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How trophic interaction strength depends on traits

A conceptual framework for representing multidimensional trophic niche spaces

Theoretical Ecology Aims and scope Submit manuscript

Abstract

A key problem in community ecology is to understand how individual-level traits give rise to population-level trophic interactions. Here, we propose a synthetic framework based on ecological considerations to address this question systematically. We derive a general functional form for the dependence of trophic interaction coefficients on trophically relevant quantitative traits of consumers and resources. The derived expression encompasses—and thus allows a unified comparison of—several functional forms previously proposed in the literature. Furthermore, we show how a community’s, potentially low-dimens ional, effective trophic niche space is related to its higher-dimensional phenotypic trait space. In this manner, we give ecological meaning to the notion of the “dimensionality of trophic niche space.” Our framework implies a method for directly measuring this dimensionality. We suggest a procedure for estimating the relevant parameters from empirical data and for verifying that such data matches the assumptions underlying our derivation.

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Acknowledgements

The authors thank Jacob Johansson for inspiring discussions. A.G.R. gratefully acknowledges support from a Beaufort Marine Research Award by the Marine Institute, under the Sea Change Strategy and the Strategy for Science, Technology and Innovation, funded under the Irish National Development Plan (2007–2013). Å.B. and U.D. gratefully acknowledge support from the European Marie Curie Research Training Network on Fisheries-induced Adaptive Changes in Exploited Stocks (FishACE), funded through the European Community’s Sixth Framework Programme (Contract MRTN-CT-2004-005578). U.D. gratefully acknowledges additional financial support from the Austrian Science Fund, the Vienna Science and Technology Fund, and the European Science Foundation.

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Appendix

Appendix

A Empirical determination of parameters relating phenotypic traits and interaction strength

In this appendix, we shortly describe a procedure for estimating the factor a 0, the vector b, and the matrix C in the expression for the interaction strength in Eq. 3, based on measured trait vectors t i and interaction strengths \(\hat a_{ij}\) for all pairs of species (i, j) with i, j = 1,..., N in a community of N species.

Presumably, the most laborious aspect of this procedure is the measurement of the empirical interaction strengths \(\hat a_{ij}\). In principle, it would be desirable to obtain these quantities from measured functional responses. Due to practical constraints, one will often assume linear functional responses and, for each pair (i, j), estimate the interaction strength \(\hat a_{ij}\) as the biomass flow density from species i to species j divided by the biomass densities of i and j (other currencies of flows and abundances can be used as well). Procedures for measuring flows and abundances at the community level have, for example, been described by de Ruiter et al. (1995) and Fath et al. (2007). A set of phenotypic traits, adequate for the community under consideration, has to be defined and the traits have to be measured for each species. One phenotypic trait vector t i is obtained for each species i.

Finding the parameters entering the expression for the interaction strength in Eq. 3 might, at first sight, appear to be a generalized linear regression problem. In practice, however, many empirical interaction strengths will be zero, which requires ad hoc assumptions when applying the logarithmic scale. Ignoring these data points is not an option, since the fact that one species does not consume another contains important ecological information. For these reasons, we propose using nonlinear regression of a ij directly to fit the parameters (Daniel and Wood 1999; Motulsky and Christopoulos 2004) by choosing parameters in such as way that, in a well-defined sense, the residual errors

$$ \varepsilon_{ij}= \hat a_{ij} -a_0 \exp \left[ \mathbf{b} \binom{\mathbf{t}_i}{\mathbf{t}_j}+ \frac{1}{2}\binom{\mathbf{t}_i}{\mathbf{t}_j}^{\mathrm{T}}C\binom{\mathbf{t}_i}{\mathbf{t}_j} \right] $$
(8)

are minimized. The standard assumption of independent, normally distributed errors suggests least-square fitting, i.e., a choice of a 0, b, and C so that the sum \(\sum_{ij} \varepsilon_{ij}^2\) is minimized. More complex error models lead to other criteria, e.g., via likelihood maximization, and might suggest weighting errors differently. However, it has to be kept in mind when solving this problem that the particular form we assumed for the dependence of interaction strength on phenotypic traits was motivated by qualitative considerations. One should therefore not generally expect more than a semiquantitative approximation of the data by this expression. The residuals ε ij thus combine contributions from measurement errors of the \(\hat a_{ij}\) with approximation errors. Rather than weighting each data point (i, j) according to the accuracy of the measured interaction strength \(\hat a_{ij}\), we therefore recommend to weight the data more evenly.

The total number of parameters to be fitted is 1 + 3m + 2m 2, where m is the number of traits considered. For m = 10, this corresponds to 231 parameters, and the number of measured \(\hat a_{ij}\) should be significantly larger than this. These data requirements might first appear daunting. However, for most resource–consumer pairs (i, j), expert knowledge will be sufficient to exclude trophic interactions, and \(\hat a_{ij}\) can then be set to zero without measurements. Expert knowledge will also be sufficient to exclude significant effects of certain phenotypic traits on either foraging or vulnerability. The corresponding entries in b and C can then be set to zero a priori, reducing the number of parameters to be estimated and, thus, the demands on data.

