Diagonal Dominance and Harmless Off-Diagonal Delays

. Systems of linear differential equations with constant coefficients, as well as Lotka--Volterra equations, with delays in the off--diagonal terms are considered. Such systems are shown to be asymptotically stable for any choice of delays if and only if the matrix has a negative weakly dominant diagonal. 1. Introduction Consider a system of retarded linear differential equations with constant coefficients of the form x i = n X j=1 a ij x j (t \Gamma ø ij ); for i = 1; : : : n (1.1) with ø ij 0 for 1 i 6= j n and ø ii = 0 for i = 1; : : : ; n: (1.2) This paper deals with the following question: (*) For which matrices A = (a ij ) is the trivial solution x = 0 of (1.1) asymptotically stable for any choice of delays satisfying (1.2)? Such type of stability has been referred to as `absolute stability' in the literature (cf. El'sgol'ts and Norkin [3, p. 175]). In the context of population modelling, the term `harmless' delays has also been used (see Gopalsamy [5]). For n = 2, thi..