Global stability of obligate mutualistm in community modules with facultative mutualists

Mutualism is a fundamental building block of ecological communities and an important driver of biotic evolution. Classic theory suggests that a pairwise two-species obligate mutualism is fragile, with a large perturbation potentially driving both mutualist populations into extinction. In nature, however, there are many cases of pairwise obligate mutualism. Such pairwise obligate mutualisms are occasionally associated with additional interactions with facultative mutualists. Here, we use a mathematical model to show that when a two-species obligate mutualism has a single additional link to a third facultative mutualist, the obligate mutualism can become permanently persistent. In the model, a facultative mutualist interacts with one of two inter-dependent obligate mutualists, and the facultative mutualist enhances the persistence not only of its directly interacting obligate mutualist, but also that of the other obligate mutualist indirectly, enabling the permanent coexistence of the three mutualist species. The effect of the facultative mutualist is strong; it can allow a three-species permanent coexistence even when two obligate mutualists by themselves are not sustainable (i.e. not locally stable). These results suggest that facultative mutualists can play a pivotal role for the persistence of obligate mutualisms, and contribute to a better understanding on the mechanisms maintaining more complex mutualistic networks of multiple species.