D. Brent Gurd. 2008. Mechanistic analysis of interspecific competition using foraging trade-offs: implications for duck assemblages. Ecology 89:495–505.

Appendix A. Additional details on the methods used to estimate the parameters V_s, V_e, R_s,e,i and Z.

I used computer-aided design software (Solidworks™ 2001) to create digital, three-dimensional replicas of a skull of a male northern pintail (Anas acuta), a male American wigeon (A. americana), a female blue-winged teal (A. discors), a female green-winged teal (A. crecca), and a male gadwall (A. strepera). The maxilla and mandible of the replicas were placed in a series of start and end positions. At each position, the shape of the gape and the space separating the maxillary and mandibular lamellae was quantified and used to calculate R_s,e,i. The spacing of the lamellae along the mandible sets the minimum size of a particle that can be retained (Gurd 2007). For each species, I used the mean of ten measurements of the spacing of the mandibular lamellae to estimate values of i for which all R_s,e,i =0 (see Table A1). The volume enclosed by the bill was also estimated at each start (V_s) and end (V_e) position. The difference between these volumes, minus the volume of the tongue, gives an estimate of the volume of water filtered per cycle. The maximum values of V_s and V_e are determined, in part, by the ability of the tongue to retract and elevate during the filtration cycle. However, tongue movement has been estimated only for mallards and shovelers (Kooloos et al. 1989). For the other species I assumed they were directly proportional to the length of the maxilla (see Table A1). Estimates of Z are also known only for mallards and shovelers. I assumed Z was 19 cycles/s for all species, which is the mean cycle rate of mallards (Kooloos et al. 1989). Although the time involved in estimating the relationship between gape, lamellar separation and filtration rate precluded a formal sensitivity analysis, these assumptions provide good first approximations and should not have strong effects on the results for three reasons. First, the optimal solution to the trade-off between prey size selection and water filtration rate is dependent on how tongue retraction and elevation are constrained by bill position rather than the maximum filtration rates (Z) ducks can achieve. Second, the shape of the maxilla and mandible primarily determines the values of gape and lamellar separation each species can achieve. Third, E_N is sensitive to variation in Z only when E_N is constrained by t rather than V_MAX, which occurs only at very low particle concentrations.

TABLE A1. Morphological characteristics of seven species of dabbling ducks.

† From Nudds et al. (1994).