Abstract
 Top of page
 Abstract
 Introduction
 Methods
 Results and discussion
 Acknowledgements
 References
1. Van Noordwijk & de Jong proposed a model to predict the sign of an intraspecific correlation between two lifehistory traits, given that a tradeoff between the two traits exists. In this paper, the model is adapted to egg size/number tradeoffs.
2. The predictions of the model are tested within various clades of waterfowl. Although the relationship between egg mass and clutch size in this group has been examined previously, prior analyses have either failed to take phylogenetic structure into account or relied upon phylogenies that lack resolution. Phylogenetic structure is adjusted for using recently constructed phylogenies and the method of phylogenetically independent contrasts.
3. Van Noordwijk and de Jong's model was successful in explaining betweenclade variation in the relationship between egg mass and clutch size. Even if a tradeoff between egg size and number exists, a negative relationship will only be observed if the variation in the pattern of allocation (i.e. few large eggs vs many small eggs) is high relative to the variation in the amount of resources invested in the clutch. Thus, this model may be useful for understanding interspecific relationships as well as intraspecific patterns.
Introduction
 Top of page
 Abstract
 Introduction
 Methods
 Results and discussion
 Acknowledgements
 References
Reproduction requires resources to be allocated among competing demands (Stearns 1992). If resources limit the production of propagules, there will be a tradeoff between the number of propagules produced and the investment in each (Bernardo 1996). However, a tradeoff within individuals will not necessarily result in a negative relationship between propagule size and number within species if individuals differ in their access to resources (van Noordwijk & de Jong 1986; Flint, Grand & Sedinger 1996). ‘Highquality’ individuals (i.e. those with more resources) will be able to produce more propagules and larger propagules than ‘lowquality’ individuals. Many species do not show negative correlations between propagule size and number (Bernardo 1996), perhaps as a result of variation in resource acquisition.
Van Noordwijk & de Jong (1986) proposed a model to predict the sign of the correlation between two lifehistory traits, given that a tradeoff between the two traits exists. In the model, the investment in the two traits is measured in the same units (e.g. energy). Hence, it is not immediately obvious how this model can be applied to the tradeoff between egg size (measured in mass or volume) and clutch size (measured in number of eggs). Furthermore, the model was developed to explain intraspecific correlations, and certain assumptions of the model (see Results and discussion) may not be realistic in interspecific relationships.
In this paper, it is shown how van Noordwijk & de Jong's (1986) model can be adapted to the egg size/number tradeoff. The predictions of this model are then tested to see if they hold for interspecific relationships by examining the correlations between egg mass and clutch size in various clades of waterfowl (Order Anseriformes; geese, swans and ducks). Although egg size/number tradeoffs in this group of birds have been examined previously by Rohwer (1988), his analyses ignored phylogenetic relationships among species, which can lead to misleading interpretations of comparative data (Felsenstein 1985). Blackburn (1991b) reanalysed Rohwer's data, adjusting for phylogenetic structure, but the phylogeny he used (i.e. Livezey 1986a) lacked resolution, such that most of Rohwer's (1988) data were discarded (data from 151 species were reduced to 37 points). As a result, Blackburn (1991b) could only test for a relationship across the entire order, and not within lower taxonomic groups. Here recently constructed phylogenies (see below) and the method of phylogenetically independent contrasts (Felsenstein 1985) are used to examine the relationship between egg mass and clutch size across the Anseriformes and within a number of tribes and subfamilies.
The predictions of van noordwijk & de jong's model with regards to the egg size/number tradeoff
 (eqn 1)
Although investment in egg mass (E) and clutch size (C) is multiplicative, i.e.
 (eqn 2)
rather than additive, an egg size/number tradeoff can easily be fit to the model by logarithmically transforming the variables, i.e.
 (eqn 3)
 (eqn 4)
Van Noordwijk & de Jong (1986) refer to A as the ‘acquisition’ of resources, but in an egg size/number scenario it more properly refers to the amount of resources invested in a clutch (i.e. log[clutch mass]). The model of van Noordwijk & de Jong (1986) incorporates an additional parameter, B, the fraction of resources allocated to R, such that
 (eqn 5)
 (eqn 6)
In an egg size/number tradeoff, resources invested in egg size and clutch size are not mutually exclusive. However, the following parameter can be used as an index of allocation (i.e. few large eggs vs many small eggs):
 (eqn 7)
 (eqn 8)
This parameter satisfies the condition of van Noordwijk & de Jong's (1986) model that 0 ≤ B ≤ 1 as long as E > 1. The latter condition is easily achieved by using appropriate units for egg mass.
The model predicts that, given a tradeoff between egg size and number, there will be a negative correlation between egg mass (i.e. log[E]) and clutch size (i.e. log[C]) when the variation in allocation of resources (i.e. B = log[egg mass] {log[clutch mass]}^{−1}) is large relative to the variation in investment of resources (i.e. A = log[clutch mass]), and vice versa (van Noordwijk & de Jong 1986).
Results and discussion
 Top of page
 Abstract
 Introduction
 Methods
 Results and discussion
 Acknowledgements
 References
Given an underlying tradeoff between egg size and number, van Noordwijk & de Jong's (1986) model predicts that a negative correlation between egg mass and clutch size will be observed when the variation in allocation of resources (i.e. log[egg mass] {log[clutch mass]}^{−1}) is large relative to the variation in investment of resources (i.e. log[clutch mass]). Among tribes and subfamilies of waterfowl, the ratio of variation in allocation to variation in investment, adjusted for body mass and phylogenetic structure, is highest in the Anatini and the Aythyini (Table 1). As predicted, negative correlations between egg mass and clutch size are only observed within these two tribes (Table 1) when body mass and phylogenetic relatedness are taken into account. Similar results were obtained when phylogenetic structure was not taken into account (Rohwer 1988). The lack of a significant negative correlation within the Tadornini, Mergini and Anserinae (Table 1) is not simply due to small sample sizes; if the ‘real’ strength of the correlation between clutch size and egg mass in these three groups was as strong as that observed in the Anatini (r^{2} = 0·51), the statistical power to detect the relationship (with type I error level, α = 0·05) would be high (0·83, 0·89 and 0·98, respectively). The lack of relationship within the Dendrocygninae and the Oxyurini (Table 1) may be due to small sample sizes.
Table 1. Variation in investment (i.e. log[clutch mass]) and allocation (i.e. log[egg mass] {log[clutch mass]}^{−1}), and the relationships between clutch size and egg mass in the Anseriformes and within lower taxonomic groups. Relationships were considered significant at type I error level, α = 0·05    Variation in investment and allocation  Relationship between clutch size and egg mass 

