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					     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
                                                                                                                                                                               Email: a.thierry@shef.ac.uk
   ¹The Department of Animal and Plant Sciences, The University of Sheffield U.K.
   ²Microsoft Research, Cambridge, U.K.

            Size structure in food webs
  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.

            Measuring size structure
  In order to quantify the different aspects of size structure within a community, and make                                    Figure 2.
  comparisons between different systems, we advocate the generalisable approach of using                                                      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
  bivariate relationships between different topological properties (as response variables) and
                                                                                                                            y-axis. If we assume that the niche axis in the cascade and niche models5 equates to
  body mass (as the explanatory variable). Stronger correlations would indicate a higher degree
                                                                                                                            body size, it is possible to see how well these simple structural models encompass
  of size structure. Two important properties of a web which we can examine in this way are the                             the range, and type of size structuring seen in real webs over a similar range of
  generality and the vulnerability of a species. Generality is the number of prey a species has (this                       connectance and species richness.
  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.
                                                                                                                                   Modelling size structure
                                                                                                                              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
     Figure 1.                                                                                                               predict if a feeding interaction occurs between two species. The model
                                                                                                                             relates foraging behavior to body size using allometric relationships.
  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                            To modify the ADBM we added two extra traits to the handling time
  the slope is expected to be reversed with larger organisms suffering less predation.                                       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:
  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                              where Hij is the handling time of predator j consuming prey i, Mi is the mass
  figure 2., we have plotted these correlations for 15 of the most highly resolved food webs for                             of an individual of prey species i, and Mj is the mass of an individual of
  which information on body sizes is currently available. It is clear that our measure shows that                            predator species j. The exponents hi and hj determine the scaling of
  many of these food webs are size structured and that the degree of size structure differs greatly                          handling time with prey and predator body mass respectively. We then
  between webs. Figure 2. can also be used to examine where webs generated by different                                      added dependence of handling times on two other traits in addition to
  models are situated in this ‘size space’. We can see that three alternative models for generating                          body mass. A prey specific handling time modifier (ti, where i denotes a
  food webs (Random, Niche and Cascade) differ considerably in the regions they occupy, this                                 prey) and a predator specific handling time modifier (tj, where j denotes a
  means that no single model can be used to explore the properties of webs throughout this                                   predator). They have a multiplicative effect on handling times.
  spectrum.
                                                                                                                             The sizes of ti and tj, these handling time modifiers were random numbers
                                                                                                                             drawn from a uniform distribution with minimum of -Wi (or -Wj) and
                                                                                                                             maximum of +Wi (or +Wj). Varying the values of Wi and Wj controlled the
        a             b             c                                                                                        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).

        d             e             f
                                                                                                                               Summar             In order to understand the importance of size
                                                                                                                                     y
                                                                                                                              structure in food webs we need to be able to quantify and model it. We
        g             h                 i                                                                                     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
   Figure 3.                                                         Figure 4.                                                diets, allows us for the first time to move through ‘size space’ within a
                   Predation matrices with different                                   The positions of the modelled
                                                                                                                              common framework. We hope that this will prove to be a useful tool as
size structures created using the modified ADBM.                  webs (Figs 3, a...i) in the same size-structure
Consumers are on the x axis and resources on the y                space as in Fig 2, illustrating the way in which the        we begin to explore the consequences of size structure.
axis. Dots represent feeding interactions, when above             model can generate webs distributed throughout              We now hope to use this tool kit to systematically examine the differences
the 1:1 diagonal they indicate that the consumer is               the same area of size-structure space as most of            between real webs to discover where size based organising ‘rules’ are
eating a species smaller than itself.                             the real webs, and existing models.
                                                                                                                              more or less important e.g. differences between aquatic and terrestrial
                                                                                                                              habitats. Our intention is to use model webs to explore the degree to
   References        1. Elton, C. (1927) Animal Ecology 2. Petchey, O.L. , Beckerman, A.P., Riede, J.O. & Warren, P.H.        which extinction cascades affect food webs as the degree of size structure
(2008) Size, foraging and food web structure, PNAS. 3. Cohen, J.E., Jonson, T. & Carpenter, S.R., (2003) Ecological           varies. Perhaps by so doing we shall also glean some insight into what
community description using the food-web, abundance and body size. PNAS.                 4. Yvon-Durocher, G., Montoya,       effect changing size structure at the internal level has on emergent whole
J.,Emmerson, M.C. & Woodward G. (2008) Macroecological patterns and niche structure in a new marine food web.                 network level properties, such as connectance and food chain length.
Central European Journal of Biology. 5. Williams, R.J. & Martinez, N. (2000) Simple rules yield complex netwroks, Nature.

				
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