A class of difference equations modeling state-dependent delay

We consider difference equations of the form \[ x_{k+1} = x_k - ag(x_k)x_k - a(1-g(x_{k-1}))x_{k-1},\quad a > 0,\quad g\colon\mathbb{R} \to [0,1]. \] We view $g$ as determining the share of a feedback response that occurs ``quickly" versus ``slowly"; thus the above equation can be viewed as a simple discrete-time model incorporating state-dependent delay. We present some preliminary results and pose several open questions.