Group  Taxonomic level  n^{a}  σ_{investment}^{2} (× 10^{−3})  σ_{allocation}^{2} (× 10^{−3})  σ_{allocation}^{2}/σ_{investment}^{2}  r^{2}  Major axis slope  P 


Anatini  Tribe  49  4·91  1·12  0·23  0·51  −0·49  0·0001 
Anseriformes  Order  145  3·68  0·42  0·11  0·18  −0·51  0·0001 
Anserinae  Subfamily  23  2·34  0·11  0·05  0·00  −0·14  NS 
Aythyini  Tribe  14  0·51  0·11  0·22  0·57  −1·21  0·001 
Dendrocygninae^{b}  Subfamily  8  –  –  –  0·32  −2·52  NS 
Mergini  Tribe  15  3·30  0·17  0·05  0·01  0·02  NS 
Oxyurini^{b}  Tribe  6  –  –  –  0·26  −0·27  NS 
Tadornini  Tribe  13  2·56  0·09  0·04  0·04  −6·16  NS 
The ratio of variation in allocation to variation in investment across the entire Anseriform order (0·11) is intermediate between the ratios in clades with significant negative correlations (≈ 0·22 in the Anatini and the Aythyini) and the ratios in clades without (≈ 0·05 in the Anserinae, Mergini and Tadornini) (Table 1). Similarly, the strength of the negative relationship between clutch size and egg mass in the Anseriformes (r^{2} = 0·18) is intermediate between these two groups (Table 1, Fig. 1). The negative correlation across the entire order is highly significant, and is robust with regards to the branch lengths used; the r^{2} of the relationship ranges from 0·092 to 0·298 in 95% of simulated trees, while the slope varies from −0·70 to −0·34. The strength of the tradeoff between egg size and number in Anseriformes is similar to that found by Rohwer (1988; r^{2} = 0·13), even though his analysis considered species as statistically independent. Blackburn's (1991b) analysis, which took phylogeny into account but included only a subset of Rohwer's (1988) data, provided an estimate that is at the upper end of the range found in this study (r^{2} = 0·29).
Van Noordwijk & de Jong's (1986) model assumes that variation in investment and allocation are independent of one another. However, the model appears to be robust to violation of this assumption; these two parameters are highly correlated in all but two groups of waterfowl (Table 2). Specifically, large clutch mass tends to be associated with clutches of many small eggs. Despite the correlation between investment and allocation, the model successfully explains betweenclade variation in the relationship between egg mass and clutch size. This study has therefore shown (1) how van Noordwijk & de Jong's (1986) model can be extended to egg size/number tradeoffs, and (2) that this model may be useful above the species level.
Table 2. Correlations between investment (i.e. log[clutch mass]) and allocation (i.e. log[egg mass] {log[clutch mass]}^{−1}) in the Anseriformes and within lower taxonomic groups. Relationships were considered significant at α = 0·05 Group  r  P 

Anatini  −0·68  0·0001 
Anseriformes  −0·51  0·0001 
Anserinae  −0·59  0·003 
Aythyini  0·09  NS 
Mergini  −0·66  0·007 
Tadornini  −0·10  NS 
The tradeoff between egg mass and clutch size is reflected by a negative correlation between these two lifehistory parameters within Anseriformes (this study) and among birds in general (Blackburn 1991a), despite many potentially confounding variables (see Blackburn 1991a). However, there is often no such inverse relationship within tribes and subfamilies (this study), and generally none within species of waterfowl (Rohwer 1988; Lessells, Cooke & Rockwell 1989; Rohwer & Eisenhauer 1989; Flint et al. 1996). A similar situation is observed in other taxa: interspecific trends often suggest tradeoffs (Godfray, Partridge & Harvey 1991; Bernardo 1996; Polishchuck & Tseitlin 1999) while intraspecific correlations do not (Bernardo 1996). The results of this study suggest that van Noordwijk & de Jong's (1986) model may prove useful in understanding the heterogeneity between taxonomic levels in the relationship between egg size and number. Tradeoffs between egg size and number will not be reflected by a negative correlation between these two parameters if the variation in investment is high relative to the variation in the patterns of allocation.