## Abstract

Nee *et al*. (Reports, 19 August 2005, p. 1236) used a null model to argue that life history invariants are illusions. We show that their results are largely inconsequential for life history theory because the authors confound two definitions of invariance, and rigorous analysis of their null model demonstrates that it does not match observed data.

Decades of research on life history invariants have identified deep symmetries in evolutionary biology that reveal fundamental and pervasive constraints upon diverse organisms (*1*, *2*). Nee *et al*. (*3*) constructed null models to argue that slopes and *R*^{2} from log-log plots cannot reliably identify invariants. This analysis has led others to conclude that life history invariants may not exist (*4*, *5*). These conclusions are erroneous because they are based on a misrepresentation of life history invariants and on a failure to recognize that properties of these null models differ significantly from empirical data. Here, we show that Nee *et al*.'s results are largely inconsequential for life history and allometric theories.

Within life history theory, the term invariance is used in two ways. In type A invariance, a biological characteristic does not vary systematically with another characteristic, such as body size (*6*); in type B invariance, a biological characteristic exhibits a unimodal central tendency and varies over a limited range (*7*, *8*). For example, the ratio of weaning mass to adult mass in mammals is a type A invariant because its value shows no trend with body size. In addition, this ratio is a type B invariant because weaning mass is typically close to 30% of adult mass. In his original work, Charnov (*2*) examines both types of invariants; however, he clearly emphasizes type A as the more important. He begins his book by stating that “Something will be called invariant (or an invariant) if it does not change under the specified transformation” and “the underlying transformation is adult body size between species.”

The null models of Nee *et al*., although presented and interpreted as posing problems for all life history invariants, are only relevant for detecting type B invariance, as we now explain. If (i) the ratio of life history characteristics, *c* = *y*/*x*, is randomly distributed and (ii) Var[ln(*x*)] » Var[ln(*c*)], then it is straightforward to show analytically that the slope and *R*^{2} of ln(*y*) versus ln(*x*) are near 1. Nee *et al*.'s null models satisfy these assumptions (*3*). By assuming condition (i), that *y*/*x* does not vary systematically with another variable, the simulated data of Nee *et al*. are type A invariants, by definition. Thus, they do not provide an alternative null model to this type of invariance. Indeed, the Nee *et al*. results demonstrate that slopes and *R*^{2} near 1 from log-log plots are valid for identifying type A invariance. Moreover, when *c* is drawn from a uniform random distribution with any reasonable choice of bounds (*9*), condition (ii) is satisfied, and therefore *R*^{2} ≈ 1. Thus, Nee *et al*.'s choice of the uniform random distribution, but not the specific bounds, is crucial for obtaining their results and, as such, the critical comparison to determine whether the observed data are described by this null is to compare the distribution of invariants to a uniform random distribution (*10*).

When the null results of Nee *et al*. are rigorously compared with existing empirical data, it becomes obvious that the null fails to predict important biological properties of life history invariants and thus fails to describe type B invariance as well. First, the intercept from a regression on their null predicts a life history ratio, given by the distribution mean, that is independent of taxa. Therefore, Nee *et al*. cannot account for the observation that different clades or taxa are described by different values of *c*, representing important evolutionary differences between taxonomic groups (Fig. 1) (*2*). Second, many observed distributions of life history invariants are unimodal with a constrained range and thus significantly differ from Nee *et al*.'s null model (Fig. 2). Third, their null model predicts *R*^{2} values that are in fact lower than those for real data (Fig. 2) (*11*, *12*). Fourth, Nee *et al*.'s testing of the ratio of annual clutch size to annual mortality rate against their null is clearly incorrect. Instead of using their null to calculate an expected *R*^{2}, they use Ricklefs' (*13*) empirical data to calculate the *R*^{2} of the data, which is a statistical fact unrelated to their null model. Nee *et al*. incorrectly cite this as evidence against life history invariants. Using a uniform random distribution, corresponding to Nee *et al*.'s null model, we find *R*^{2} = 0.44, much lower than that of Ricklefs' empirical data (*R*^{2} = 0.84) and thus, evidence against Nee *et al*.'s null.

Further, contrary to claims by Nee *et al*. and others that life history theory implies that invariants show no variation, Charnov [e.g., pp. 5 and 15 of (*2*)] clearly states that a distribution is expected for any life history invariant. Indeed, life history theory endeavors to understand how natural selection sets both the central tendency and the distribution of these ratios (*2*, *7*, *11*). Finally, Nee *et al*.'s null model can only produce slopes near 1 and, thus, contrary to concerns raised by de Jong and others (*4*, *5*), cannot explain the ubiquitous quarter-power slopes observed in allometry or, by extension, the life history invariants formed from them (*1*, *2*, *7*).

Life history invariants are governed by nonrandom processes and form the cornerstone of a general framework that mechanistically links variation in organismal form, function, ecology, and evolution across differing environments [e.g., (*1*, *2*, *8*)]. For more than 40 years, fishery science has used them with great success [e.g., (*14*)]. Life history invariants are certainly not illusions.