(a) A license granting the privilege to harvest kelp or other aquatic plants shall be issued upon application and the payment of a fee of one hundred dollars ($100) to the department. The license shall be valid from January 1 to December 31, inclusive, or, if issued after the beginning of that term, for the remainder thereof. (b) This chapter does not apply to aquatic plants grown on private land or on state water bottoms leased pursuant to Division 12 (commencing with Section 15000).