Stability of large systems

We address a long-standing dilemma concerning stability of large systems. MacArthur (1955) and Hutchinson (1959) argued that more ``complex'' natural systems tended to be more stable than less complex systems based upon energy flow. May (1972) argued the opposite, using random matrix models; see Cohen and Newman (1984, 1985), Bai and Yin (1986). We show that in some sense both are right: under reasonable scaling assumptions on interaction strength, Lyapunov stability increases but structural stability decreases as complexity is increased (c.f. Harrison, 1979; Hastings, 1984). We apply this result to a variety of network systems. References: Bai, Z.D. \& Yin, Y.Q. 1986. Probab. Th. Rel. Fields 73, 555. Cohen, J.E., \& Newman, C.M. 1984. Annals Probab. 12, 283; 1985. Theoret. Biol. 113, 153. Harrison, G.W. 1979. Amer. Natur. 113, 659. Hastings, H.M. 1984. BioSystems 17, 171. Hastings, H.M., Juhasz, F., \& Schreiber, M. 1992. .Proc. Royal Soc., Ser. B. 249, 223. Hutchinson, G.E. 1959. Amer. Natur. 93, 145, MacArthur, R. H. 1955. Ecology 35, 533, May, R.M. 1972. Nature 238, 413.