Thank you for your question.
Absent a contrary agreement, each contributor owns an equal share of the final composition, even if one contributor only wrote a few notes or words. Jerry Wexler wrote two words of Natural Woman but owns 1/3 of the copyright with ***** ***** and Gerry Goffin.
You can negotiate with the lyricist to try to agree on a fairer split of the ownership of the copyright and can then register the copyright according to the agreement, for example, you own 75% and the lyricist owns 25%. A written agreement would be enforceable, but the default position of the copyright law is that you each own half.
As stated at http://copyright.universityofcalifornia.edu/ownership/joint-works.html
"Absent an agreement to the contrary, authors own the work jointly and equally. Each joint author, therefore, has the right to exercise any or all of the exclusive rights inherent in the joint work. This means that each author can grant third parties permission to use the work on a nonexclusive basis without the consent of other joint authors. Each author may also transfer his or her entire ownership interest to another person without the other joint authors' consent. Each author may also update the work for his or her own purposes. Additionally, each joint author has a duty to account to the other joint authors for any profits received from licensing the joint work."
You might benefit from getting a copy of This Business of Music
It is best to straighten this out before the song is generating money to be fought over.
I hope this information is helpful.