongly depend upon and trust each other: A good theory that remains untested and unused is just as sad to see as empirical efforts to test a theory that is botched from the start. The more empiricists and theorists understand each other's work, the easier this trust is built. The present work is that of a theorist, and hopefully of use for other theorists to read. But it is also meant as an invitation to empiricists to dwell in the world of theoretical thinking, to be inspired by empirical questions arising from theoretical consideration, and to begin a new dialogue with theory.

Note

1. This argument makes use of the principle of induction, which states that we may generalize from the observed to the unobserved. For a philosophical discussion, see, e.g., Vickers (2010).

3

Some Basic Concepts

This chapter introduces elementary concepts and vocabulary. Definitions are indicated by printing the defined term in italic. The first part defines basic terminology of food-web studies, the second part briefly recalls conventions related to physical quantities.

3.1 Basic Concepts of Food-Web Studies

Ecological communities are groups of populations of living organisms that interact with each other. According to a common definition, community food webs describe the network of flows of energy and biomass between the populations of different species, as resulting from feeding interactions in ecological communities. Empirical studies have sometimes considered only parts of community food webs, e.g. the part that supplies food to a particular species (sink webs) or that depends on particular species for food (source webs). Nowadays, however, the majority of studies of food webs considers community food webs, and the short-hand food web generally means the latter. Figure 3.1a is a graphical illustration of a hypothetical food web. Circles represent species, and arrows point into the direction of biomass flows between species.

Figure 3.1 Basic concepts in food-web studies. A food web (a), and two kinds of food chains (b),(c).

A consumer denotes a species that feeds on other species. Feeding is in food-web studies usually understood as feeding by killing living individuals or feeding on living bodies. Scavengers and detrivores are not at the centre of attention, except for cases where entire communities are fueled by detritus, e.g. soil or benthic aquatic food webs. This can be motivated by a general argument provided in Section 21.5. A producer is a species that is not a consumer. In Figure 3.1a, for example, species 1–5 are producers and 6–13 consumers. By convention, producers are drawn at the bottom of food-web graphs, so that energy flows mostly upward and top-predators are on the top. From the perspective of food-web studies, producers are sources of mass and energy in the system, while consumers are partially sinks and partially just pass mass and energy through to other consumers. Thus, the roles of producers and 15 consumers in food webs are fundamentally different, calling for a distinction of these two types of species even in the most abstract models.

In food-web studies, species not feeding on other species are sometimes referred to as basal species rather than producers. This notion takes into account that empirical food webs may be incomplete, so that some species may end up at the bottom of the food web simply because their resources have not been recorded, and that some species, such as detrivores, do not feed on living resources but aren't producers in a stricter sense, either. In the strict ecological sense, detrivores are consumers, so that food-web graphs can contain consumers without resources. For the clarity of argument, I will here stick with the strict consumer/producer dichotomy defined above based on whether a species feeds other species or not, keeping in mind that it is a simplification of ecological convention.

What makes a food web a network is the fact that consumers generally do not feed on all species present in a community. A species that a consumer feeds on is called its resource, and the species feeding on a given species are called its consumers. For example, the resources of species 9 in Figure 3.1a are 2, 3, and 6, its consumers 12 and 13. A feeding relation between a consumer and a resource in a food web is called a trophic link (arrows in Figure 3.1a).

These basic concepts characterizing network structure build on the assumption that one can sharply distinguish between those species that are in the diet of a consumer, and those that are not. In Section 11.2 below it is argued that this is not actually the case. Some links are, in some sense, strong, with relatively large amounts of energy and mass flowing through them, others are weak, and yet others are so very weak that trophic interactions of the corresponding consumer-resource pair are in practice never observed. A network structure arises only when considering all links as absent that are weak by some threshold criterion. The set of all nodes of a network, that is, all species in a community, together with the set of all trophic links between them together describe the topology of the network. The metaphor of a food web as a network with well-defined topology is useful for verbal arguments and empirical studies. In mathematical descriptions of food webs, it becomes less important.

Three elementary characterizations of a food-web topology are the number of species S it contains, the number of consumers Sc among these, and the total number of links L in a food web. The number S is also called species richness. For the food web in Figure 3.1a, S = 13, Sc = 8, and L = 36.

The notion food chain describes two related but different concepts. The first is a sequence of species in a food web where each species consumes its predecessor. Figure 3.1b, for example, illustrates the food chain 1 → 6 → 8 → 12 that is embedded in the food web of Figure 3.1a. A food chain in this sense does not necessarily start at a producer. By some accounts, 6 → 8 → 12 would be considered a food chain as well. Food webs tend to contain many food chains of this kind. In fact, the three-species food chains tend to be overrepresented in food webs when compared with random expectation (Bascompte and Melian, 2005; Stouffer et al., 2007). Another kind of configuration of three connected species (a motif, Bascompte and Melián, 2005) with a tendency to occur rather frequently consists of two food chains of different lengths that are linked at both ends, such as the chains 5 → 7 → 11 and 5 → 11 in Figure 3.1a. This motif is called omnivory or intra-guild predation (Arim and Marquet, 2004) in food-web studies, irrespective of the more specific ecological roles of the species involved. Loops, that is food chains for which start and end point are identical, appear to be comparatively rare in food webs, except for the simple case of two species eating each other (Stouffer et al., 2007).

