Increasing stability with complexity in a system composed of unstable subsystems

Abstract

AbstractWe examine stability of Hoffman's symmetric model of the immune system ẋ = Si − xi∑j=1n Kji xj; xi > 0; i= 1,2, …, n; (1) where Si > 0, Kij = Kji ⩾ 0. This paper gives necessary and sufficient conditions on {Si} and {Kij} for Eq. (1) to have a unique, stable, steady-state solution. Determining existence of a steady-state solution requires a theorem delimiting the range R of a function F: D ⊆ Rn → R ⊆ Rn, where D is a (possibly proper) subset of Rn. This theorem may be new.If off-diagonal elements {Kij: i ≠ j} are non-zero with probability C and 0 < Smin ⩽ Si ⩽ ϱSmin, ϱ a fixed integer, we let P(n, C) be the probability that Eq. (1) does not have a stable, steady-state solution. Let T(n) = (ϱ + 1)2ϱln nn (2) As n → ∞, CT(n) → r > 1 implies P(n, C) → 0. If we set {Kii = 0; i = 1, 2,…, n}, this result shows that accumulating more unstable subsystems increases the probability of stability of this system