When can few-species models describe dynamics within a complex community?

Dynamics of species’ abundances in ecological communities are often described using models that only account for a few species. It is not clear when and why this would be possible, as most species form only a part of diverse ecological communities, with many species that are not included in these few-variable descriptions. We study theoretically the circumstances under which the use of such models is justified, by considering the dynamics of a small set of focal species embedded within a diverse, sparsely-interacting community. We find that in some cases many species fluctuate chaotically and a few-variable description becomes impossible. Yet in other cases, where the abundance fluctuations are concentrated on the focal species and those close to them on the interaction network, we show that such a description exists, even when the interactions with the surrounding community are not weak, and its effect on the focal species’ dynamics is nontrivial. We show that in many cases the time-dependent abundances of the focal species can be well-described by dynamical equations that explicitly model only the focal species abundances, with the effect of the surrounding community incorporated merely by changing the model parameters to effective ones. We give two different methods for achieving this, cast as approximations that become exact in two different regimes, depending on the focal species and their relation to the rest of the community. Both methods work surprisingly well in many of the cases that we check, with effective dynamics that are often very similar and sometimes indistinguishable from the true dynamics, even when the effect of the community on the focal species is significant.