Appendix F. Fitting a Type I functional response to literature and experimental data.
We have taken two approaches to assess the fit of a Type I functional response. First, we asked whether there was a significant positive relationship between clearance rate and food concentration in our experimental data for food concentrations between 13 and 95 µg C/L (Fig. 4). The linear fit was highly significant (R^{2} = 0.31, P < 0.005) with a slope of 0.01 (SE = 0.003). Thus, the null (Type I) hypothesis of zero slope can be rejected, and a Type III functional response is statistically indicated by the positive slope. The value of the slope seems small but this is a function of the units of the axes. We discuss the magnitude of the change in clearance rate in the Discussion.
The second approach was to explicitly fit a Type I model to data from (1) the secondary analysis of mass-specific clearance rates (residuals for the no exhaustion data set in Table 2) and (2) the data from our experiment. This approach is somewhat problematic, because the Type I model with saturation is comprised of two functions joined by a conditional statement, and so must be fit by an iterative non-linear procedure (piecewise regression, see below). This precludes a rigorous statistical comparison between Type I and the other response types.
We fit the following model using the non-linear routine in JMP (SAS Institute):
If C < q, f = a |
If C ≥ q, f = (aq)/C |
where C is food concentration, f is clearance rate, q is the incipient limiting concentration (where f begins to decline with increasing C) and a is the maximum clearance rate. The function looks like this:
We iteratively fit models with various combinations of values for a and q and chose the model with the lowest error mean square (MS_{e}) as the best fitting model (JMP does not provide R^{2} estimates). We then compared the MS_{e} of the best fitting Type I model to that of the corresponding polynomial fit. The MS_{e} for the best fitting Type I models were 0.143 and 0.213 for the secondary analysis and the experiment, respectively. These values are higher than the MS_{E} for the polynomial fits (0.096, 0.196, respectively), meaning that the Type I fit was poorer than the Type III fit in both cases.
LITERATURE CITED
JMP, The Statistical Discovery Software. 2003. SAS Institute Incorporated.