# Equation of the Month

A blog run by the

Theoretical Population Ecology and Evolution Group,

Biology Dept.,

Lund University

The purpose of this blog is to emphasize the role of theory for our understanding of natural, biological systems. We do so by highlighting specific pieces of theory, usually expressed as mathematical 'equations', and describing their origin, interpretation and relevance.

# S = cAz

What it means

The equation states the relationship between an area (A) and the expected number of extant species (S) within that area. The constants c and z define the shape of the nonlinear relationship (Rosenzweig 2000). With an increasing area the expected number of species inhabiting that area is also increasing at a rate mainly dictated by z.

Where does it come from?

Originally the relationship, presented above, was theoretically derived from a species-abundance framework (Preston 1962). Given the assumption of a lognormal distribution of species abundances in a community, Preston derived the equation and calculated the z-value to be 0.27. This provided an empirically testable theory of biodiversity in island biogeography as well as mainland regions of different size.

Explanation and implications

The species-area relationship (mainly described by the exponent z above) can be explained by fundamental eco-evolutionary processes such as migration, speciation and extinctions which ultimately are driven by mechanisms such as niche availability, density dependence and species ranges (McGlade 1999). Although all mechanisms possibly are ubiquitous, some may be more important under certain conditions than others.. For example, large geographical areas include more diverse habitats, and hence more niches, facilitating high species diversity. In addition the degree of migration to and from the island, dictated by island area and isolation, has been identified as an important factor affecting the relationship.
In mainland areas with similar conditions the relationship can be explained by population size and geographical range of the species (McGlade 1999). As geographical area is decreased, population sizes and species ranges also decrease. This may give rise to an increase in extinction rate. Conversely, increasing population size and range facilitate speciation as large populations with large ranges often contain large genetic variation and are split into allopatry more often.

It has been shown that the coefficient c is often dependent on the taxon and biogeographical region, whereas z is more stable and has been estimated to fall between 0.20-0.35 for mainland biogeography and 0.12-0.17 for island biogeography (MacArthur 1969). These parameter estimations often fall below the theoretical value derived by Preston. Lower z-values than predicted can, for example, indicate high immigration of transient species from surrounding areas. Conversely, large z-values may indicate large islands or geographical areas which include several biomes whose species can evolve as independent assemblages. The species-area relationship has often been used in conservation biology (Krebs 1999), but not always without problems (see e.g., He & Hubbell 2011)

Mikael Pontarp

He, F. & Hubbell, S.P. (2011) Species-area always overestimate extinction rate from habitat loss. Nature, 473

Krebs, C.J. (1999) Ecological methodology. Addison-Welsy Educational Publishers. Menlo Park

MacArthur, H.R & Wilson, E.O. (1968) The theory of island Biogeography. Princeton university press. Princeton