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Part I
Preliminaries
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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:
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).
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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 parame...
