Mutualisms are ubiquitous in nature, provide important ecosystem
services, and involve many species of interest for conservation.
Theoretical progress on the population dynamics of mutualistic
interactions, however, comparatively lagged behind that of trophic and
competitive interactions, leading to the impression that ecologists
still lack a generalized framework to investigate the population
dynamics of mutualisms. Yet, over the last 90 years, abundant
theoretical work has accumulated, ranging from abstract to detailed.
Here, we review and synthesize historical models of two-species
mutualisms. We find that population dynamics of mutualisms are
qualitatively robust across derivations, including levels of detail,
types of benefit, and inspiring systems. Specifically, mutualisms tend
to exhibit stable coexistence at high density and destabilizing
thresholds at low density. These dynamics emerge when benefits of
mutualism saturate, whether due to intrinsic or extrinsic
density-dependence in intraspecific processes, interspecific processes,
or both. We distinguish between thresholds resulting from Allee effects,
low partner density, and high partner density, and their mathematical
and conceptual causes. Our synthesis suggests that there exists a robust
population dynamic theory of mutualism that can make general
predictions.