The Ymodel
The concept that tradeoffs between fitnessrelated traits are fundamental in shaping both evolutionary trajectories and equilibrium trait values has proved to be a powerful heuristic tool, with strong theoretical and empirical support (e.g. Roff, 1992, 2002; Stearns, 1992; Reznick et al., 2000). Nevertheless, predicted tradeoffs, generally measured by their statistical correlation, are not always found. In fact, positive phenotypic correlations between traits predicted to be involved in fitness tradeoffs (and hence expected to be negatively correlated) are uncovered not infrequently in laboratory and natural populations (Reznick et al., 2000; Roff, 2002). One of the most influential models explaining how such positive correlations can arise in the presence of tradeoffs is the acquisition–allocation model of Van Noordwijk & de Jong (1986), which was first formulated by James (1974) (his analysis contains an arithmetic error, which, unfortunately, is reproduced in Roff, 1992 and Roff, 2002) and also derived by Riska (1986). James called the model the ‘partitioning of resources model’, whereas Riska called it a ‘variable parts model’ and De Jong & van Noordwijk (1992) referred to it as the ‘Ymodel’, which, for simplicity, is the name we use in this paper. This model posits that within an individual two traits (X_{1},X_{2}) are determined by the allocation of resources from a common pool, T: T = X_{1} + X_{2} (note that we here use the symbols X_{1} and X_{2} rather than X and Y, because we are referring to the functional form, rather than the statistical manifestation, of the tradeoff). Given this mathematical relationship we can state that the value of trait X_{2} is restricted in its possible values by the allocation of some proportion, P, of the acquired resource T to X_{1}, leaving (1 − P)T to be allocated to X_{2}. Using this model, and assuming that variation exists among individuals, we can scale up to the level of the population and calculate the covariance between the two traits X_{1} and X_{2}, denoted as σ_{X1}_{X2}, as:
 (1)
where μ designates mean values and σ^{2} designates variances. For a fixed acquisition () the covariance between X_{1} and X_{2} is negative and the correlation is −1, indicating the functional tradeoff. However, when there is variability in acquisition it is possible for the covariance between X_{1}and X_{2} to be positive, giving the false impression, as measured by the statistical relationship between X_{1} and X_{2}, that there is no tradeoff. This result applies equally to phenotypic and genetic covariances (Houle, 1991; De Jong & van Noordwijk, 1992).
The basic Ymodel has provided a valuable insight into the pitfalls of trying to deduce evolutionarily important functional tradeoffs from patterns of variation in contemporary populations but, unfortunately, it has been interpreted in an oversimplified manner in much of the literature. Numerous authors have inferred from this model that the sign of the correlation between the two traits depends only upon the relative variances in acquisition and allocation (e.g. Glazier, 1999; Christians, 2000; Reznick et al., 2000; Jordan & Snell, 2002; Brown, 2003; Ernande et al., 2004). This is not mathematically correct and the misunderstanding appears to come from interpreting the range in the two traits given in Fig. 1 of Van Noordwijk & de Jong (1986) as the allocation variance rather than P. It is clear from eqn 1 and as illustrated schematically in Fig. 3, that the sign of the covariance depends on both the mean values and variances of acquisition and allocation.
An alternate statement of the covariance in the Y model that emphasizes the importance of the coefficients of variation is:
 (2)
where CV_{T} is the coefficient of variation in acquisition and CV_{P} is the coefficient of variation in allocation. Thus the sign of the covariance depends upon the relative sizes of the coefficients of variation rather than the variances. This dependency is not simple and a mere comparison of the relative sizes of the CVs is insufficient to determine the sign of the covariance. We illustrate this with data on the covariance between ovary mass and somatic mass in the sand cricket, Gryllus firmus (Table 1). The relative size of the covariances bears no relationship to the observed correlation: for example, for shortwinged females aged 1–3 days the CV for acquisition is considerably less than that for allocation (19.83 vs. 121.61 respectively) but the correlation is positive. Eqn 2 correctly predicts the signs of the correlations. These results illustrate that the effects of variation in acquisition and allocation on the expression of bivariate tradeoffs are complex and little can be inferred from simple ‘rules of thumb’ for comparing variances or CVs.
