## Abstract

The literature on modelling a predator’s prey selection describes many intuitive indices, few of which have both reasonable statistical justification and tractable asymptotic properties. Here, we provide a simple model that meets both of these criteria, while extending previous work to include an array of data from multiple species and time points. Further, we apply the expectation–maximisation algorithm to compute estimates if exact counts of the number of prey species eaten in a particular time period are not observed. We conduct a simulation study to demonstrate the accuracy of our method, and illustrate the utility of the approach for field analysis of predation using a real data set, collected on wolf spiders using molecular gut-content analysis.

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## Acknowledgments

The information reported in this paper (No. 15-08-008) is part of a project of the Kentucky Agricultural Experiment Station and is published with the approval of the Director. Support for this research was provided by the University of Kentucky Agricultural Experiment Station State Project KY008055 and the National Science Foundation Graduate Research Fellowship Program.

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Handling Editor: Bryan F. J. Manly.

## Appendix: Varying trap efficiency

### Appendix: Varying trap efficiency

When there is concern about trap efficiency varying across time, we recommend the following reparameterization of \(c_{st} = \lambda _{st}/\gamma _{st}\). Let

such that \(\sum _{s=1}^S k_{st} = 1\), for all *t*. The parameter \(d_t\) allows for changes in trap efficiency or in actual abundance over time. If such changes are believed to be present, this reparameterization allows for a more natural interpretation. The parameter \(k_{st}\) models relative consumption to encounter rates, as expected.

Since this reparameterization amounts to a linear transformation of the parameters, the asymptotic distribution for \(\hat{d}_t\) can be evaluated directly. Because of the constraint \(\sum _{s=1}^S k_{st} = 1\), we find that \(d_t = \sum _{s=1}^S c_{st}\) or written in matrix notation \(\mathbf {d} = \mathbf {1}' \mathbf {c}\), where \(\mathbf {d} \in \mathbb {R}^T\). The asymptotic distribution of \(\hat{\mathbf {d}}\) follows immediately, \(\hat{\mathbf {d}} \mathop {\sim }\limits ^{\cdot } \mathcal {N}(\mathbf {1}'\mathbf {d}, \mathbf {1}'\varSigma \mathbf {1})\), where \(\varSigma \) is the covariance matrix of the asymptotic distribution of \(\hat{\mathbf {c}}\).

More importantly, we can find the asymptotic distribution of \(\hat{\mathbf {k}}_t = (k_{1t}, \ldots , k_{St})' \in \mathbb {R}^S\). We make use of the delta method (Serfling 2001). The asymptotic variance relies on the derivatives, \(d k_{rt} / d c_{st}\). Let \(c_{\cdot t} = \sum _{s=1}^S c_{st}\). Because \(k_{st} = c_{st}/ c_{\cdot t}\), we can find the necessary derivatives in two cases

Thus, the asymptotic distribution of \(\hat{\mathbf {k}}_{t}\) is \(\mathcal {N}(\frac{c_{st}}{c_{\cdot t}}, (\frac{d \mathbf {k}_t}{d \mathbf {c}_t}) \varSigma (\frac{d \mathbf {k}_t}{d \mathbf {c}_t})')\).

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Roualdes, E.A., Bonner, S.J., Whitney, T.D. *et al.* Formal modelling of predator preferences using molecular gut-content analysis.
*Environ Ecol Stat* **23, **317–336 (2016). https://doi.org/10.1007/s10651-016-0341-3

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### Keywords

- Electivity
- Expectation–maximisation
- Food web analysis
- Generalist predators
- Predator–prey interactions