1. The Large and Small of it: Quantifying Size Structure in Ecological Networks Aaron Thierry¹ & ², Andrew Cole¹, Owen Petchey¹, Andrew Beckerman¹, Phil Warren¹ & Rich Williams² ¹The Department of Animal and Plant Sciences, The University of Sheffield U.K. ²Microsoft Research, Cambridge, U.K. Researchers often treat ecological communities as networks, webs of interactions such as those between predator and prey (food webs) or pollinator and plant. These networks capture some of the complexity of an ecosystem and allow us to make comparisons between different habitats. Back in 1927 Charles Elton, one of the first people to look at patterns of structure in food webs, stated that “[body] size has remarkably great influence on the organisation of animal communities” 1. While this insight had long been overlooked there has recently been much interest in trying to understand how body size structures ecological networks 2-4. Here we propose a method by which we can quantify the size structure of an ecological network and demonstrate that by adapting an existing model for generating food webs, we can systematically explore the characteristics of webs through a spectrum of size structures. In order to quantify the different aspects of size structure within a community, and make comparisons between different systems, we advocate the generalisable approach of using bivariate relationships between different topological properties (as response variables) and body mass (as the explanatory variable). Stronger correlations would indicate a higher degree of size structure. Two important properties of a web which we can examine in this way are the generality and the vulnerability of a species. Generality is the number of prey a species has (this is also referred to as its in-degree). Whereas vulnerability is the number of predators a species has (also known as its out-degree), see figure 1. a) An example of a highly size structured web with a strong correlation, b) a barely size structured web with a weak correlation. The principle is the same for vulnerability except that the slope is expected to be reversed with larger organisms suffering less predation. We can then take the strength of Pearson’s measure of the generalism-mass correlation and vulnerability-mass correlation and plot a position for the web in a two dimensional plane. In figure 2., we have plotted these correlations for 15 of the most highly resolved food webs for which information on body sizes is currently available. It is clear that our measure shows that many of these food webs are size structured and that the degree of size structure differs greatly between webs. Figure 2. can also be used to examine where webs generated by different models are situated in this ‘size space’. We can see that three alternative models for generating food webs (Random, Niche and Cascade) differ considerably in the regions they occupy, this means that no single model can be used to explore the properties of webs throughout this spectrum. Size structure in food webs Measuring size structure Figure 1. The size structure of 15 real food webs is examined by plotting the Pearson's correlations of generalism-mass on the x-axis and vulnerability-mass on the y-axis. If we assume that the niche axis in the cascade and niche models 5 equates to body size, it is possible to see how well these simple structural models encompass the range, and type of size structuring seen in real webs over a similar range of connectance and species richness. Figure 2. In order to explore the role of size structure in shaping webs we need a model that allows us to generate a large range of size structures, a mechanistic model in particular would aid greatly in disentangling the determinants of different dimensions of size structure. In order to try and construct such a model we decided to modify the Allometric Diet Breadth Model (ADBM) 2, which constructs webs using optimal foraging theory to predict if a feeding interaction occurs between two species. The model relates foraging behavior to body size using allometric relationships. To modify the ADBM we added two extra traits to the handling time function ( handaling time is the length of time it takes to capture and digest a prey item). It depends on prey mass and predator mass such that: where H ij is the handling time of predator j consuming prey i, M i is the mass of an individual of prey species i, and M j is the mass of an individual of predator species j. The exponents h i and h j determine the scaling of handling time with prey and predator body mass respectively. We then added dependence of handling times on two other traits in addition to body mass. A prey specific handling time modifier (t i, where i denotes a prey) and a predator specific handling time modifier (t j, where j denotes a predator). They have a multiplicative effect on handling times. The sizes of t i and t j, these handling time modifiers were random numbers drawn from a uniform distribution with minimum of -W i (or -W j ) and maximum of +W i (or +W j ). Varying the values of W i and W j controlled the importance of these additional traits for handling times relative to the importance of body mass. Examples of the resulting webs are depicted in figures 3 & 4). Modelling size structure d ab c g f e ih The positions of the modelled webs (Figs 3, a...i) in the same size-structure space as in Fig 2, illustrating the way in which the model can generate webs distributed throughout the same area of size-structure space as most of the real webs, and existing models. Predation matrices with different size structures created using the modified ADBM. Consumers are on the x axis and resources on the y axis. Dots represent feeding interactions, when above the 1:1 diagonal they indicate that the consumer is eating a species smaller than itself. Figure 4. Figure 3. 1. Elton, C. (1927) Animal Ecology 2. Petchey, O.L., Beckerman, A.P., Riede, J.O. & Warren, P.H. (2008) Size, foraging and food web structure, PNAS. 3. Cohen, J.E., Jonson, T. & Carpenter, S.R., (2003) Ecological community description using the food-web, abundance and body size. PNAS. 4. Yvon-Durocher, G., Montoya, J.,Emmerson, M.C. & Woodward G. (2008) Macroecological patterns and niche structure in a new marine food web. Central European Journal of Biology. 5. Williams, R.J. & Martinez, N. (2000) Simple rules yield complex netwroks, Nature. References Summar y We now hope to use this tool kit to systematically examine the differences between real webs to discover where size based organising ‘rules’ are more or less important e.g. differences between aquatic and terrestrial habitats. Our intention is to use model webs to explore the degree to which extinction cascades affect food webs as the degree of size structure varies. Perhaps by so doing we shall also glean some insight into what effect changing size structure at the internal level has on emergent whole network level properties, such as connectance and food chain length. In order to understand the importance of size structure in food webs we need to be able to quantify and model it. We suggest that the measure of size structure used here, provides one good way of comparing size structure between webs. We have also shown that a new version of the ADBM model, which addresses a known limitations of other models by being able to generate non-contiguous diets, allows us for the first time to move through ‘size space’ within a common framework. We hope that this will prove to be a useful tool as we begin to explore the consequences of size structure. Email: a.thierry@shef.ac.uk