Competition and mutualism are inevitable processes in ecology, and a central question is which and how many taxa will persist in the face of these interactions. Ecological theory has demonstrated that when direct, pairwise interactions among a group of species are too numerous, or too strong, then the coexistence of these species will be unstable to any slight perturbation. The stability of these systems is also strongly influenced by the structure of the interaction networks. In the case of mutualistic networks, the nested structure often found in nature, has been shown to make the system less locally stable. Here, we consider two different ecological models. The first is a model of evolution in a mutualistic system, where competitive and mutualistic interactions are modeled as direct, pairwise interactions using a system of Lotka-Volterra equations. This mutualistic network is allowed to evolve according to a fixed set of rules. We prove results about the final, nested, structure of the evolved network. The second model is a consumer-resource model where competitive and mutualistic interactions are indirect results of the consumption and exchange of resources. We are able to prove stability of positive equilibria based on the structure of the exchange network. We show that systems based on general rules for the consumption and exchange of resources are guaranteed to be stable when exchange of resources is reciprocated by each pair of partners. These cooperative, mutualistic interactions can be arbitrarily strong and yet not disrupt stability. Although more general modes of exchange can lead to instability when supply rates are low, we show that when resource supply from outside the system is sufficiently high, arbitrary exchange is consistent with a stable equilibrium.
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Network structure and stability criteria for complex ecological systems
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