DI Models

The Maths of DI Models

Welcome to the maths of Diversity-Interactions (DI) models. Here you will find ...

Different interaction structures for Diversity-Interactions models

Null Model
$$\large y= \mu + \alpha A + \epsilon$$
No effect of changing diversity on ecosystem function.
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Species Identities Only
$$\large y = \sum_{i=1}^{S} \beta_{i} p_{i} + \alpha A + \epsilon$$
Ecosystem function is affected only by species identites.
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Average Pairwise Interaction
$$\large y = \sum_{i=1}^{S} \beta_{i} p_{i} + \delta_{AV} \sum_{\substack{i,j=1 \\ i < j}}^{S} p_{i} p_{j} + \alpha + \epsilon$$
Strength of interaction is same for all pairs of species so a single interaction term is sufficient.
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Functional Groups
$$\large y = \sum_{i=1}^{S} \beta_{i} p_{i} + \sum_{q=1}^{T} \omega_{qq} \sum_{\substack{i,j \in FG_k \\ i \lt j}} p_i p_j + \sum_{\substack{q,r = 1 \\ q \lt r}}^{T} \omega_{qr} \sum_{i \in FG_q} \sum_{j \in FG_r} p_i p_j + \alpha A + \epsilon$$
Species can be grouped into functional groups based on the functions they perform in the ecosystem.
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Additive Species Effects
$$\large y = \sum_{i=1}^{S} \beta_{i} p_{i} + \sum_{\substack{i,j=1 \\ i < j}}^{S} (\lambda_{i} + \lambda{j}) (p_{i} p_{j}) + \alpha A + \epsilon$$
The contribution a species makes in its interaction is the same regardless of the species it interacts with.
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Full Pairwise Interactions
$$\large y=\sum_{i=1}^{S} \beta_{i} p_{i} + \sum_{\substack{i,j=1 \\ i < j}}^{S} \delta_{ij} (p_{i} p_{j}) + \alpha A + \epsilon$$
Each pair of species has a unique interaction effect on the ecosystem function.
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