Phylogenetic Correlations (I)

Dear Foodwebs List Members

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******* First of all a happy year 2009 for all of you! *********
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With this message I would like to initiate a small tutorial on

     Phylogenetic correlations in food-web topology.  

Cheers,
Axel

** PREFACE

While it has long been recognized that phylogenetic correlations can
induce structure in ecological data in general, the role of
phylogenetic correlations for food webs topology is often
underestimated.  Some of you might not even be sure what "phylogenetic
correlation" in this context means.  The purpose of this first message
is to clarify this question.  Later messages, following in loose
sequence, will address other issues, e.g., empirical evidence for
phylogenetic correlations and typical patterns characteristic for
phylogenetically structured food webs.

I am writing this tutorial because, in my view, unawareness of the
explanatory power of phylogenetic correlations has become a major
obstacle for progress in food-web science.  Progress which is critical
if food-web science is to contribute to the mitigation of biodiversity
loss, either by informing management or by contributing to a better
founded valuation of biodiversity, morally or monetarily.  For those
of us, including myself, who have put efforts into the investigation
of phylogenetic correlations in food webs already, this is of course
also an opportunity to showcase our work.  But I hope to convince you
that the subject is too important to be left alone, independent of
whether you have worked on it before or not.

A mailing list seems to be the ideal medium for such a tutorial,
especially because of its strongly interactive nature.  If you are
following the tutorial and you find points that are unclear (which is
then entirely my fault) or you can offer additional or alternative
perspectives on a topic, I strongly encourage you to share your
questions and views on the list (simply by replying to
<foodwebs@...>) or otherwise to write me directly
(<axel@...>).  Hopefully, many of you will find these
messages interesting.  To the others I would like to extend my
apologies already now, in case that messages become too frequent or
too long.

** Part I: WHAT ARE PHYLOGENETIC CORRELATIONS IN FOOD-WEB TOPOLOGY? 

PHYLOGENETIC CORRELATIONS

Related species (or individuals) have similar ecological (or
biological) characters.  More precisely: When ecological characters
are described by quantitative measures, these measures are
statistically more correlated in ensembles of species pairs with close
phylogenetic (evolutionary) relationships than for ensembles of
distantly related species pairs.  This phenomenon is a keystone of
Darwin's theory.  It is known under various names, such as
"phylogenetic constraints", "phylogenetic signal", or "phylogenetic
correlation".

Phylogenetic correlations can lead to a violation of the hypothesis of
statistical independence of data points that underlies most standard
methods of statistical data analyses.  Much work has therefore been
devoted to the question how the effect of phylogenetic correlations
can be taken into account in order to isolate those correlations
attributable to causal relationships alone.  A common approach is the
use of "phylogenetically independent contrasts".  Another option is to
generate the distributions of test statistics under the null
hypotheses in question in Monte Carlo simulation that mimic evolution,
and thus naturally produce phylogenetic correlations.  For an
extensive discussion, see Garland et al.'s introductory review
http://findarticles.com/p/articles/mi_qa3746/is_199904/ai_n8829021.

PHYLOGENETIC CORRELATIONS IN FOOD WEB TOPOLOGY

How does the similarity of related species affect food-web topology?
Intuitively, we expect that similar species have similar consumers and
resources.  Apart from subtleties discussed below, this implies that
related species tend to share consumers and resources.  As a simple
example, a bird eating one kind of seed is likely to eat other kinds
of seeds but perhaps no insects, and a bird eating one kind of insect
will eat other insects but perhaps no seeds.  

In order to make this idea precise, consider food-web topologies
represented by adjacency matrices a_ij (with the indices i,j running
over all species in the web), where a_ij=1 when i is eaten by j and
otherwise a_ij=0.  The intuition that related species share consumers
can be formalized by the statement that a_ik and a_jk are, for fixed
k, more correlated for ensembles of pairs of closely related resources
(i,j) than for ensembles of distantly related resource pairs.  The
corresponding statement for similarity of resources refers to
correlations between a_ki and a_kj.  Yes, correlations can be computed
for binary variables such as the matrix elements a_ij.  But of course
one expects the same kind of correlation structure also if, instead of
a_ij, a suitable quantitative measure of link strength l_ij is used.In
short, trophic similarities of related species would directly be
reflected in phylogenetic correlations of trophic links or link
strengths.

