High-order species interactions shape ecosystem diversity

Abstract

Classical theory shows that large communities are destabilized by random interactions among species pairs, creating an upper bound on ecosystem diversity. However, species interactions often occur in high-order combinations, whereby the interaction between two species is modulated by one or more other species. Here, by simulating the dynamics of communities with random interactions, we find that the classical relationship between diversity and stability is inverted for high-order interactions. More specifically, while a community becomes more sensitive to pairwise interactions as its number of species increases, its sensitivity to three-way interactions remains unchanged, and its sensitivity to four-way interactions actually decreases. Therefore, while pairwise interactions lead to sensitivity to the addition of species, four-way interactions lead to sensitivity to species removal, and their combination creates both a lower and an upper bound on the number of species. These findings highlight the importance of high-order species interactions in determining the diversity of natural ecosystems.

Figure 1. Species can exhibit high-order interactions, whereby the interaction between two species is affected by other species.

(a) Pairwise interactions in a community: species affect each other directly. For instance: species ‘1' could be producing an antibiotic, inhibiting the growth of species ‘2'. (b) Three-way interactions in a community: one species can modulate the interactions between two others. For instance, species ‘3' might degrade the antibiotics produced by species ‘1', thereby attenuating the inhibitory effect of species ‘1' on species ‘2'. (c) Four-way interactions in a community. For example, species ‘4' might produce a compound that inhibits the antibiotic-degrading enzyme produced by species ‘3'.

Figure 2. The destabilizing effect of diversity is inverted when interactions are high-order.

(a) The critical strength of interactions beyond which the community becomes unfeasible (defined as the value at which 5% of random communities exhibit extinctions; error-bars indicate the range of 2–10%). While the critical strength of pairwise interactions decreases with community diversity (α_c, red, slope=−1.20±0.05), it remains unchanged for three-way interactions (β_c, blue, slope=−0.07±0.04) and increases for four-way interactions (γ_c, green; slope=0.95±0.04). (b) Example simulations of two communities with pairwise interactions of the same strength but with different numbers of species (N=7, left; N=20, right; red square and circle in a). While the small community shows convergence to a stable fixed-point with all species coexisting, the large community exhibits species extinction. (c) As shown in b, but for communities with four-way interactions. Here the trend is reversed, and the small community exhibits extinctions, while the large community exhibits stable coexistence.

Figure 3. The combination of high-order and pairwise interactions determines an optimally stable range of diversities.

(a) Regions of stability for 5, 8 and 18 species communities in the space of pairwise α- and four-way γ-interaction strengths (assuming no three-way interactions, β=0). Dots show the critical thresholds at which 5% of the simulations become unstable when the interaction strengths increase along radial lines in the logα-logγ space (see the ‘Methods' section). (b–d) The fraction of stable communities over the initial number of species is shown for three combinations of pairwise and four-way interaction strengths (at the points labelled by ‘x' in panel a); pairwise-dominated in b, mixed pairwise and four-way in c, and four-way dominated in d. When pairwise interactions dominate, diversity is bounded from above; when four-way interactions dominate, a lower bound on diversity appears; with both types of interactions, the level of diversity is defined to a narrow range, and within this range the community is sensitive not only to the addition but also to the removal of species.

Figure 4. Range of stable diversities over interaction strength for a community with pairwise and four-way interactions.

As the total interaction strength is increased, the lower bound on the number of species increases, while the upper bound decreases, until the range of allowed diversities narrows around a defined number of species N*. Here, the relative strengths of pairwise and four-way interactions is fixed so that both will have a significant effect (γ=10³α). A similar behaviour appears when accounting for three-way interactions (Supplementary Fig. 8). Note that an ecologically relevant lower bound (N>>1) requires the coefficient of the high-order interactions to be much larger than the pairwise interactions (γ⩾α).