To measure biodiversity, we draw on the species-area relationship — one of community ecology's few genuine laws — which defines the relationship between the expected number of species and habitat area. This approach to modelling biodiversity is also used by, for example, Nelson et al. (2009). If one graphs the number of species st supported by a particular habitat (i.e. an agricultural land use) i, against its area at, then the data are well approximated by a power function (Rosenzweig 1995):
where the parameter ci is the species productivity of land use i. The higher is c, the more
species a habitat is likely to support. In contrast, z is a scale parameter that determines how species productivity changes in response to habitat area. We then calculated biodiversity or species value as the expected number of unique species in the landscape, ^ st, the upper value of which is constrained by the total area of agricultural
land, A, such that ^ at < A. Since ct and z are positive constants, the marginal
diversity value of habitat is positive (dsi/dai > 0) but decreasing in area
(2s(da2 < 0) since z < 1. Hence any reduction in habitat area will be negative for its
contribution to biodiversity -which follows common perception - but the strength of the impact will depend on the relative scarcity of the habitat and its species productivity. In other words, a relatively large reduction in a common habitat would imply a relatively small reduction in biodiversity value, whereas a small decrease in relatively scarce, productive habitat would imply a relatively large loss in value. The impact of a land use change at the landscape level on biodiversity could therefore be either positive or negative depending on the marginal biodiversity value of competing farmland habitat (e.g. grassland or arable crops).
This indicator has a number of characteristics that are both appealing and useful for policy analysis. Firstly, given observations of species and habitat area, the species productivity factor can be calibrated by rearranging Equation 1 and plugging in the relevant data, that is
Secondly, Equation 1 is a homothetic function because it is homogeneous of degree z. This implies that only relative values of c are needed to rank different land allocations in terms of their contribution to biodiversity. Hence, given some information about the relative values of c for different habitat, the species-area relationship can be used to rank the impacts of changes in agricultural habitat on biodiversity. This is important because few surveys tally all species (Magurran, 2004). Since z typically falls within a narrow range (0.18-0.25) for a diverse suite of ecosystems, we set it to 0.19 (Rosenzweig, 1995).
In the model, we use the number of threatened or red-listed species as a proxy for uniqueness and hence value (IUCN, 2001) — fundamental to the nature of biodiversity value is the number of different or unique species present in the landscape (Weitzman, 1992). Red-listing considers a range of characteristics that are relevant to value (in particular regional and global scarcity) and is the central indicator in international conventions on biodiversity (e.g. Countdown, 2010). Red-listed species represent as well a subset of total species, and, given that the most species-rich habitats in our case-study regions are also those supporting most red-listed species (i.e. pasture and grasslands compared to intensive arable crops), our biodiversity measure can be considered a weighted index of biodiversity value, which is what we require.