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

Abstract

Mays celebrated theoretical work of the 70s contradicted the established paradigm by demonstrating that complexity leads to instability in biological systems. Here Mays random-matrix modelling approach is generalized to realistic large-scale webs of species interactions, be they structured by networks of competition, mutualism or both. Simple relationships are found to govern these otherwise intractable models, and control the parameter ranges for which biological systems are stable and feasible. Our analysis of model and real empirical networks is only achievable upon introducing a simplifying Google-matrix reduction scheme, which in the process, yields a practical ecological eigenvalue stability index. These results provide an understanding on how network topology, especially connectance, influences species stable coexistence. Constraints controlling feasibility (positive equilibrium populations) in these systems, are found more restrictive than those controlling stability, helping explain the enigma of why many classes of feasible ecological models are nearly always stable.