"Stick insects are a group of plant-feeding insects characterised by a remarkable similarity to their host vegetation. S tick insects feed on vegetation and are usually active after dark. ...When disturbed, stick insects will often fall to the ground and 'play dead' for hours. Another bizarre behaviour is the 'dance', where the stick insect sways back and forwards for hours in a peculiar motion, the function of which is a mystery. Many stick insect species, including some New Zealand species, can reproduce without males, a mode of reproduction known as parthenogenesis."
from Landcare Research's The New Zealand Stick Insect Website. Much more there on New Zealand's stick insects.
A Wikipedia entry on stick insects suggests they may be kept as pets. Might be a bit difficult to take one for a walk - the collar would have to be awfully small, no neck after all. Knew a dog that used to play dead when it had had enough walking, imagine waiting hours for your pet "stickie" to decide it had had enough playing dead or dancing.
Stick with it...
Stick insect in the Amuri district, North Canterbury, 2009