Yes, it is normal behavior if one parent dies for the oldest cygnet to vie for the position of the mate. Do you know which parent died (Male or female) and the gender of the cygnets, especially the one vying for the position of mate?

If the cause is vying for position, there is a good chance that the other cygnet which is being chased is seen as competition from the other cygnet. If this is not the cause, then there is a possibility that there is something wrong with the cygnet (disease or injury). These conditions will cause family and flock members to shun affected members as they do not want to catch a disease from a sick member or followed to the family by an injured member attracting predators. Do you know how the parent died? Was it from a disease or illness? If so, there is a possibility that the cygnet may have been affected and this could lead to further spreading to the remaining members.

You probably do not want to bring any undue attention to this swan family as the Michigan DNR has a bounty on all Mute Swans and wants to kill all of them by 2030. So, if you bring attention to these swans they may be killed by your state wildlife officials. The Regal Swan