Elephants have been observed showing signs of grieving or mourning for deceased members of their herd in various ways. They may stay close to the body of the deceased elephant, showing signs of distress, such as trumpeting, touching, or caressing the body. Elephants have also been seen displaying behaviors like standing vigil over the deceased, refusing to leave the body, and even covering the body with leaves, branches, or dirt. These behaviors suggest a deep emotional connection and mourning process among elephants when one of their herd members passes away.