Abstract
Minimal models for complex ecosystems often assume random interactions, whose statistics suffice to predict dynamical and macroecological patterns. However, interaction networks commonly possess large-scale structures, such as hierarchies or functional groups. Here, we ask how conclusions from random interaction models are altered by the presence of such large-scale network structures. We consider a simple superposition of structured and random interactions in a classic population dynamics model, and study macroscopic observables, abundance distributions and dynamical regimes. Randomness and structure combine in a surprisingly yet deceptively straightforward way: contributions from each component to the patterns are largely independent, and yet their interplay has non-trivial consequences, notably out of equilibrium. We conclude that whether interaction structure matters depends on the pattern: conclusions from randomly interacting models are less robust when considering static patterns of species presence and abundance, and more robust in the dynamical nature of their ecological regimes.
Competing Interest Statement
The authors have declared no competing interest.
Footnotes
Several paragraphs shortened; figure 6 updated
1 The most significant difference with prior studies is that, while DMFT in the usual setting yields a low-dimensional model, i.e. a single equation representing the distribution of possible trajectories of any species picked at random in the system, here equations eq. (6) are still as many as the initial model, because species are intrinsically different from each other due to their distinct u’s and v’s; what has been gained is that the equations are now uncoupled.
2 The equations above are strongly nonlinear at
, when species go extinct. Therefore, they generally admit multiple equilibrium solutions, which may differ in abundance as well as stability.
3 In the case without disorder, eq. (12), the interaction matrix Aij was low-rank, allowing direct coarse-graining, whereas here a low-dimensional description exists despite (Aij) being full-rank. Despite accounting for most of the dimensionality of the matrix, random interactions thus modify the system in a minimal way, by adding a single macroscopic relation on top of those provided by structure.