Influence of the resource acquisition-allocation Y-model's parameters on the detection of phenotypic trade-offs

Phenotypic trade-offs, predicted to occur due to resource limitation, are not commonly detected. For this, the Y-model provides a powerful heuristic by showing that greater variation in resource acquisition than variation in allocation, masks trade-offs. However, the Y-model contains several other parameters beyond variation in acquisition and allocation, whose independent and interactive influences remain unclear. We simulate the means of, variation in, and correlation between, acquisition and allocation, to systematically explore how these parameters influence the phenotypic association between two traits. We find that the mean resources acquired by a population has no direct mathematical influence on phenotypic correlations. Instead: the mean of the resource allocation of a population; the correlation between allocation and acquisition; and the ratio of variation in acquisition to variation in acquisition plus allocation, directly impact phenotypic correlations. Importantly, a three-way interaction between these parameters provides a better prediction of the phenotypic correlation than their independent effects. We validate our simulations using an empirical dataset and analytical solutions to demonstrate their robustness. Using several biological examples such as resource limitation, we show how Y-model parameters might manifest their influence on phenotypic correlations in empirical data. Despite the simplicity of our simulations and assumptions (e.g., sampling from normal distributions), our study provides a quantitative extension of the Y-model paradigm to understand the detectability of phenotypic trade-offs.