Table 1. Illustrative analysis of the tradeoff between allocation to ovaries (O) vs. soma (S) in the wing dimorphic sand cricket, Gryllus firmus. Acquisition is approximated by the total body mass, T, and P is the proportion allocated to the ovaries. For further details see Crnokrak & Roff (2002). Wing morph  Age span (days)  CV_{T}  CV_{P}  Eqn 2*  Eqn 3*  r†  n‡  P 


S  1–3  19.83  121.62  0.3706  0.0005  0.1419  42  0.3701 
S  5–7  17.27  16.56  0.0563  0.0061  0.5238  37  0.0009 
L  1–3  13.95  99.85  0.1016  0.0001  0.0508  39  0.7588 
L  5–7  13.33  68.90  −0.3723  −0.0035  −0.2104  32  0.2478 
  Mean values, variances and covariances   
 V_{T}   V_{P}  Cov_{OS} 
S  1–3  0.7756  0.0237  0.0202  0.00060  0.00049   
S  5–7  1.1698  0.0408  0.2608  0.00186  0.00604   
L  1–3  0.9484  0.0175  0.0171  0.00029  0.000118   
L  5–7  1.2132  0.0261  0.1380  0.00904  −0.00348   
An unrelated but perhaps more difficult problem with testing predictions of the Ymodel is the problem of operationally defining and measuring ‘acquisition’. This is not a simple task: proposed measures are size at some given age (Biere, 1995; MitchellOlds, 1996; Glazier, 1998, 1999; Dudycha & Lynch, 2005), particular body stores such as lipids (Chippindale et al., 1998) and the rate of food acquisition (Ernande et al., 2004). Whether these are appropriate will depend upon the organism under study and the question addressed. Body mass may be appropriate for a capital breeder (an organism that meets its reproductive effort from stored reserves) but possibly not for an income breeder (an organism that meets its reproductive effort from energy gathered during the course of reproduction). Even in the former case body mass does not take into account variation in energy content of the component tissues or the energy expended in developing and maintaining them. Feeding rate supposes that rate of utilization does not change, but selection experiments on response to crowding in D. melanogaster suggest that fast feeding larvae are less efficient at utilizing the food (Foley & Luckinbill, 2001; Prasad & Joshi, 2003; Mueller et al., 2005), which would generally make feeding rate a poor index of resource acquisition.
Expanding the scope of the Ymodel
It is possible to recast the Ymodel in a form that is closer to the spirit of the statement that the sign of the correlation between the two traits depends upon the relative variances in acquisition and allocation. The variance in a sum of two variables is equal to the sum of the two variances plus twice the covariance, which can be rearranged to give the covariance of the two variables as:
 (3)
From this it can be seen that a negative covariance between any two traits in an allocationbased tradeoff will occur if the sum of the variances in the allocated components exceeds the variance in acquisition. Eqn 3 correctly predicts the signs of the correlations between somatic mass and ovary mass in the sand cricket (Table 1).
The above equation is a more general statement and can be applied in those cases in which the distribution of resources is dichotomous but does not fit the assumptions of the Ymodel. In general, eqn 3 will be the appropriate statistical model when the components of the tradeoff are measured in different units. An example of this is the hypothesized functional tradeoff between egg size and number:
 (4)
where C is clutch mass, E is individual egg mass and N is the number of eggs. This can be made into a simple arithmetic model by taking log, giving:
 (5)
By converting to a log scale the allocation between ln(egg mass) and ln(egg number) can be mathematically equated to the Ymodel, by writing the proportional contribution, B, to ln(egg mass) as B = ln E/ ln C, the allocation model thus being ln C = B ln C + (1 − B) ln C (Christians, 2000; also adopted by Brown, 2003). For the reasons advanced above, the analysis of Van Noordwijk & de Jong (1986) cannot be used to justify the claim by Christians (2000) that given a tradeoff between egg size and number, there will be a negative correlation between egg mass and clutch size when the variation in ln B is large relative to variation in ln C. A much simpler approach is available by noting, as above, that:
 (6)
from which we can derive the statement that a negative correlation will occur when the sum of the variances in ln E and ln N exceed the variance in ln C. Whereas ln C can be regarded as acquisition, there is no reasonable index of allocation, nor is it useful in this case to assign an index of allocation. Unfortunately, the data presented in Christians (2000) and Brown (2003) are insufficient to use eqn 7 to determine if the conclusions reached in these two studies are valid.
