Egg mass in Glaucous-winged Gulls (Larus glaucescens) as a function of length and width.
For colonial seabirds, egg mass can be determined using a portable scale carried through the colony during nest surveys. In order to receive an accurate reading, however, the scale must rest on a level surface, remain sheltered from the wind, and be recalibrated at each new site. Furthermore, because eggs lose moisture over time and get progressively lighter, an egg should be weighed on the day that it is laid (Romanoff and Romanoff 1949). It is not always possible or convenient to fulfill these requirements for measuring egg mass. In contrast, egg length and width, which remain constant from laying to hatching, can be measured rapidly and accurately in the field with calipers.
Several equations for estimating egg mass or volume from length and width have appeared in the literature. Hoyt (1979) estimated fresh egg mass M from length L and width W with the model M=[k.sub.M]L[W.sup.2] , in which [k.sub.M] is a species-specific parameter. He provided [k.sub.M] values for 26 species, including Western Gulls (Larus occidentalis), for which [k.sub.M] was calculated to be 0.000531 g/[mm.sup.3] (n = 8). For chicken eggs, Narushin (2005) used geometric considerations to obtain the volume model V=(a-bW)L[W.sup.2], and Abanikannda and others (2007) used a model of the form M = a + bL + cW + d(W/L).
In this study, we tested these 3 alternative models as well as 3 allometric models for Glaucous-winged Gulls (L. glaucescens). In particular, we: (1) parameterized the models with data collected from a Glaucous-winged Gull colony; (2) selected the best model(s) using the Akaike Information Criterion (AIC); and (3) validated the selected model(s), without refitting, on a data set that was not used for parameter estimation.
We collected data during the 2009 and 2010 breeding seasons at Protection Island National Wildlife Refuge (48[degrees]07'40" N, 122[degrees]55'0" W), Jefferson County, Washington, USA. Violet Point, a gravel spit extending to the southeast of the main island, contained a breeding colony of 3000 (in 2009) and 2469 (in 2010) pairs of Glaucous-winged Gulls (Cowles and others 2012, and unpubl, data). We selected 5 rectangular sample subareas of the Violet Point colony as shown in Figure 1 of Henson and others (2010; Plots A-E), with a combined sample area of 4205 [m.sup.2]. Plot A bordered the beach on the north-central edge of the colony; Plot B was landlocked in the center of the colony; Plot C was at the west end of the colony and bordered the north beach of a small marina closed to the public; Plot D was in the center of the colony and also bordered the north beach of the marina; Plot E was in the center of the colony and bordered the south beach of the marina.
Daily, throughout the 2009 and 2010 laying seasons (late May-early July), we monitored all nest cups in the 5 sample areas. When clutch initiation occurred in a nest, we placed a numbered wooden stake near the nest. Each staked nest was checked daily for new eggs. The mass of each egg was measured on the day it was laid using a 400-g capacity Ohaus Scout Pro SP401 portable electronic balance, and each egg was measured with vernier calipers at its widest and longest points. One egg, a 'dwarf' whose length (55.9 mm) and width (38.6 mm) were 22% smaller than that of a typical egg, was excluded from the data set. In total, 1474 eggs were recorded, with 751 eggs measured in 2009 and 723 eggs measured in 2010. The average length was 71.7 mm (range 61.4-85.8, s = 2.80), and the average width was 49.5 mm (range 38.1-58.9, s = 1.67).
We considered the 6 alternative models: M = aL[W.sup.2], (1) M = (a - bW)L[W.sup.2], (2) M = a + bL + cW + d(W/L), (3) M = a[L.sup.b] [W.sup.c], (4) M = a[L.sup.b], (5) M = a[W.sup.c], (6)
where a, b, c, and d > 0 are parameters to be estimated from data.
We parameterized the 6 alternative models on the 2009 data using the method of nonlinear least squares (LS), reserving the 2010 data for validation. In order to account for demographic noise due to the variability between individual eggs, we parameterized on the square root scale; that is we minimized RSS= [SIGMA] [([square root of observed] - [square root of predicted]).sup.2] as a function of the parameters (Dennis and others 2001; Cushing and others 2003; Hayward and others 2005).
For each of the parameterized alternative models, we computed the [DELTA]AIC, a model-theoretic selection index. Models with [DELTA]AIC < 2 were considered (indistinguishably) the 'best' models, and models with [DELTA]AIC > 10 were considered inferior to the best models and were discarded (Burnham and Anderson 2002).
We assessed the goodness-of-fit with a generalized [R.sup.2]:
[R.sup.2] = 1 - RSS/[SIGMA] [([square root of observed] - mean [square root of predicted]).sup.2] (7)
(Hayward and others 2005); here RSS denotes the fitted residual sum of squares. We validated the selected model(s) by comparing its [R.sup.2] on the 2009 data (to which it was fitted) to its [R.sup.2] on the 2010 data (without re-fitting).
