Abstract
A goal of complex system research is to identify the dynamical implications of network structure. While early results focused mainly on local or global structural properties, there is now growing interest in mesoscale structures that comprise more than one node but not the whole network. A central challenge is to discover under what conditions the occurrence of a specific mesoscale motif already allows conclusions on the dynamics of a network as a whole. In this paper, we investigate the dynamics of ecological food webs, complex heterogeneous networks of interacting populations. Generalizing the results of MacArthur and Sánchez-García (2009 Phys. Rev. E 80 26117), we show that certain mesoscale symmetries imply the existence of localized dynamical modes. If these modes are unstable the occurrence of the corresponding mesoscale motif implies dynamical instability regardless of the structure of the embedding network. In contrast, if the mode is stable it means that the symmetry can be exploited to reduce the number of nodes in the model, without changing the dynamics of the system. This result explains a previously observed dynamical equivalence between food webs containing a different number of species.
Highlights
Making progress in the investigation of complex systems requires finding concepts that allow reducing the complexity, while leaving emergent-level features of the system intact
We show that the nodes of a symmetry carry localized dynamical modes that characterized solely by the mesoscale structure of their symmetry
The application considered below builds on previous results that have been obtained with a generalized model (GM) of ecological food webs [18, 19]
Summary
Making progress in the investigation of complex systems requires finding concepts that allow reducing the complexity, while leaving emergent-level features of the system intact. By representing the system as a set of discrete nodes connected by links, networks neglect much of the internal complexity of the system’s constituents, but retain the complexity of their interaction Thereby, they offer a suitable intermediate level of description on which a qualitative understanding of the systems dynamics can be gained. In the present paper we consider the example of ecological food webs [5], i.e. the networks of who-eats-who in ecology In these networks, the nodes represent distinct populations, whereas the links represent predator-prey interactions. Food webs offer a highly simplified description of ecological systems, neglecting for instance non-predatory interactions and intra-population dynamics They capture the complexity of the network of predatory interactions that forms the backbone of most natural ecosystems. IV we establish the dynamical modes for several example food web symmetries and derive the coarse-graining rules allowing to remove dynamical modes of on the mesoscale
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