These observations motivate the assignment of species to trophic levels. Intuitively, a species' trophic level is the length of a food chain linking it to a producer. However, different food chains of this type can have different lengths, leading to different values for the trophic level of a species. This phenomenon is also called omnivory, and the omnivory motif is a special case. There are different schemes for resolving the ambiguities of trophic level resulting from omnivory, all coming with their own strengths and weaknesses (Williams and Martinez, 2004a). A simple solution is to consider only the shortest food chains. By this rule, the trophic level of the producers in Figure 3.1a is 1, that of species 13 is 3, and all other species are at trophic level 2. The rule assigns species 12 to level 2, even though it is linked to species 5 through the long chain 5 → 6 → 7 → 9 → 12.

A more sophisticated approach, attributed to Heald (1975) and Levine (1980), is to define the trophic level of a consumer as 1 plus the mean trophic level of its resources, where resources are weighted according to the flows of energy or biomass that they contribute to the diet of the consumer population. For food webs without loops this calculation is straightforward, in the presence of loops the assignment of trophic levels requires solving a linear system of equations (Levine, 1980). Obviously, this diet-weighted trophic level can attain non-integer values in the presence of omnivory. Nevertheless, values close to integers are preferred (Section 19.2), so that rounding trophic levels to the nearest integer number can be legitimate simplification.

The idealization that all species in a food web have a well-defined integer trophic level leads to the second conception of food chain. The food chain of an ecological community is the sequence of sets of species grouped by trophic level. Highly simplifying, one could, for example, assign species 1–5 in Figure 3.1a to level 1, species 6–11 to level 2, species 12 and 13 to level 3, and then summarize the structure of the community by the food chain shown in Figure 3.1c. An application of this concept of food chain is the derivation of simplified budgets for the flows of mass and energy through communities, carried out in Section 21.

3.2 Physical Quantities and Dimensions

A physical quantity is the product of a number and a unit of measurement:

(3.1) equation

A practice that proved useful in the physical sciences and that might find its place in ecology as well is to distinguish clearly between the physical dimensions of a quantity and the units in which it is measured. A physical quantity can be expressed using different units, but it always has the same dimensions. Dimensions of physical quantities can be expressed as products of a few basic physical dimensions raised to some power. The notation for basic dimensions varies; I will here use capitalized English words. In ecology, the three basic dimensions Length, Time, and Mass are generally sufficient.

The specification of a point in time such as

(3.2) equation

is a quantity of dimensions Time, as is the time it takes to boil an egg

(3.3) equation

The dimensions of other physical quantities can be obtained by combining the fundamental dimensions: An area has dimensions Length², a velocity Length/Time. The dimensions of the product (quotient) of two quantities are the product (quotient) of the dimensions of the two factors. If the dimensions of a product or quotient of physical dimensions fully cancel each other, the resulting quantity is said to be dimensionless. Two quantities can be added, subtracted, or compared to each other if and only if they have the same dimensions.

Part II

Elements of Food-Web Models

4

Energy and Biomass Budgets

The idea of a food web as a network of flows of energy or biomass through an ecological community requires some clarification to become a well-defined concept. This chapter first addresses the problem of how, conceptually, to quantify the flows, and then briefly summarizes important applications. Later chapters will complement this conception of food webs as flow networks with the idea of food webs as interaction networks.

4.1 Currencies of Accounting

The biomass flowing through food webs carries both material to build consumer tissue from resource tissue and energy to be used by consumers for various purposes. Much can be learned from disintegrating these flows into various components, e.g., from considering only the energy flows, or flows of biomass split into different elemental components [ecological stoichiometry, see Sterner and Elser (2002)], by distinguishing between water, fat, carbohydrates, proteins, and fibres, or by tracing the amounts of some more complex molecules, such as vitamins or pollutants passed from resources to consumers.

To keep things simple, none of this shall be done here. Consistent models can be built without taking these complications into account. In the approximation that the composition of the biomass of each population does not vary much through time, mismatches between the needs of consumers and the composition of their resources can be modeled by reduced efficiencies at which consumers utilized the biomass of their resources. This means, of course, that this efficiency will generally be different for each consumer-resource pair.

Even after deciding to trace flows through food webs in terms of biomass, there is still a choice to be made: One can do the accounting either in terms of dry or in terms of wet biomass. Many arguments speak in favour of using dry mass or even just carbon content. For example, artifacts might arise when using wet biomass, such as jellyfish that grow by 2 gram by feeding on 1 gram of resource. Besides, when discussing the flow of resource biomass through a consumer's body, as briefly done below, the notion of wet mass loses its meaning. However, many empirical results are reported in terms of wet mass, 23 for the obvious reasons that it can be measured more easily and non-lethally. Most models to be developed here can be interpreted in terms of both, wet and dry mass, so that the decision can be postponed to specific applications of these models. In illustrative examples I will refer to wet mass, for simplicity.