Table 1 lists the [DELTA]AIC, the [R.sup.2], and the parameter estimates for the 6 alternative models. The best models were Model (3) with AAIC = 0 and Model (4) with [DELTA]AIC < 1. On the validation data set, both of these models were able to explain >94% of the variability with average absolute error <1.27 g and average relative error <1.35%. Of the 2 best models, Model (4) is the most parsimonious, having one less parameter than Model (3).
Each of the remaining 4 models had [DELTA]AIC > 10. Although Model (2) fit the data well and had a fairly low [DELTA]AIC (14.82), it can attain negative mass values, a fact that is troublesome both theoretically and numerically when estimating parameters. Although Hoyt's Model (1) lags well behind the first 3 models, it still explains 87% of the variability in the estimation data with only 1 species-specific parameter. Interestingly, our study, which fitted n = 751 Glaucous-winged Gull eggs to Model (1), produced the same [??] value (Table 1) as Hoyt's study, which was based on only 8 Western Gull eggs "obtained on various field expeditions and from zoos and commercial sources" (Hoyt 1979). As functions of single egg dimensions, Models (5) and (6) were not able to describe the data. Note, however, that the width measurement alone described 73% of the variability in egg mass, whereas the length measurement alone described only 41% of the variability. This is consistent with Abanikannda and others (2007), who showed that, in chicken eggs, width exhibits a stronger correlation with mass than does length.
The superiority of the allometric Model (4) with length and width exponents parameterizing somewhat close to I and 2, respectively, and the reasonable fit of Hoyt's Model (1) are not surprising given that egg mass is related to egg volume and the volume of a prolate spheroid (an ellipse revolved about the long axis) is V = ([pi]/6)L[W.sup.2]. Because eggs are not elliptical in the planes parallel to the long axis, model accuracy would improve by incorporating an egg shape parameter. Preston (1974) found that egg volume depends on the shape of the longitudinal contour, which is determined by the 4 parameters length, width, asymmetry, and bicone. Egg shape can be measured photographically in the field; Bridge and others (2007) found that digital photographic analysis of avian eggs better accounted for variations in volume than a model utilizing only measurements of length and width. For a practical method of estimating the initial masses of large numbers of eggs in the field, however, Models (3) and (4), which require only caliper measurements of length and width, may be an acceptable simplification.
The near-identical parameterization of Hoyt's Model (1) on eggs from the closely-related Glaucous-winged and Western Gulls measured from various colonies across a 30-y time span suggests that our parameterized models are likely robust for these 2 species. The models would require re-parameterization for other species.
In conclusion, the estimation of Glaucous-winged Gull initial egg mass from length and width measurements can describe 89 to 94% of the variability in mass. A mathematical model that accurately predicts egg mass from length and width may eliminate the need to use a portable scale during nest surveys. Such a model also may provide realistic estimates of the original masses of empty shells and museum specimens.
Key words: allometric model, egg length, egg mass, egg width, Glaucous-winged Gull, Larus glaucescens, Protection Island, Washington
Acknowledgments.--Thanks to K Ryan, project leader of the Washington Maritime National Wildlife Complex, US Fish and Wildlife Service, for permission to work at Protection Island National Wildlife Refuge; L Megna, A Moncrieff, and B Payne for assistance in data collection; and the Rosario Beach Marine Laboratory for logistical support. Financial support was provided in part by National Science Foundation grant DMS 1022494 (SMH and JLH). MAM conducted this research as an undergraduate member of the JN Andrews Honors Program at Andrews University.
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Department of Biology, Andrews University, 4280 Administration Drive, Berrien Springs, MI 49104; email@example.com (MAM), firstname.lastname@example.org (JLH); Department of Mathematics, Andrews University, 4260 Administration Drive, Berrien Springs, MI 49104; email@example.com (SMH). Submitted 11 September 2012, accepted 30 January 2013. Corresponding Editor: Joan Hagar.
TABLE 1. Results for model selection, parameterization, and validation. Estimation (n = 751) Model [DELTA]AIC [R.sup.2] [??] (3) 0 0.8947 -203.0 (4) 0.9244 0.8943 2.444 x [10.sup.-3] (2) 14.82 0.8920 6.982 x [10.sup.-4] (1) 123.8 0.8748 5.311 x [10.sup.-4] (6) 690.4 0.7345 0.03438 (5) 1286 0.4135 0.2629 Validation (n= 723) Model [??] [??] [??] [R.sup.2] (3) 1.936 2.107 77.36 0.9433 (4) 0.8979 1.721 N/A 0.9455 (2) 3.361 x [10.sup.-6] N/A N/A N/A (1) N/A N/A N/A N/A (6) N/A 2.027 N/A N/A (5) 1.375 N/A N/A N/A
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|Title Annotation:||GENERAL NOTES|
|Author:||McCormick, Melissa A.; Hayward, James L.; Henson, Shandelle M.|
|Publication:||Northwestern Naturalist: A Journal of Vertebrate Biology|
|Date:||Sep 22, 2013|
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