To improve statistical power, practical analyses will often consider
correlations computed over links to all consumers and/or all resources
k at the same time.  One can expect that high values of
corr(a_ik,a_jk) and/or corr(a_ki,a_kj) computed for fixed (i,j) over
all k can directly be associated with the phylogenetic distance
between i and j.  This is how Cattin et al. did the first statistical
demonstration of phylogenetic correlations in food webs
(http://www.unifr.ch/biol/ecology/bersier/publications/Nature_Cattin+_2004.pdf).
Analyses making more direct use of the established statistical
techniques have, to my knowledge, so far remained preliminary (see
Ives and Godfray,
http://ora.ouls.ox.ac.uk/objects/uuid:e57495e5-9d2e-4858-8e24-8fc7b7445615/datastreams/ATTACHMENT01).

PHYLOGENETIC CORRELATIONS OF TRAITS DETERMINING TROPIC LINKS

Trophic interactions between species are not immediately hereditary
traits of either consumers or resources.  For example, we expect two
sympatric, closely related species to interact similarly with an
invading alien species, even though such interactions have never
occurred before.  This expectation is based on the intuition that
trophic interactions are determined by traits of consumers and
resources, and phylogenetic correlations in these traits lead to
correlated interaction strengths.

The subtlety mentioned above now occurs when consumers are highly
specialized to particular resources, and consumers and resources
co-evolve (e.g., in the case of parasites).  This can lead to
situations where species A eats B but not B', and A' eats B' but not
B, that is, there is no overlap in resources or consumer sets, and
yet, from the resource set of A and the close phylogenetic
relationships between A and A' as consumers and B and B' as resources,
one may guess that A' eats B'.  This is an example of a manifestation
of phylogenetic correlations in food webs which is not reducible to
phylogenetic correlations of link presences/absences.  It has to be
explained in terms of correlations in traits determining links.

The relationship between traits and trophic link strength (or
presence/absence of links) is, however, not well understood.  To
illustrate, nevertheless, how phylogenetic correlations of traits can
affect correlations of link strength, consider a simple example where
the link strength l_ik for consumption of different resource species i
by some fixed consumer k is determined additively by contributions
from three traits of the resource species.  Link strength can then be
decomposed as

  l_ik = (some constant) + u_ik + v_ik + w_ik,

with the value of each addend determined by one trait.  One can also
consider multiplicative contributions of the three traits to link
strength.  In this case l_ik as given above is interpreted as the
logarithm of link strength.

Now, assume that the contributions of the three components are
uncorrelated (a change of trait variables to the principal components
contributing to l_ik might achieve this).  Then

  cov(l_ik,l_jk) = cov(u_ik,u_jk) + cov(v_ik,v_jk) + cov(w_ik,w_jk) 

and hence

  var(l_ik) = var(u_ik) + var(v_ik) + var(w_ik) 

giving the correlation

  corr(l_ik,l_jk) = cov(l_ik,l_jk)/var(l_ik).

In the simple case that the variances of all components are equal, that
is, if var(u_ik) = var(v_ik) = var(w_ik), the correlation of link
strength is easily seen to reduce to

  corr(l_ik,l_jk) = [corr(u_ik,u_jk) + corr(v_ik,v_jk) + corr(w_ik,w_jk)]/3,

that is, the size of phylogenetic correlation of links strength is
simply the mean over all traits of the of phylogenetic correlations of
their contributions to l_ik.  When the variances of the contributions
differ, one obtains a corresponding weighted mean with the weights
equal to the variances.

To estimate the phylogenetic correlations of the component
contributions to link strength, it may often be justified to assume
these contributions to depend linearly on the trait values in a first
approximation.  In such a case, the phylogenetic correlations of the
contributions to link strength equal the phylogenetic correlations of
the corresponding trait values.

This simple calculation suggests that the order of magnitude of
phylogenetic correlations of link strength (1) is given by the
magnitude of the phylogenetic correlations of the main traits
determining link strength, and (2) is independent of the number of
traits determining link strength.

OUTLOOK

In the next message, I will discuss the question how strong
phylogenetic correlations in food webs typically are.