For the validation of our approximation of interaction strength, we recommend, again keeping its semiquantitative nature in mind, to inspect graphs relating measured interaction strengths to the predictions by our model with fitted parameters. Strong outliers that are not explained by measurement errors in \(\hat a_{ij}\) do not necessarily show that the model Eq. 8 is inappropriate, they could also indicate that important phenotypic traits have not been included in the analysis. Inspection of the underlying ecology and inclusion of the relevant phenotypic traits should improve the correlation between measurements and model prediction. If the trophic niche space of a community is high-dimensional, that is, if the number of relevant trophic traits is large, the number of relevant phenotypic traits must be large, too. It may then be difficult in practice to achieve good model fits, even when the underlying model (Eq. 8) is a good description. This situation can be identified by verifying that (1) the correlation between predicted and measured interaction strength improves as the number m of phenotypic traits considered increases, even when carefully guarding against over-fitting (e.g., by employing cross-validation procedures, Efron 1987) and (2) the dimensionality of trophic niche space continues to increase with m. When only enhanced correlations (1) are observed with increasing m, but not increased dimensionality (2), this is a signature of a low-dimensional niche space with the few relevant trophic traits depending on a large number of phenotypic traits. When the correlation between model and data does not improve with the number of traits considered, this indicates that the models given by Eq. 8 or, equivalently, by Eq. 3 may be inappropriate. The main assumption underlying our theory is, thus, open to empirical falsification.

B Identification of trophic traits

Below we describe the detailed procedure for deriving Eq. 4 from Eq. 3.

By the spectral decomposition theorem, the symmetric matrix C can be represented as \(C=\sum_i \mathbf{e}_i \lambda_i \mathbf{e}_i^{\mathrm{T}}\) in terms of an orthonormal set of 2 m eigenvectors e i and the corresponding real eigenvalues λ i . Without loss of generality, we assume that the eigenvalues are sorted so that |λ 1| ≥ |λ 2| ≥ ... ≥ |λ 2m |. We, define for each i = 1,...,2m partial eigenvectors \(\mathbf{e}_i',\mathbf{e}_i''\in R^m\) representing the components of e i referring to resource and consumer, respectively, that is,

$$ \mathbf{e}_i=\binom{\mathbf{e}_i'}{\mathbf{e}_i''}. $$
(9)

The 2m-dimensional vector v can now be represented as

$$ \mathbf{v}=\sum\limits_{i=1}^{2m} \mathbf{e}_i \left(\mathbf{e}_i^{\mathrm{T}} \mathbf{v}\right)=\sum\limits_{i=1}^{2m} \mathbf{e}_i (V_i-G_i), $$
(10)

with vulnerability traits V 1,...,V 2m of r and corresponding (raw) foraging traits G 1,...,G 2m of c defined by

$$ V_i:={\mathbf{e}_i'}^{\mathrm{T}} \mathbf{t}_r,\quad G_i:=-{\mathbf{e}_i''}^{\mathrm{T}} \mathbf{t}_c. $$
(11)

Note that, since the 2m values V i are determined by linear projections of the m components of t r , at most m values V i are independent. Similarly, there are at least m linear relationships between the 2m values G i .

Inserting Eq. 10 into expression 3 for ln a rc yields

$$ \ln a_{rc}=\ln a_0+\sum\limits_{i=1}^{2m} \mathbf{b}^{\mathrm{T}}\mathbf{e}_i(V_i-G_i) +\frac{1}{2}\sum\limits_{i=1}^{2m} \lambda_i (V_i-G_i)^2. $$
(12)

To simplify this expression, consider the last sum first. Note that the upper bound \(|\mathbf{t}|<t_{\text{max}}\) (see the section “Theory”) implies an upper bound on |V i  − G i |: with \(|\mathbf{v}|^2=|\mathbf{t}_r|^2+|\mathbf{t}_c|^2\le 2 t_{\text{max}}^2\), one obtains

$$ |V_i-G_i|=|{\mathbf{e}_i}^{\mathrm{T}} \mathbf{v}|\le \sqrt{2}\, t_{\text{max}}. $$
(13)

Dropping terms with small |λ i | from the second sum will therefore often yield good approximations for ln a rc .

We denote by n the number of terms in the sum that need to be retained to maintain a given level of model accuracy. We define the trophic baseline traits (note the lower bounds of the sums) as

$$ V^*:=\sum\limits_{i=n+1}^{2m} \mathbf{b}^{\mathrm{T}}\mathbf{e}_i V_i \quad\text{ and }\quad F^*:=-\sum\limits_{i=n+1}^{2m} \mathbf{b}^{\mathrm{T}}\mathbf{e}_i G_i, $$
(14)

the offsets between matched vulnerability and raw foraging traits as

$$ d_i:=\frac{\mathbf{b}^{\mathrm{T}}\mathbf{e}_i}{\lambda_i}\quad(i=1,\ldots,n), $$
(15)

the (adjusted) foraging traits as

$$ F_i:=G_i-d_i\quad(i=1,\ldots,n), $$
(16)

and the scaling factor as

$$ a_1=a_0 \exp\left(-\sum\limits_{i=1}^n \lambda_i d_i^2/2\right). $$
(17)

Inserting these definitions into the logarithmic form of Eq. 4, i.e., into

$$ \ln a_{rc}=\ln a_1+V^*+F^*+\frac{1}{2}\sum\limits_{i=1}^{n} \lambda_i (V_i-F_i)^2, $$

it is readily verified that this is equivalent to Eq. 12 with the second sum truncated at i = n. As discussed in the section “Theory,” chances are good that the first 2n values V i and F i (i = 1, ...n) of a species are all numerically (but not necessarily statistically) independent of each other, even though this is certainly not the case for the original set of 2 × 2m values V i and G i (i = 1 ..., 2m) defined by Eq. 11.

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Rossberg, A.G., Brännström, Å. & Dieckmann, U. How trophic interaction strength depends on traits. Theor Ecol 3, 13–24 (2010). https://doi.org/10.1007/s12080-009-0049-1

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