Louisiana Civil Code Tit. VIII, Art. 250. Persons entitled to tutorship

Upon the death of either parent, the tutorship of minor children belongs of right to the other. Upon divorce or judicial separation from bed and board of parents, the tutorship of each minor child belongs of right to the parent under whose care he or she has been placed or to whose care he or she has been entrusted; however, if the parents are awarded joint custody of a minor child, then the cotutorship of the minor child shall belong to both parents, with equal authority to act alone, on behalf of the child, and with equal privileges and responsibilities, unless modified by order of the court or by an agreement of the parents, approved by the court awarding joint custody. In the event of the death of a parent to whom joint custody had been awarded, the tutorship of the minor children of the deceased belongs of right to the surviving parent.

All those cases are called tutorship by nature.