Stability of large systems.

We use the May-Wigner Stability Theorem (Geman (1984) preprint, Brown University; Hastings (1984) preprint, Hofstra University), to study the Lyapunov and structural stability of "real" large systems. Here are our new main results. For large systems which satisfy certain natural scaling relations (Harrison, Am. Natur., 113 (1979) 659; May (1979) Blackwell Scientific, Oxford), Lyapunov stability tends to increase with increasing complexity. However, at least one aspect of structural stability decreases: both competitive and cooperative effects can rapidly destabilize such a system. Finally, we observe that random matrices which satisfy the hypotheses and stability criterion of the May-Wigner theorem are asymptotically of the form 'rotation followed by multiplication by lambda,lambda less than 1'. This allows an easy analysis of the effects of noise in these systems. We conclude by briefly discussing applications to analysis of stability of systems such as the world economy, power networks, and the immune system.