  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.
