How Many Eggs Are Too Many? Utilizing an Under-Dispersed Count Data Model to Gain Insights Into Evolutionary Productivity Constraints on Bird Species

Changes in productivity are primary mechanisms via which bird populations change and understanding how these processes operate is key to monitoring their populations in a changing environment. A major component of productivity is fecundity, the number of propagules produced, which for birds is the number of eggs laid (clutch size) and chicks that hatch from these (brood size). There are evolutionary constraints on the size of these fecundity measures and, therefore, variation tends to be smaller than other forms of count data. Using data on clutch and brood sizes for 55 and 52 UK bird species respectively we show these are consistently under-dispersed with respect to the standard Poisson model, which is often used to fit such data. A three-parameter exponentially weighted Poisson (EWP₃) model fits substantively better than either a Poisson or under-dispersed variants. We provide an R package to enable easy fitting of such models. The EWP₃ is characterized by two dispersion parameters, ${\beta }_1$ and ${\beta }_2$, and we suggest that these can quantify evolutionary constraints on incubation. We show that ${\beta }_2$ is generally greater than ${\beta }_1$, indicating a greater compression at the right hand end of the distribution. This suggests that the cost of having an extra egg or chick is higher than the cost of having one too few. Although we consider avian reproduction this method should be suitable for any species which has a small number of offspring in each reproductive event.