A delay logistic equation with variable growth rate

Abstract

A logistic equation with distributed delay is considered in the case where the growth rate oscillates sinusoidally about a positive mean value. A delay kernel is chosen which admits bifurcation of the equilibrium state into a periodic solution when the growth rate is constant. It is shown that the fluctuations in growth rate modulate the bifurcation into a quasiperiodic solution. In certain circumstances, however, it is shown that frequency locking can occur but that this is a local phenomenon which does not persist outside the immediate vicinity of the bifurcation point.