Erratum: “Controlling species densities in structurally perturbed intransitive cycles with higher-order interactions” [Chaos 32(10), 103122 (2022)]

The impact of localized perturbation on a single pair of interaction: We have perturbed the first and second species’ density by $β≠0$⁠. Due to our choice of $α=0.75>0.50$⁠, $β$ can be varied within $[−0.25,0.25]$⁠. However, we ignore the case for $β=−0.25$ to avoid the rank deficient matrix $A ~$⁠. If we choose $β<−0.25$⁠, the interaction between the first and the second species gets reversed, so we disregard these values of $β$⁠. For $β>0.25$⁠, $α$ exceeds the value $1$⁠. (a) External perturbation (⁠ $β>0$⁠). Here, the unperturbed species is marked by blue diamonds, and first two perturbed species are marked by red circles. Here, the abundance of unperturbed species increases slowly, and the abundances of perturbed species are decreased slowly. (b) Internal perturbation (⁠ $β<0$⁠). Here, the abundance of unperturbed species (blue diamonds) is dropped rapidly. The analytical calculation fits perfectly with the numerically simulated values shown by the markers.

Impact of a single perturbation on the first-second species interaction: (a) and (d) A common convergence for all the species in an identical density is observed for the unperturbed case with $β=0$⁠. (b) and (e) For $β=0.1$⁠, the external perturbation reduces their respective densities of first and second species. The emergence of two clusters is represented in subfigure (e) after the initial transient. (c) and (f) For $β=−0.1$⁠, the internal perturbation enhances the densities of two perturbed species. The unperturbed three species maintain a coherent state. All these observations remain valid for any permissible choice of initial conditions. Here, $α=0.75$ and $N=5$⁠.