Global stability and periodicity in a delay differential model

Simple form delay differential equation (DDE) is considered as a mathematical model of several biological processes. The problems of the global asymptotic stability (GAS) of the unique positive equilibrium and the existence of periodic solutions slowly oscillating about this equilibrium are studied. Sufficient conditions for the GAS are derived in terms of the global attractivity of the unique fixed point of induced interval maps, one set being delay independent conditions and the other one dependent on the size of the delay. Slowly oscillating periodic solutions always exist when the linearized about the equilibrium DDE is unstable. The theoretical results are demonstrated by extensive numerical simulations.