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Mays celebrated theoretical work of the 70s contradicted the establishedparadigm by demonstrating that complexity leads to instability in biologicalsystems. Here Mays random-matrix modelling approach is generalized to realisticlarge-scale webs of species interactions, be they structured by networks ofcompetition, mutualism or both. Simple relationships are found to govern theseotherwise intractable models, and control the parameter ranges for whichbiological systems are stable and feasible. Our analysis of model and realempirical networks is only achievable upon introducing a simplifyingGoogle-matrix reduction scheme, which in the process, yields a practicalecological eigenvalue stability index. These results provide an understandingon how network topology, especially connectance, influences species stablecoexistence. Constraints controlling feasibility (positive equilibriumpopulations) in these systems, are found more restrictive than thosecontrolling stability, helping explain the enigma of why many classes offeasible ecological models are nearly always stable.

Determinants of Structural Stability in Complex Ecological and  Biological Networks: the Google Matrix Approach Determinants of stability in  biological and ecological networks: the Google matrix approach