Appendix B. Statistics used for functional response analysis.
Data analysis for this experiment was a three step process and followed the procedures outlined by Juliano (2001). We first determined the shape of the functional response curves for each of the three predator densities of each crab species (six curves total) using separate polynomial logistic regressions. All six were saturating curves, indicating that predation followed either a type II or type III functional response. We fit a cubic model to each curve and observed the sign of the linear term in the polynomial equation to differentiate between type II and type III curves (a negative term indicates type II and a positive term together with a negative quadratic term indicates type III response). When the cubic term was not significant, it was removed and the analysis was repeated (there were no instances when quadratic polynomials had insignificant terms). Conclusions from these analyses were verified by visual inspection of plots of proportion of prey eaten vs. initial prey density.
We next estimated the parameters of the functional response equation using nonlinear least squares regression for each of the six curves. This was done to understand how predator density influenced the mechanisms of predation, specifically prey handling time and searching efficiency, for C. maenas and H. sanguineus. As logistic regression indicated type III functional responses for both predators at all three densities, we fit the data to a type III functional response model that accounts for prey depletion, as occurred in our experiments (this is the integrated form of the type III functional response equation given by Hassel 1978 where searching efficiency is a function of prey density, and is equation 10.5 from Juliano 2001):
Where Ne and N0 are the number of mussels consumed and the initial number of mussels offered, T is the duration of the experiment, Th is the handling time (the time required to consume a single mussel), and b, c, and d are constants that relate the attack rate to prey density (i.e., the searching efficiency). The purpose of the analysis was to estimate values for Th, b, c, and d.
Parameters values that were not significantly different from zero (based on 95% CI) were removed and the analysis was repeated. For each of the six functional response curves, c and d were not different from zero and were thus removed, resulting in the minimal form of the type III functional response equation. Nonlinear regression thus resulted in an estimate (mean and SE) of Th and b for each curve.
Estimates of Th and b were then compared between each pair of predator. Because we wanted to compare parameter estimates between three different predator densities for each species, an ANOVA would have been ideal. However, nonlinear regression analyses provide a single estimate and standard error for each of the parameters, rather than replicate estimates that are necessary to perform an ANOVA. We therefore made pair wise comparisons using individual t tests (Glantz and Slinker 1990, Juliano 2001, Fussmann et al. 2005).
Finally, our overarching goal was to determine whether C. maenas’ and H. sanguineus’ predation was explained better by the prey dependent model or by the ratio dependent model. The model in Eq. B.1 represents the prey dependent model. The ratio dependent model was obtained by replacing N0 with N0/Pm (Hassell and Varley 1969), where P is the number of predators in an enclosure, and m is an interference coefficient. When m = 0, the model reduces to the prey dependent form. Ratio dependence is modeled when m = 1. Intermediate values of m represent varying degrees of predator dependence. We fit the data for each predator species (across all predator densities simultaneously) to models ranging from prey dependence to ratio dependence at intervals of m = 0.1. We determined which of these models fit the data best by choosing the model with the smallest residual sum of squares (i.e., the one with the least amount of variability that was not explained by the model) (Fussmann et al. 2005).
All statistical analyses were conducted in SAS version 9.1. Program code for logistic and nonlinear regressions was modified from that given in Juliano (2001 supplementary material).
Fussmann, G. F., G. Weithoff, and T. Yoshida. 2005. A direct, experimental test of resource vs. consumer dependence. Ecology 86:29242930.
Glantz, S. A., and B. K. Slinker. 1990. Primer of Applied Regression and Analysis of Variance. McGraw-Hill, New York< New York, USA.
Hassel, M. P. 1978. The Dynamics of Arthropod Predator-Prey Systems. Princeton University Press, Princeton, New Jersey, USA.
Hassel, M. P., and G. C. Varley. 1969. New inductive population model for insect parasites and its bearing on biological control. Nature 223:11331137.
Juliano, S. A. 2001. Nonlinear curve fitting: predation and functional response curves. Pages 178196 in S. M. Scheiner and J. Gurevitch, editors. Design and Analysis of Ecological Experiments. Oxford University Press, New York, New York, USA.