Linear stability criteria for population models with periodically perturbed delays

We examine some simple population models that incorporate a time delay whic h is not a constant but is instead a known periodic function of time. We ex amine what effect this periodic variation has on the linear stability of th e equilibrium states of scalar population models and of a simple predator p rey system. The case when the delay differs from a constant by a small ampl itude periodic perturbation can be treated analytically by using two-timing methods. Of particular interest is the case when the system is initially m arginally stable. The introduction of variation in the delay can then have either a stabilising effect or a destabilising one, depending on the freque ncy of the periodic perturbation. The case when the periodic perturbation h as large amplitude is studied numerically. If the fluctuation is large enou gh the effect can be stabilising.