Transcript The large and small of it: Quantifying size structure in

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.
Email: [email protected]
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
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.
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.
a
b
Figure 2.
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 models5 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.
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
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 Hij is the handling time of predator j consuming prey i, Mi is the mass
of an individual of prey species i, and Mj is the mass of an individual of
predator species j. The exponents hi and hj 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 (ti, where i denotes a
prey) and a predator specific handling time modifier (tj, where j denotes a
predator). They have a multiplicative effect on handling times.
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
importance of these additional traits for handling times relative to the
importance of body mass. Examples of the resulting webs are depicted in figures
c
3 & 4).
d
e
g
h
Figure 3.
f
Summar
In order to understand the importance of size
structure
y in food webs we need to be able to quantify and model it. We
i
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.
References
Figure 4.
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.
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.
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.
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.