Although insects don't have eyelids, there is a lot of scientific evidence that suggests they need to sleep just like most animals on the planet do.
Because we don't have brain wave equipment small enough to monitor that of tiny little bugs, we have to rely on studying behavioral patterns and observing what happens when insects are disturbed for long periods of time to keep them from falling asleep.
Without much surprise, when insects were kept awake for long periods of time at night, they would simply become less involved during the day, just like you would if you stayed up all night and kept dozing off during the day.
In some cases, insects that were kept awake for too long would actually die of sleep deprivation, which is similar to what would happen if we were unable to sleep for extended periods of time.
So as it turns out, insects can indeed sleep and they need to do so. If they don't it can have a negative impact on their life.
