Mutualistic networks, such as plant–pollinator networks, have attracted increasing attention in the ecological literature in the last decades, not only because of their fascinating natural history, but also because mutualistic interactions have been shown to play a key role in the maintenance of biodiversity. Although inter-specific competition has long been known to be a crucial driver of species coexistence as well, there is a lack of theory investigating the interplay between the structures of competitive and mutualistic networks when jointly considered. Here, we develop an analytical framework to study the structural stability — the range of conditions under which all species coexist stably, i.e. where the community is both feasible and stable — of ecological communities in which both mutualistic interactions between plants and pollinators and competitive interactions among plants and among pollinators are present. Using the structure of 50 real networks for mutualistic interactions, combined with analytical and numerical analyses, we show that the structure of the competitive network radically alters the necessary conditions for species coexistence in these communities. Our mathematical framework also allows to accurately characterize the structural stability of these systems. Moreover, we introduce a new metric that accurately links the network structures of competitive and mutualistic interactions to species coexistence. Our results highlight the joint role of the structures of different interaction types to understand the stability of ecological communities and facilitate the analysis of similar natural and artificial systems in which mutualism and competition coexist.@en

Networks are often made up of several layers that exhibit diverse degrees of interdependencies. An interdependent network consists of a set of graphs G that are interconnected through a weighted interconnection matrix B, where the weight of each intergraph link is a non-negative real number p. Various dynamical processes, such as synchronization, cascading failures in power grids, and diffusion processes, are described by the Laplacian matrix Q characterizing the whole system. For the case in which the multilayer graph is a multiplex, where the number of nodes in each layer is the same and the interconnection matrix B=pI, I being the identity matrix, it has been shown that there exists a structural transition at some critical coupling p∗. This transition is such that dynamical processes are separated into two regimes: if p>p∗, the network acts as a whole; whereas when p<p∗, the network operates as if the graphs encoding the layers were isolated. In this paper, we extend and generalize the structural transition threshold p∗ to a regular interconnection matrix B (constant row and column sum). Specifically, we provide upper and lower bounds for the transition threshold p∗ in interdependent networks with a regular interconnection matrix B and derive the exact transition threshold for special scenarios using the formalism of quotient graphs. Additionally, we discuss the physical meaning of the transition threshold p∗ in terms of the minimum cut and show, through a counterexample, that the structural transition does not always exist. Our results are one step forward on the characterization of more realistic multilayer networks and might be relevant for systems that deviate from the topological constraints imposed by multiplex networks.

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