Zero-Inflated Distributions of Lifetime Reproductive Output

Lifetime reproductive output (LRO), also called lifetime reproductive success (LRS) is often described by its mean (total fertility rate or net reproductive rate), but it is in fact highly variable among individuals and often positively skewed. Several approaches exist to calculating the variance and skewness of LRO. These studies have noted that a major factor contributing to skewness is the fraction of the population that dies before reaching a reproductive age or stage. The existence of that fraction means that LRO has a zero-inflated distribution. This paper shows how to calculate that fraction and to fit a zero-inflated Poisson or zero-inflated negative binomial distribution to the LRO. We present a series of applications to populations before and after demographic transitions, to populations with particularly high probabilities of death before reproduction, and a couple of large mammal populations for good measure. The zero-inflated distribution also provides extinction probabilities from a Galton-Watson branching process. We compare the zero-inflated analysis with a recently developed analysis using convolution methods that provides exact distributions of LRO. The agreement is strikingly good.