Modification speed alters stability of ecological higher-order interaction networks

Abstract

Higher-order interactions (HOIs) have the potential to greatly increase our understanding of ecological interaction networks beyond what is possible with established models that usually consider only pairwise interactions between organisms. While equilibrium values of such HOI-based models have been studied, the dynamics of these models and the stability of their equilibria remain underexplored. Here we present a novel investigation on the effect of the onset speed of a higher-order interaction. In particular, we study the stability of the equilibrium of all configurations of a three-species interaction network, including transitive as well as intransitive ones. We show that the HOI onset speed has a dramatic effect on the evolution and stability of the ecological network, with significant structural changes compared to commonly used HOI extensions or pairwise networks. Changes in the HOI onset speed from fast to slow can reverse the stability of the interaction network. The evolution of the system also affects the equilibrium that will be reached, influenced by the HOI onset speed. This implies that the HOI onset speed is an important determinant in the dynamics of ecological systems, and including it in models of ecological networks can improve our understanding thereof.

Figures (5)

• Figure 1: Different 3-species systems with one HOI. The large arrowhead represents the negative side of the interaction, the small arrowhead represents the positive side. (A-C) The three systems where the pairwise interactions form a transitive system. $A > B > C$ and $A > C$ in the pairwise system. (D) A system where the pairwise interactions form an intransitive system. $A > B > C > A$ in the pairwise system.
• Figure 2: Three different possibilities for one HOI to manifest, shown for the 3-species intransitive system. (A) Symmetric modification $\overleftrightarrow{AB}_{C}$: both sides of the pairwise interaction get modified. (B) Asymmetric modification $\overleftarrow{BA}_{C}$: only the negative effect of $A$ on $B$ gets modified. (C) Asymmetric modification $\overleftarrow{AB}_{C}$: only the positive effect on $A$ from $B$ gets modified.
• Figure 3: The effect of the modification speed $\omega$ and modification strength $\beta$ on the intransitive system stability and coexistence of species. (A) The effect on stability. In the blue region, parameter combinations result in convergence to a stable equilibrium. In the yellow region, parameter combinations result in convergence to a stable limit cycle. (B) The effect on coexistence. The light green region is the region where all species coexist. The dark green region is the region where only one species survives. (C) The superposition of A and B. (D-H) Time series of the species abundances for different parameter combinations. (D) $\omega = 0.1, \beta = -3$ (E) $\omega = 1, \beta = -3$. (F) $\omega = 10, \beta = -3$. (G) $\omega = 1, \beta = -7$. (H) $\omega = 1, \beta = -12$.
• Figure 4: The oscillation probabilities $\xi$ for non-identical pairwise interaction coefficients, where $\alpha_{AB} \neq \alpha_{AC} = \alpha_{BC}$. (A) The intransitive system. (B) Transitive system $\mathcal{C}$. One pixel corresponds to the oscillation probability $\xi$ for the given combination of $\alpha_{AB}$ and $\alpha_{AC}=\alpha_{BC}$. $\xi$ is defined as the area of oscillations compared to the total area of the parameter space as in Fig. \ref{['fig: oscillations coexistence intransitive double']}. The region of oscillations is defined as the area where the system converges to a stable limit cycle. One pixel in this figure captures $27\times 17 = 459$ parameter combinations of $\beta$ and $\omega$, with domains $[10^{-3}, 10^2[$ and $[-80, 0[$ respectively.
• Figure 5: How oscillations occur in the intransitive system ($\beta = -3$, $\omega = 1$). (A) The species abundances $n_i$ over time. (B) The modifier value $m$ over time. (C) When $m < 0$, the system evolves from intransitive to transitive due to modification.