The generalized replication principle and the partitioning of functional diversity into independent alpha and beta components

The replication principle was first proposed by Hill (1973, Ecology 54: 247- 432) as an advantageous property of his family of diversity indices. Later Jost (2007, Ecology 88: 2427- 2439) discovered that diversity measures satisfying this principle allow partitioning of gamma diversity into independent alpha and beta components by simple multiplicative partitioning. Despite the emerging agreement on measuring taxonomic beta-diversity by multiplicative partitioning of Hill diversity, there is no consensus on how to measure functional beta diversity. Two different generalizations of Hill numbers for measuring functional diversity were proposed by Leinster and Cobbold (2011, Ecology 93: 477- 489) and Chiu and Chao (2014, PLoS One 9: e1000014). Both generalizations attempted to satisfy the generalized replication principle, but they formulate it in different ways. The aims of this paper are 1) to review approaches for measuring functional diversity in units of equivalent numbers without explicit reference to replication principle; 2) to compare the two proposed replication principle and to point out some important differences in the behavior of diversity families derived from the two principles; 3) to explore the conditions necessary for partitioning functional diversity of Leinster and Cobbold into meaningful alpha and beta components; 4) and, finally, to explore how transformation of among-species distances into similarities influences the sensitivity of functional diversity to the scale parameter.