Application of a Negative Multinomial Model Gives Insight into Rarity-Area Relationships

The distribution of individuals of different species across different sampling units is typically non-random. This distributional non-independence can be interpreted and modelled as a correlated multivariate distribution. However, this correlation cannot be modelled using a totally independent and random distribution such as the Poisson distribution. In this study, we utilized the negative multinomial distribution to overcome the problem encountered by the commonly used Poisson distribution and used it to derive insight into the implications of field sampling for rare species’ distributions. Mathematically, we derived, from the negative multinomial distribution and sampling theory, contrasting relationships between sampling area, and the proportions of locally rare and regionally rare species in ecological assemblages presenting multi-species correlated distribution. With the suggested model, we explored the cross-scale relationships between the spatial extent, the population threshold for defining the rarity of species, and the multi-species correlated distribution pattern using data from two 50-ha tropical forest plots in Barro Colorado Island (Panama) and Heishiding Provincial Reserve (Guangdong Province, China). Notably, unseen species (species with zero abundance in the studied local sample) positively contributed to the distributional non-independence of species in a local sample. We empirically confirmed these findings using the plot data. These findings can help predict rare species–area relationships at various spatial scales, potentially informing biodiversity conservation and development of optimal field sampling strategies.

Asymptotic Normality Through Factorial Cumulants and Partition Identities

In the paper we develop an approach to asymptotic normality through factorial cumulants. Factorial cumulants arise in the same manner from factorial moments as do (ordinary) cumulants from (ordinary) moments. Another tool we exploit is a new identity for ‘moments’ of partitions of numbers. The general limiting result is then used to (re-)derive asymptotic normality for several models including classical discrete distributions, occupancy problems in some generalized allocation schemes and two models related to negative multinomial distribution.