Biodiversity, functional redundancy and system stability: subtle connections.

The relationship between biodiversity and functional redundancy has remained ambiguous for over a half-century, likely due to an inability to distinguish between positivist and apophatic (that which is missing) properties of ecosystems. Apophases are best addressed by mathematics that is predicated upon absence, such as information theory. More than 40 years ago, the conditional entropy of a flow network was proposed as a formulaic way to quantify trophic functional redundancy, an advance that has remained relatively unappreciated. When applied to a collection of 25 fully quantified trophic networks, this authoritative index correlates only poorly and transitively with conventional Hill numbers used to represent biodiversity. Despite such a weak connection, the underlying biomass distribution remains useful in conjunction with the qualitative diets of system components for providing a quick and satisfactory emulation of a system's functional redundancy. Furthermore, an information-theoretic cognate of the Wigner Semicircle Rule can be formulated using network conditional entropy to provide clues to the relative stability of any ecosystem under study. The necessity for a balance between positivist and apophatic attributes pertains to the functioning of a host of other living ensemble systems.