## Introduction

The relationships between life history traits are hypothesized to be caused by two underlying processes: the acquisition of resources and the subsequent allocation of those resources among competing traits (James, 1974; Riska, 1986; van Noordwijk & de Jong, 1986; Reznick *et al.*, 2000; Angilletta *et al.*, 2003; Roff & Fairbairn, 2007). Differential allocation to competing structures and functions generates ‘trade-offs’ in the expression of these traits (Gadgil & Bossert, 1970; Bell & Koufopanou, 1986; Reznick, 1985; Roff, 1992; Stearns, 1989; Roff, 2002). Allocation of more energy to one trait is hypothesized to reduce the energy available for the other traits, resulting in negative correlations between the competing traits. However, individuals also vary in the acquisition of resources (Ricklefs, 1991; Weiner, 1992; Hammond & Diamond, 1997). Individuals that are able to acquire more resources than others will have more resources available to allocate to both traits. In the case where trait values depend more strongly on the total resource pool than on the proportion allocated to competing traits, the correlation between traits will be positive. This concept has been formalized in a mathematical framework known as the ‘Y model’ (James, 1974; Riska, 1986; van Noordwijk & de Jong, 1986; Houle, 1991).

The Y model was most clearly articulated by van Noordwijk & de Jong (1986). Their model consists of two traits (*x*_{1} and *x*_{2}) drawing from a common resource pool and includes variation in both acquisition (the size of the total resource pool = *T*) and proportional allocation (proportion of resources allocated to *x*_{1} = *P*) of resources. For a fixed acquisition value, variation in allocation leads to a negative covariance between traits. However, if some individuals in a population are able to acquire more resources than others, they will have a larger resource pool and can allocate more resources to both traits involved in a trade-off. This variation in acquisition can lead to a positive correlation between these traits when measured across individuals in a population even while there is a functional trade-off within individuals. The Y model makes the important assumption that variances in acquisition and in allocation are independent. Specifically, this model predicts that the strength and sign of the covariance between the two traits depends on relative variation in acquisition and relative variation in allocation and can be predicted from the following equation (van Noordwijk & de Jong, 1986):

where is the covariance between *x*_{1} and *x*_{2}, is the variance in acquisition, is the variance in allocation, *μ*_{T} is the mean of acquisition and *μ*_{P} is the mean of allocation. From this equation, it can been seen that if the variance of acquisition () is zero, the covariance will be negative (). If the reverse is true and variance of allocation () is zero, the covariance will be positive (). We do not generally expect populations to show zero variance in either acquisition or allocation, therefore, two more informative predictions are as following:

- 1 For a constant , , and increasing variance in acquisition () makes the correlation between
*x*_{1}and*x*_{2}more positive. - 2 For a constant , , , and increasing variance in allocation () makes the correlation between
*x*_{1}and*x*_{2}more negative.

The Y model has been influential and is commonly cited as a possible explanation when expected trade-offs are not observed (e.g. Spitze *et al.*, 1991; Genoud & Perrin, 1994; Yampolsky & Ebert, 1994; Reznick *et al.*, 2000; Jordan & Snell, 2002; Messina & Fry, 2003; Ernande *et al.*, 2004; Vorburger, 2005). Despite its impact, the Y model has rarely been rigorously tested. This deficiency stems in part from the difficulties associated with accurately measuring the complex processes of acquisition and allocation, which requires quantifying both an individual’s total resource pool and the proportion of those resources allocated to various traits.

For most life history trade-offs, it is assumed that energy is the major limiting resource, and in this case, total acquisition will be the total energy acquired by an organism and allocation will be the proportion of that energy allocated to various traits. Estimating energy acquisition can be challenging. Energy acquisition is a complex trait that is potentially influenced by many factors, including food availability, time spent foraging, the rate of digestion and absorption of nutrients, and the efficiency of digestion and absorption of nutrients (for reviews see Ricklefs, 1991; Weiner, 1992; Hammond & Diamond, 1997). In the past, energy acquisition has been estimated by a number of different methods, including total mass (Christians, 2000; Brown, 2003; Uller & Olsson, 2005), size at a given age (Biere, 1995; Dudycha & Lynch, 2005), growth rate (Tessier & Woodruff, 2002), absorption efficiency of macronutrients (Zera & Brink, 2000) and feeding rate (Ernande *et al.*, 2004). Estimating allocation includes similar challenges. Studies of differential resource allocation typically do not measure allocation in units of energy. For example, a study of the trade-off between reproduction and survival might use egg number and/or egg size as an estimate for reproductive output and lifespan as a measure of survival. These types of measures are assumed to be correlated with the amount of energy allocated to different functions. However, life history traits are complex and will often involve many factors not accounted for by these simple measures. Any test of the Y model requires a more complete estimate of acquisition and allocation in units of energy.

This task is difficult enough in a single population let alone for many different populations or species. Therefore, researchers aiming to test this model typically must make assumptions regarding acquisition and/or allocation. For example, Glazier (1999) attempted to test the Y model by comparing laboratory and field studies. He made the assumption that variation in acquisition will be minimized in the less variable conditions of the laboratory, and therefore, negative correlations will be found more often in laboratory studies. This hypothesis was consistent with what he found in a review of the literature, and he concluded the Y model was supported. However, the opposite claim has also been made. Variation in acquisition may be minimized in low resource environments (Messina & Fry, 2003; Ernande *et al.*, 2004), and negative correlations may be found more often in harsher, more resource-stressed conditions, such as those in the field (Tuomi *et al.,* 1983; Bell & Koufopanou, 1986; Reznick, 1985). Additionally, because the Y model predicts that the trade-off function will depend on the mean and variance of allocation as well, Glazier (1999) also assumes that the mean and variance of allocation are not also changing under laboratory conditions. Without explicitly measuring acquisition and allocation, we can only make hypotheses regarding the causes of changes in observed trade-off patterns.

In this study, we take the approach of utilizing the well-studied trade-off between flight capability and reproduction in the sand cricket, *Gryllus firmus* as a case study to test the Y model. We first estimate energy acquisition and allocation for a set of *G. firmus* individuals and use these data to formulate a model to predict these quantities from organ masses alone. Subsequently, we experimentally alter variation in acquisition by rearing individuals on three different food levels and utilize our predictive model to estimate acquisition and allocation to directly test the assumptions and predictions of the Y model.