Positivity and stability of coupled differential-difference equations with time-varying delays

This paper studies the asymptotic stability of a special class of coupled delay differential-difference equations with internal positive property. An explicit characterization on the positivity of coupled differential-difference equations is firstly given. Then, based on the positivity of coupled differential-difference equations with constant delays, we investigate the entrywise monotonicity and asymptotic property of their state trajectories starting from appropriately chosen initial conditions. Furthermore, the time-varying delay system is analyzed through comparing with the corresponding constant delay system. It turns out that an internally positive coupled differential-difference equation with bounded time-varying delays is asymptotically stable as long as the corresponding delay-free system is asymptotically stable.