Statistical methods for paleodemography on fossil assemblages having small numbers of specimens: an investigation of dinosaur survival rates

We describe statistical methods to formulate and validate statements about survival rates given a small number of individuals. These methods allow one to estimate the age-specific survival rate and assess its uncertainty, to assess whether the survival rates during some age range differ from the survival rates during another age range, and to assess whether the survivorship curve has a particular shape. We illustrate these methods by applying them to a sample of 22 Albertosaurus sarcophagus individuals. We show that this sample is too small to provide any confidence in the claim that this species had a “convex” survivorship curve arising from age-specific survival rates that decreased monotonically with age. However, we show that a sample of 50 to 100 individuals has reasonable statistical power to support such a claim. There is evidence for the much weaker claim that average survival rates for ages 2 to 15 were higher than survival rates for later ages. Finally, we describe one way to account for size-dependent fossilization rates and show that a plausible positively-size-dependent fossilization rate results in a substantially non-convex survivorship curve for A. sarcophagus.

A revised life table and survivorship curve for Albertosaurus sarcophagus based on the Dry Island mass death assemblageThis article is one of a series of papers published in this Special Issue on the theme Albertosaurus.

In 1910 a crew from the American Museum of Natural History discovered a bone bed composed primarily of the large tyrannosaurid Albertosaurus sarcophagus in what is now Dry Island Buffalo Jump Provincial Park in Alberta, Canada. Study of the remains from the site allowed the first life table and survivorship curve for a non-avian dinosaur to be created. These have served as a model for subsequent studies of dinosaurian population biology. Since 2006, the discovery and preparation of hundreds of new elements from the bone bed stand to substantially increase the minimum number of individuals (MNI) represented. This would allow testing of previous conclusions regarding the population biology of these animals and refinement of our understanding of the patterns of survivorship. Here, four formerly unrecognized individuals from the Dry Island assemblage are revealed and a revised life table presented. As in the previous analysis, a left skewed age distribution and sigmoidal survivorship pattern were found. Annual mortality rates averaged 3.47% between ages two and 13 and then increased to a mean of 19.5% prior to extinction of the cohort after 28 years of age. Mean life expectancy for individuals surviving to two years of age was 15.19 years. Mid-life increase in attrition corresponds to entrance into the breeding population. The MNI is unlikely to substantially increase, and new individuals are unlikely to affect the pattern of survivorship inferred here. Nevertheless, future excavations stand to reveal more about the anatomical and pathological variance within the Dry Island Albertosaurus population.

Invalid estimates of rate of population increase from Glossina (Diptera: Glossinidae) age distributions

Several published reports have presented estimates of the rate of increase, r, based on sampled ovarian age distributions from Glossina populations throughout Africa. These estimates are invalid, because an age distribution sampled at one point in time can be equated to a survivorship curve only if r = 0. When such a survivorship curve and a corresponding fecundity schedule are then used to estimate r via the Euler-Lotka equation, the result is a value of r near zero, regardless of the population's true rate of increase. Synthetic sampling from a hypothetical tsetse population confirmed that estimates computed in this fashion are entirely the products of sampling error. Valid estimates of r can sometimes be obtained from an age distribution, using an alternative method, but such estimates are highly sensitive to sampling errors in the distribution.

Survivorship in the Bivalvia: comparing living and extinct genera and families

George Gaylord Simpson was one of the first paleontologists to apply survivorship analysis to the study of fossil taxa. His finding that the survivorship curve for extant bivalve genera plotted above that for extinct genera led him to conclude that bivalve genera are drawn from at least two distinct distributions of longevities, and formed the fundamental basis for his influential concepts of horotelic and bradytelic evolutionary rates. Survivorship curves presented in this paper show the same pattern of disjunct survivorship in genera from the Treatise on Invertebrate Paleontology and in families from Sepkoski's compendium.Some of the observed differences between survivorship curves are artificial, occurring because long-lived genera and families are more likely to survive to the Recent than short-lived genera and families. The living fauna thus contains a disproportionate number of long-lived genera and families, and the survivorship curve for the living fauna is expected to lie above that for the extinct fauna for this reason alone—even if all longevities are drawn from the same distribution. Recognition of this bias led Raup (1975) to question the significance of the survivorship patterns presented by Van Valen (1973), and Stanley's (1984) acceptance of Raup's argument led him to dismiss the survivorship pattern discovered by Simpson. But statistical analysis using bootstrapping shows that this bias accounts for only a small proportion of the difference between survivorship curves. Other biases considered, such as “pull of the Recent,” “asymmetrical range truncation,” and erroneous concatenation of stratigraphic ranges, do not account for the pattern either. Although still other biases, as yet unknown, cannot definitively be ruled out, it appears that the longevities of extinct and living bivalve taxa are meaningfully different, and that the fundamental causes are biological.

Forest fire cycles and life tables: a case study from interior Alaska

The negative exponential and Wiebull distributions were used to estimate stand survivorship curves for forested sites in the Porcupine River drainage of interior Alaska. The survivorship curve of Piceaglauca (Moench) Voss sites was best described by a Wiebull function, while both functions adequately described the Piceamariana (Mill.) Britton, Sterns & Poggenburg hardwood and all sites stand survivorship curve. Fire cycles calculated from the Wiebull distribution were 43, 113, 36, and 26 years for the entire study area, P. glauca, P. mariana, and hardwood sites, respectively. Fire frequencies estimated from a life table analysis were 48, 105, 43, and 30 years, respectively. The relationship between fire cycle and fire frequency calculations is discussed and various management implications are given.