An automatic model is presented for animating gaze shifts of virtual characters towards target locations in a virtual environment. Two connected components are described: an eye-head controller and a blinking controller. The gaze control model is based on results from neuroscience, and dictates the contributions of the eyes and head to a gaze shift according to an individual's head movement propensity; that is, their tendency to recruit their head when making gaze motions under different conditions. The blink controller simulates gaze-evoked blinking, a specific category of behaviours that accompany gaze shifts. The probability of occurrence of such blinks, and their amplitude, is related to the gaze shift. The model forms the basis for a number of experiments investigating the impact of blinking, eye-head ratio and direction of head movements on user perception. In addition to other application domains, the findings are of significance to serious games environments, where the perceived quality of a character's gaze may affect engagement, immersion and learning outcomes.