Recurrent violations of invariant rules for offspring size: evidence from turtles and the implications for small clutch size models.

Smith and Fretwell's classic model predicts that parents can maximize fitness by dividing the energy available for reproduction into offspring of an optimal size. However, this model breaks down when clutch size is small (~1-10 offspring). Invariant rules are an extension of the Smith-Fretwell model, and these rules predict how offspring size will vary among and within individuals that produce small clutch sizes. Here, we provide a narrow test of invariant rules using three turtle species, then we synthesize and re-analyze existing data from 18 different species (comprising five Orders) to evaluate whether invariant rules are followed across broad taxa. We do not find support for most invariant rules in turtles, and our re-analysis demonstrates a general mismatch between observed and expected values across all taxa evaluated, suggesting that invariant rules fail to predict reproductive patterns in nature. Morphological constraints on offspring size and reproductive effort may be important reasons for disparities between theory and observation both in turtles and other taxa. Paradoxically, morphological constraints are most common in small-bodied species and individuals, but these same candidates are also those which produce the small clutch sizes that are necessary to test invariant rules, such that a fair test of invariant rules will often be challenging. Mismatches between theory and observation might also occur because theory assumes that mothers exert control over resource allocation to offspring. In fact, there is evidence of widespread genetic correlations among investment per offspring and reproductive effort, such that these traits are not independent.