The impact of structured higher-order interactions on ecological network stability

Abstract

The impact of higher-order interactions, those involving more than two species, is increasingly appreciated as having the potential to strongly influence the dynamics of complex ecological systems. However, although the critical importance of the structure of pairwise interaction networks is well established, studies of higher-order interactions still largely assume random structures. Here, we demonstrate the strong impact of structured higher-order interactions on simulated ecological communities. We focus on effects caused by interaction modifications within food webs, where a consumer resource interaction is modified by a third species, and for which plausible structures can be hypothesised. We show how interaction modifications introduced under a range of non-random distributions may impact the overall network structure. Local stability and the size of the feasibility domain are critically dependent on the inter-relationship between trophic and non-trophic effects. Where interaction modifications are structured into mutual interference motifs (associated with consumers switching between resources) synergistic signs and topological effects have particularly consequential impacts. Furthermore, we show that previous results of the impact of higher-order interactions on diversity-stability relationships can be reversed when higher-order interactions are structured, not random. Empirical data on interaction modifications will be a key part of improving understanding the dynamics of communities, particularly the distribution of interaction modification signs across networks.

Supplementary information: The online version contains supplementary material available at 10.1007/s12080-025-00603-0.

Keywords: Feasibility; Food web; Higher-order interaction; Interaction modification; Interaction network; Non-trophic effect; Stability.

Fig. 1

Clarification of key terms. a Schematic representation of the relationship between a facilitatory trophic interaction modification (TIM, blue dashed line) and consequent two non-trophic effects (NTE, brown solid lines). We label this TIM ‘facilitatory’ (rather than ‘interfering’) because an increase in the modifier species increases the strength of the interaction. b Resultant community matrix, where each column shows the effect of an increase in a species on the growth rate of species in each row (i.e. ${\mathit{A}}_{\mathit{i}\mathit{j}}=\frac{\partial }{\partial x_j}\frac{d{x}_i}{\mathit{dt}}$), where $x_i$ is the abundance of species $i$. Here, an interaction modification results in two non-trophic effects (in red). The modifier species $k$ acts to strengthen a trophic interaction between consumer $j$ on resource $i$. This results in a positive NTE on $j$ and a negative NTE on $i$. c An example 4-way higher-order interaction where species $j$, $k$ and $l$ have a combined impact on the dynamics of focal species $i$

Fig. 2

Illustration of the structured TIM distribution models used in this study. Cartoons illustrate the distinctive properties of each model compared to the baseline model that introduced interaction modifications such that each potential modification was equally and independently likely to occur (see the main text for a full description of each model)

Fig. 3

Illustration of construction of community matrices that specify the linearized interactions between species as quantified by the impact on population growth rates. At an assumed equilibrium, a combined community matrix ($\mathbf{A}$) is constructed from the sum of the impact of trophic ($\mathbf{B}$) and non-trophic effects ($\mathbf{C}$). Here, the species are arranged approximately in trophic height order, with basal species in the first rows and columns, and top predators in later rows and columns. The underlying trophic network depicted was generated with the niche model (Williams and Martinez 2000) with species number = 20, target connectance = 0.2 and parameterised with a bivariate Gaussian distribution. Intra-specific interactions are shown here in grey

Fig. 4

Impact of different distributions of non-trophic effects on network structures. a Connectance (fraction non-zero entries) of overall interaction matrix $\mathbf{A}$. b Connectance of non-trophic interaction matrix $\mathbf{C}$. c Variance of the off-diagonal non-trophic elements $\mathbf{C}$. d Degree heterogeneity of $\mathbf{A}$ (variance of the normalised in and out-degree distribution. e Correlation of the pairwise elements within $\mathbf{A}:{\left({\text{A}}_{\mathit{ij}},,,{\text{A}}_{\mathit{ji}}\right)}_{\text{i}\ne \text{j}}$. f Covariance between elements of trophic and non-trophic effects ($\text{cov}(\mathbf{B},\mathbf{C})$). Colours denote different distributional models, lines are loess fits through 100 replicates at each TIM frequency

Fig. 5

Impact of structured non-trophic effects on dynamic properties. a Effect of increasing frequency of TIMs on instability $\text{log}\left(\mathfrak{R},\left({\mathrm{\lambda }}_1^{\mathbf{A}}\right)\right)$, the degree of self-regulation necessary for local asymptotic stability. b The size of the feasibility domain $\mathrm{\omega }\left(\mathbf{A}\right)$. Loess fitted lines have been added to highlight differences

Fig. 6

Dependence of critical strength of interactions on HOI order and distribution scenarios. Critical strength was the highest coefficient scaler ($\alpha$, $\mathrm{\beta }$, $\mathrm{\gamma }$) where > 90% of 50 random communities showed no extinctions. a) Reproduction of results in Bairey et al. (2016), where interactions at each order (2, 3, 4) cause qualitatively different responses to increasing diversity. b) introducing an underlying trophic network structure with random HOIs maintains the original result c) Structured HOIs (3 or 4 way) reduce stability as diversity increases