Global dynamics of a state-dependent delay model with unimodal feedback

We obtain the global attractivity of nonnegative stationary states for a parameterized state-dependent delay equation with unimodal feedback. Under certain mild conditions, we show that the equation with unimodal type nonlinearity can generate rich dynamics as the parameter varies. To be specific, global attractivity of the positive stationary state is obtained in a set of nonnegative bounded continuous functions, when the stationary state is less than a value implicitly determined by a condition on the unimodal feedback. The general results of global attractivity are illustrated through two examples arising from population dynamics. Moreover, Hopf bifurcations are demonstrated in the examples when the positive stationary states lose global attractivity.