A second example of a bivariate tradeoff that does not fit simply into the Ymodel is that between development time and adult body size (Berven, 1987; Roff, 2000). Because increases in development time increase generation time and decrease fitness, the tradeoff in this case appears as a positive relationship between body size and development time. However, it can be recast into a negative relationship by using the reciprocal of development time. Now, for the purpose of illustration, suppose that all individuals follow the same linear growth trajectory and initial size can be ignored. Adult size, Y, is then proportional to development time, X: Y = RX, where R is the rate of growth, which is equivalent to the rate of acquisition of energy. Taking log and rearranging gives ln R = ln Y − ln X = ln Y + ln X^{−1}. This model can now be examined in the same manner as for the egg sizenumber model. The immediate result is that if there is variation in growth rate (i.e. variation in R) then the tradeoff between body size and development time, as measured by the correlation, can be obliterated, which has been observed in some studies (Roff, 2000).
The hierarchical Ymodel
The allocation of resources may be made at a hierarchy of levels, with a binary split at each node. Even if the allocation is tripartite the allocation can be described mathematically by a pathway with two splits, the first being between one of the traits and the other two. The several numerical and analytical examinations of the hierarchical model have found, perhaps not surprisingly, that correlations at the end of the hierarchy depend critically upon allocation patterns at the beginning of the hierarchy (De Laguerie et al., 1991; De Jong, 1993; Worley et al., 2003; Bjorklund, 2004).
Agespecific fecundity provides a good example of a hierarchical pattern of allocation in which a proportion P_{x} of the total fecundity is realized at age x leaving (1 − P_{x}) to be allocated over the remaining ages. Tanaka (1996) analysed agespecific fecundity in the bruchid beetle Callosobruchus chinensis using this approach. In Tanaka's model, fecundity at age x, m_{x}, is given by:
 (8)
where Tanaka defined M as resource acquisition and s as allocation. While we agree with the definition of M, we believe that is a better interpretation of allocation. In the present model, at any given age, the amount remaining for future allocation is M e^{−sx} (i.e. the solution to the integral ): thus the partitioning at age x is Ms e^{−sx} and M e^{−sx}, giving a total fecundity of M(s e^{−sx} + e^{−sx}), and the proportion allocated at age x is thus . The fecundity function considered by Tanaka was unusual in its shape, more typical shapes being uniform or, most generally, triangular. This unusual fecundity function is responsible for the unexpected result that allocation was predicted to be independent of age. More generally, we would expect allocation to change with age. For example, if fecundity is a uniform function such that m_{x} = c, where c is a constant, then the total fecundity is cω where ω is the reproductive lifespan and the proportion allocated at age x is c/[c + c(ω − x)] = 1/(1 + ω − x), which means that the proportional allocation increases with age.
The Ymodel as an explanation for the sign of observed tradeoffs
Glazier (1999) argued that the variance in resource acquisition in natural populations would, in general, be greater than in captive or domestic populations and hence tradeoffs would be more often observed in the laboratory than the field. Because, as discussed above, the conditions under which a tradeoff will be observed are defined by the means as well as the variances, this hypothesis implicitly assumes no differences in the means between lab and field, or that the coefficients of variation change in parallel with the variances, which need not be the case.
A further critical assumption of this hypothesis is that the genetic component of the variance in resource acquisition is small. For his analysis Glazier (1999) selected the tradeoff between a measure of reproductive investment (clutch size, number of clutches per unit time, clutch mass, or fat content of ovaries or clutch) and a measure of somatic investment (various measures of body fat content or sizecorrected maternal body mass). At least some of the indexes of acquisition, such as body fat content, are likely to be highly correlated to total body size for which there is an abundant evidence of considerable genetic variance under laboratory conditions (Mousseau & Roff, 1987; for plants see Geber & Griffen, 2003). Coefficients of genetic variation for morphological traits are typically about 6% but can exceed 30% (see Fig. 1 of Houle, 1992; Messina, 1993; Imasheva et al., 2000; Hermida et al., 2002; Loh & BitnerMathé, 2005) and demonstrate that this source of variation cannot be ignored. Given this, it would come as no surprise not to find tradeoffs being expressed under laboratory conditions, as illustrated by the sand cricket data (Table 1). Indeed, Tuomi et al. (1983) argued the opposite of Glazier, namely that field conditions will be harsh, resource acquisition less variable in the field and tradeoffs more likely to be observed in the field than in the laboratory.
