Figure 7. Interaction chain but not interaction modification may be represented by a multispecies pairwise model.

We examine three-species communities engaging in indirect interactions. Each species pair is representable by a two-species pairwise model (saturable L-V or alternative pairwise model, purple in the right columns of B, D, and F). We then use these two-species pairwise models to construct a three-species pairwise model, and test how well it predicts the dynamics known from the mechanistic model. In B, D, and F, left panels show dynamics from the mechanistic models (solid lines) and three-species pairwise models (dotted lines). Right panels show the difference metric $\overline{D}$. (A–B) Interaction chain: S₁ affects S₂, and S₂ affects S₃. The two interactions employ independent mediators C₁ and C₂, and both interactions can be represented by the saturable L-V pairwise model. The three-species pairwise model matches the mechanistic model in this case. Simulation parameters are provided in Figure 7—source data 1. (C–F) Interaction modification. In both cases, the three-species pairwise model fails to predict reference dynamics even though the dynamics of each species pair can be represented by a pairwise model. (C–D) S₃ consumes C₁, a mediator by which S₁ stimulates S₂. Parameters are listed in Figure 7—source data 2. Here, S₁ changes the nature of interaction between S₂ and S₃: S₂ and S₃ do not interact in the absence of S₁, but S₃ inhibits S₂ in the presence of S₁. The three-species pairwise model makes qualitatively wrong prediction about species coexistence. As expected, if S₃ does not remove C₁, the three-species pairwise model works (Figure 7—figure supplement 1A–B). (E–F) S₁ and S₃ both supply C₁ which stimulates S₂. Here, no species changes ‘the nature of interactions’ between any other two species: both S₁ and S₃ contribute reusable C₁ to stimulate S₂. S₁ promotes S₂ regardless of S₃; S₃ promotes S₂ regardless of S₁; S₁ and S₃ do not interact regardless of S₂. However, a multispecies pairwise model assumes that the fitness effects from the two producers on S₂ will be additive, whereas in reality, the fitness effect on S₂ saturates at high . As a result, the three-species pairwise model qualitatively fails to capture relative species abundance. As expected, if C₁ affects S₂ in a linear fashion, the community dynamics is accurately captured in the multispecies pairwise model (Figure 7—figure supplement 1C–D). Simulation parameters are listed in Figure 7—source data 3.

DOI: http://dx.doi.org/10.7554/eLife.25051.026

Figure 7—figure supplement 1. A multispecies pairwise model can work under special conditions.

(A–B) As a control for Figure 7C, if S₃ does not remove the mediator of interaction between S₁ and S₂, a three-species pairwise model accurately matches the mechanistic model. Simulation parameters are provided in Figure 7—source data 4. (C–D) As a control for Figure 7E, we ensured that fitness effects from multiple species are additive. In this case, a three-species pairwise model can represent the mechanistic model. To ensure the linearity and additivity of fitness effects, we have used a larger value of half saturation concentration ($K_{S_2{C}_1}$=10³ μM, instead of 10⁻¹ μM in Figure 7E–F). We have adjusted the interaction coefficients accordingly such that the overall interaction strength exerted by S₁ and S₃ on S₂ is comparable to that in Figure 7E–F (as evident by comparable population compositions). Since the interaction influences under these conditions remain in the linear range, the three-species pairwise model accurately predicts the reference dynamics. Simulation parameters are provided in Figure 7—source data 5.

DOI: http://dx.doi.org/10.7554/eLife.25051.032