Logistic Equation with Treatment Function and Discrete Delays

The logistic equation with a periodic or asymptotically periodic treatment has a delay either in the per head growth rate or in the net growth rate. When the treatment is constant over time, there exists at most one supercritical Hopf bifurcation for some critical value of the delay. We provide conditions that guarantee the global stability of the trivial steady state when the treatment is an asymptotically periodic function. For the single delayed model and asymptotically periodic drug administration, these are necessary and sufficient conditions. For the double delayed model, given conditions are only sufficient. Simulations for a pharmacokinetic treatment with various periods of drug administration show that the double delayed model is more sensitive than the single delayed model on drug dosage and on the starting time of treatment.