Another decision to be made is whether to consider absolute biomasses of species (or their population sizes) in a given, fixed geographic area, or whether it is more appropriate to operate with biomass densities and, correspondingly, biomass flow densities through food webs. However, this decision is not crucial. Spatial structure is not explicitly taken into account in the models developed below. Rather, the models build on the idealization of well-defined, distinct ecological communities occupying distinct habitats. Within this paradigm, formulations in terms of population biomasses or numbers are sufficient, and usually these will be used for brevity. Occasionally, however, formulations in terms of densities are favoured when these are more intuitive. For empirically parametrised models, for example, data normalized to area is better available and comparable. Future work might fruitfully put more emphasis on developing and analysing general, spatially explicit food-web models.

4.2 Rates and Efficiencies

Flows through food webs are quantified by flow rates. However, the word rate can refer to physical quantities of different kinds of dimensions (Section 3.2). For example, it can mean a flow rate of dimension Mass/Time, or it can refer to a proportional rate, such as an interest rate on a loan, with dimension 1/Time. It is therefore good practice to specify the dimensions of rates when these are not clear from the context.

To avoid ambiguities, flow rates (e.g., Mass/Time) generally need to be defined with respect to some interface through which the flow goes. The rate of flow through a pipe, for example, can be defined and measured by specifying a cross-section through the pipe and measuring the amount of mass m+ passing through it in one direction in a given time interval of length Δt, subtracting the amount m− passing through it in the opposite direction, and computing the flow rate as (m+ − m−)/Δt. Under the special circumstances that the material flowing through the pipe is incompressible and there is no leakage, the measured flow rate will be independent of where the cross section is located in the pipe, so that one can speak of the rate of flow through the pipe. In general, however, this is not the case.

In ecology, such special circumstances do not hold either, so that flow rates need to be defined relative to particular interfaces. The complexity of living organisms, however, requires defining more abstract kinds of interfaces than in the case of pipe flow. Flows through three interfaces are both of interest to modelling and comparatively easy to determine empirically. The first is the mouth or another corresponding body opening of a consumer. The flow rate defined by this interface is called the ingestion rate. The second interface is given by the walls of the digestive tract of an organism, with the corresponding flow rate called the assimilation rate. This flow is entirely decoupled from the flow of the water through an organism, and so it makes little sense to define it in terms of wet biomass. However, it can be quantified in terms of a nominal equivalent wet biomass by multiplying with an appropriate conversion factor (Hendriks and Mulder, 2008). Assimilation of consumers can be measured as the difference between ingestion and defecation, with the defecation rate defined by the cross section through another body opening. For most producers, in particular autotrophs, one can define an assimilation rate by the amount, per unit time, at which the organism generates organic matter from inorganic matter.

The third interface is the body opening towards an organism's respiratory organ (which may be the organism's entire surface), where the flow of interest is only that of carbon in the form of CO2. This is organic carbon that was oxidised in the body to support metabolic activity, and so is not available for the formation of body tissue anymore. The corresponding flow rate is called the respiration rate. As for the assimilation rate, equivalent flows of dry or wet biomass can be computed by multiplying with appropriate conversion factors (Hendriks, 1999). Material that is assimilated and not lost through respiration is transformed into body tissue of either the organism itself or its unborn offspring. The corresponding flow rate is called the production rate, or just production.

For any of the rates defined above one can define a corresponding proportional rate by dividing by the body mass of the organism or by the biomass of the population considered. This rate is often denoted by the same name. If not clear from the context, this ambiguity needs to be removed by specifying the dimensions.

By comparing different rates, one can compute dimensionless indices specifying the efficiencies of processes. The proportion of mass ingested that is assimilated is called the assimilation efficiencies¹ (below denoted by ε), the proportion of mass assimilated that is transformed into body tissue is called the production efficiency. If one assumes that each organism in a community can be assigned a sharply defined integer trophic level, one can also define an ingestion efficiency as the proportion of production at one trophic level ingested by the next higher trophic level.

Lindeman (1942) defined the progressive efficiency as the ratio of assimilation rates (Mass/Time) at adjacent trophic levels. It is given by the product of production efficiency at one level, and ingestion efficiency and assimilation efficiency at the next higher trophic level. The choice of comparing assimilation efficiencies at adjacent trophic levels can be motivated by noting that assimilation is the first processing step at which producers and consumers become comparable. It is also plausible from a biological point of view, considering that different species ingest indigestible parts of their resources to different degrees, depending on the mechanics of their foraging strategy. Yet, modern literature usually considers instead the ratio of production at subsequent trophic levels, which is called transfer production efficiency (Brey, 2001), or trophic transfer efficiency. Trophic transfer efficiency is empirically more easily accessible than Lindeman's progressive efficiency. Transfer efficiency equals the product of ingestion-, assimilation-and production efficiency at a given trophic level.

4.3 Energy Budgets in Food Webs

For practical reasons, flows through food webs are nearly always characterized in terms of ingestion rates apportioned to the different resources of a