Sparseness in ecological systems

 Interactions in ecosystems can be highly heterogeneous, with any given species interacting appreciably with only a few others, a situation commonly represented by sparse interaction networks. We study sparse ecosystems using the Lotka-Volterra model. For symmetric interactions, where dynamics always reach fixed points, we study their structure. When interactions are strong, the network of coexisting species breaks up into small subgraphs, while for weaker interactions these graphs are larger and more complex. This process is driven by the emergence of new allowed subgraphs as interaction strength decreases, leading to sharp transitions in community properties. For asymmetric interactions, we study the dynamical phases of the system in the space of interaction strength and system connectivity. In addition to the fixed-point and extensive fluctuations phases known to exist in systems with all-to-all interactions, we find that a new phase of local fluctuations, where fluctuations are localized around small centers, and occur with finite probability.