Logistic type equations with discrete delay and quasi-periodic suppression rate

In this paper a single species dynamics governed by the logistic type delayed equation with an external influence (more precisely, suppression) on the population size is studied. Conditions guaranteeing global stability of the zero steady state are proved. These conditions are necessary and sufficient for a periodic or quasi-periodic suppression rate. Moreover, if the external influence is periodic and the zero steady state is repulsive, the existence of a periodic solution is shown. From the applications point of view, the logistic type equations are often used in the description of underlying tumour dynamics. In this case the external influence reflects the tumour treatment and the global stability of the zero steady state can be interpreted as the cure.