Models of Biodiversity-Ecosystem-Function relationships: Pitfalls hidden in the mathematical form, and in transformation of the functional response

Background/Question/Methods

The impact of species loss on the delivery of ecosystem functions is sometimes estimated through Biodiversity-Ecosystem Function (BEF) relationships. We highlight issues in modelling data to estimate this relationship: (i) is the mathematical model used a good framework within which to quantify BEF relationships and (ii) transformations of data prior to analysis may imply BEF relationships that might be less acceptable if explicitly stated. A saturating response is most widely assumed in BEF relationships, i.e. that the maximum/minimum ecosystem function is bounded (Hooper et al 2005, Ecological Monographs, 75, 3-35). We show that a widely accepted mathematical model of BEF does not saturate but rather changes linearly with diversity and when applied to the log of function the BEF increases/decreases exponentially.

In recent work (Hector et al, 2007 Nature, 448, 188-190, Isbell et al 2011, Nature, 477, 199–202), the presence-absence (PA) model was used to model  y=log(function). The PA model for the expected community response (E(y)) for a community drawn from an s-species pool is E(y)=α+Σ₁^s β_iX_i, where variables Xi, i = 1…s are 1 if the ith species is present and 0 otherwise. Properties of this model are outlined.

Results/Conclusions

In the PA model the expected average of all monocultures is E(y)=α+β where β is the average of all β_i. The diversity effect, the difference between the expected mixture response and the average monoculture response, is (r-1)β when averaged across all r-species mixtures (r<s). Thus, the diversity effect changes linearly with species richness at a rate of β per unit increase in richness and does not saturate.

If the functional response variable is log transformed before analysis then the PA model predicts that the proportional change in response per unit increase in diversity is exp(β) and so the BEF function increases/decreases exponentially.

We recognize that while a saturating relationship is more appropriate for theoretical and physical reasons, many transformations of richness used to produce a linear BEF relationship give a decelerating but not saturating relationship. If the true relationship only saturates at an unrealistically high number of species, then it may be reasonable to model it with a decelerating non-saturating mathematical model, or even, perhaps, a model predicting linear or exponentially changing function over the richness range of interest. However, the ecologist should be aware of, and be prepared to justify, the implicit properties of the model in making this choice.