Glazier (1999) reported a significantly higher proportion of negative correlations in laboratory studies than field studies (52% vs. 24% respectively), concluding that the results supported his hypothesis. Unfortunately, this result is suspect for two reasons: first there was a taxonomic sampling bias: mammals and birds contributed the majority (85%) of the field data, whereas crustaceans and fish contributed the majority (75%) of the laboratory data. Secondly, Glazier did not correct for multiple estimates from each study, the average being 1.74 estimates per study, which could lead to pseudoreplication and inflation of the degrees of freedom. Nevertheless, the results are certainly suggestive.
An alternative method of exploring the Ymodel is to restrict acquisition experimentally by limiting intake (Glazier, 1999). Unless the restriction is extreme, some individuals will be able to satisfy their genetic propensity for acquisition whereas others will not, the result being a right truncation of the acquisition distribution realized under ad libitum rations. The result is a reduction in both the variance and the mean and thus it cannot be assumed that a covariance will become more accentuated with reduced rations. For a righttruncated normal distribution the coefficient of variation decreases monotonically with the severity of the truncation (i.e. if the distribution is truncated at k then CV, the coefficient of variation, increases with k). Therefore, under this distribution, provided that the pattern of allocation does not change, CV_{T} will decline with a decrease in ration and could convert a positive correlation to a negative (see eqn 2). Even if the correlation is negative at ‘high’ ration, under the scenario just described, the strength of the correlation should increase as ration is decreased. A critical assumption of these predictions is that the pattern of allocation of resources among components does not change with resource availability. Unfortunately, this assumption may not be valid (Sgro & Hoffmann, 2004; and see the example below).
To illustrate the effect of truncation of the resource distribution on patterns of allocation, we present data on the allocation to ovary mass and the main flight muscles (the dorsolongitudinal muscles, hereafter DLM) in the macropterous (longwinged) morph of the sand cricket, G. firmus. This experiment is described in detail elsewhere (Roff & Gelinas, 2003) and we note here only the pertinent features: nymphs were raised on ad libitum food but adults were fed either ad libitum or at a rate that had been shown to reduce seven day fecundity (measured by ovary mass) by approximately one half. On the ad libitum diet the combined mass of ovaries plus DLM was 0.162 g (SE = 0.004, n = 442, CV = 48.31%), whereas on the low ration it was 0.089 g (SE = 0.002, n = 414, CV = 13.3%). (These values differ slightly from those calculated from Table 1 of Roff & Gelinas, because in the present analysis we included only females for which we had both variables.) For the present analysis we are concerned with how the food allocated to the combination of ovaries plus DLM is distributed between these two components and therefore we can consider the combined mass as the total acquisition. The low ration reduced total acquisition by 45% and the CV of acquisition by 72%. The percentage of this total acquisition allocated to DLM averaged 15.5% for females on the ad libitum diet, and 23.7% for females on the low ration, a difference that is highly significant (t_{854} = 8.1335, P < 0.0001, Kruskal–Wallis test, = 57.80, P < 0.0001; proportions transformed using arcsine squareroot). In contrast, the correlation between ovary mass and DLM mass was −0.591 and −0.565 on ad libitum and low rations respectively, a difference that is not significant (t_{∞} = 0.5697, P > 0.5). Thus a severe reduction in acquisition did not result in a change in the correlation (i.e. the tradeoff), but did change the pattern of allocation.
Phenotypic plasticity in resource allocation is a common phenomenon (e.g. Reznick, 1983; Smith & Davies, 1997; Billerbeck et al., 2000; Li et al., 2001; Jordan & Snell, 2002; Bochdanovits & de Jong, 2003; Sgro & Hoffmann, 2004) and invalidates predictions of the sign of the correlation between two traits based solely upon the observed change in the variance or coefficient of variation in resource acquisition. The assumption that a restriction in acquisition will either convert a positive covariance to a negative covariance or increase the magnitude of the negative correlation is incorrect both on theoretical and empirical grounds. That such observations have been made in some cases (e.g. Biere, 1995; Glazier, 1999; Messina & Slade, 1999; Donohue et al., 2000) but not others (e.g. Glazier, 1999; Lardner & Loman, 2003) argues for a greater need for an understanding of the functional basis of tradeoffs, particularly with respect to the adaptive significance of patterns of allocation under different acquisition regimes. The Ymodel is a powerful conceptual and analytical tool but a misinterpretation of its predictions has led to an unfortunate number of incomplete empirical investigations.