LAUSR

We consider a class of stage-structured differential equations with unimodal feedback. By using the time delay as a bifurcation parameter, we show that the number of local Hopf bifurcation values is finite. Furthermore, we analytically prove that these local Hopf bifurcation values are neatly paired, and each pair is jointed by a bounded global Hopf branch. We use the well-known Mackey–Glass equation with a stage structure as an illustrative example to demonstrate that bounded global Hopf branches can induce interesting and rich dynamics. As the delay increases over a finite interval, the stage-structured Mackey–Glass equation exhibits certain symmetric dynamic patterns: the solutions evolve from a stable equilibrium to sustained stable periodic oscillations, to chaotic-like aperiodic oscillations and back to sustained stable periodic oscillations, to a stable equilibrium.