A Diversity-Interactions (DI) model typically includes three components in the linear predictor and a random error term:

\[\Large y = Identities + Interactions + Structures + \epsilon \quad\quad\quad\mbox{eqn (1)}\]

where \(y\) is a community-level response (e.g., biomass for a plant ecosystem); \(Identities\) are the effects of species identities and enter the model as the initial individual species proportions (relative abundances), e.g. the proportions at the beginning of the experiment or observational study; the \(Interactions\) are the pairwise effects of interspecific interactions between the initial species proportions; while \(Structures\) include other experimental structures, such as blocks, treatments or environmental gradients. The response or ‘ecosystem function’ is measured a fixed time after the initial proportions are set. \(\epsilon\) is the error term, typically assumed independent, and normally distributed with mean 0 and constant variance.

As a simple example, in a three-species randomised complete block design experiment, a possible DI model is:

\[\Large y=\sum_{i=1}^{S=3} \beta_{i} p_{i} + \sum_{\substack{i,j=1 \\ i<j}}^{S=3} \delta_{ij} (p_{i} p_{j}) + \alpha_k + \epsilon \quad\quad\quad\mbox{eqn (2)}\]

where \(p_i\) is the proportion of species \(i\), and \(\alpha_k\) is the effect of block k. The errors (\(\epsilon\)) are usually assumed to be independent and normally distributed with mean zero and constant variance \(\sigma^2\). The interpretation of this model is:

• In a monoculture of the i-th species, the expected performance of Y in block k is \(\beta_i + \alpha_k\).

• In a mixture, ignoring the block effects, the \(Identities\) component provides the expected performance as a weighted average of monoculture responses, and the \(Interactions\) component is added to it to give the overall expected performance of the mixture community.

The following figure illustrates how the parameter estimates in a DI model combine to predict the response (at time \(t+1\)) based on the species proportions in monoculture or mixture communities at time \(t\).

The model (eqn 2) assumed each species had its own identity effect estimate (\(\hat{\beta}_1\), \(\hat{\beta}_2\) and \(\hat{\beta}_3\) for species 1, 2 and 3 respectively), while each pair of species also had a unique interaction estimate (\(\hat{\delta}_{12}\), \(\hat{\delta}_{13}\) and \(\hat{\delta}_{23}\)). The predicted response for each monoculture and for the four mixture community with equal initial proportions are shown.

• The identity effects (\(\hat{\beta}_i\)’s ) estimate the performance of each species in monoculture and are also used in predicting mixture performances (as illustrated for the 0.5:0.5:0 community).

• The ‘community’ values on the x-axis represent the initial proportions p1:p2:p3.

• The predictions for the response at time \(t+1\), are generated based on the initial species proportions measured or manipulated at time \(t\).

One question of interest is whether mixture communities perform differently from what might be expected from the weighted averaging of monoculture performances. This is answered by inspecting the \(Interactions\) component (which may also be called the ‘diversity effect’, see Kirwan et al. 2009). The identity effects measure the species-specific contribution to a mixture; however, the interactions component allow for the potential for two species to interact and either increase or decrease the overall community performance compared to the weighed identity effects. For example, in a two-species community with species 1 and 2, if the species are combined in a 50:50 binary community but do not interact (\(\delta_{12} = 0\)), then the expected performance is \(\beta_1 \times 0.5 + \beta_2 \times 0.5\); however, if \(\delta_{12} \ne 0\), then the expected performance is \(\beta_1 \times 0.5 + \beta_2 \times 0.5 + \delta_{12} \times 0.5 \times 0.5\), where the estimate of \(\delta_{12}\) may be positive or negative. The community with the best overall performance will depend on both the \(Identities\) and the \(Interactions\) and will rarely be the community with the largest net interactions effect because the identity effects can be strongly influential in determining a mixture performance.

Equation (2) has a unique interaction term for each pair of species. It is possible, however, to test various forms of species interactions, some of which may be motivated by the context of the experiment and biological assumptions (Kirwan et al., 2009). For example, when the number of species is large, the number of pairwise interactions increases quickly and may be difficult to interpret or may yield non-estimable effects due to study design. Imposing different constraints in the Interactions term can make DI models more biologically informative and estimable. The Interactions component may also include a non-linear exponent parameter \(\theta\) on each \(p_ip_j\) term, where a value different to one allows the importance and impact of interaction terms to be modelled (Connolly et al. 2013). The table below shows a list of DI models with varying (and more parsimonious) interaction specifications. In addition, one may use random effects to account for some of the variation due to species interactions (Brophy et al. 2017, McDonnell et al. 2023).

This table shows five types of DI models, each showing the \(Identities\) component and the \(Interactions\) component of the respective DI models. The forms of the interactions across the models are: no interactions (ID model), all pairwise interactions assumed equal (AV model), interactions dictated by functional group membership (FG model), species specific contributions to interactions (ADD model) and a unique interaction for each pair of species (FULL). The notation used includes: \(s\) = the number of species in the pool and where the species are categorised by T functional groups (= FG) and \(p_i\) = the initial proportion of species \(i\). See Kirwan et al. (2009) for further explanation of the biological interpretations.

References

Brophy et al. 2017. Journal of Agricultural, Biological, and Environmental Statistics, 16, 409-421.

Connolly et al. 2013. Journal of Ecology, 101, 344-355.

Kirwan et al. 2009. Ecology, 90, 2032-2038

McDonnell et al. 2023. Journal of Agricultural, Biological and Environmental Statistics; 28, 1-19.