M-matrix-based globally asymptotic stability criteria for genetic regulatory networks with time-varying discrete and unbounded distributed delays

The problem of globally asymptotic stability for nonnegative equilibrium points of genetic regulatory networks (GRNs) with time-varying discrete delays and unbounded distributed delays is considered. So far, there are very few results concerning the problem; and in which the nonnegativity of equilibrium points is neglected. In this paper, the existence of nonnegative equilibrium points is firstly presented. Then, by using the nonsingular M-matrix theory and the functional differential equation theory, M-matrix-based sufficient conditions are proposed to guarantee that the kind of GRNs under consideration here has a unique nonnegative equilibrium point which is globally asymptotically stable. The M-matrix-based stability criteria derived here can be easily verified, since they are to check whether a constant matrix is a nonsingular M-matrix. Several numerical examples are offered to illustrate the effectiveness of the approach proposed in this paper.