When in any city and county as described in section 184.350 it shall have been decided by vote in the manner provided in section 184.350 that a tax shall be levied for any of the aforesaid described subdistricts, then the chief executive officer of the city shall appoint four members to the metropolitan zoological park and museum district board, and the chief executive officer of the county shall appoint four members to the metropolitan zoological park and museum district board. Said board members shall serve without compensation for a term of four years and until their successors are appointed and qualified; provided, however, that of the initial appointments by each of said chief executives one shall be for a term of one year, one shall be for a term of two years, one for a term of three years, and one shall be for a four year term respectively. Any vacancy shall be filled by the chief executive officer for the unexpired term. Each member appointed by the chief executive officer of the city shall have been a qualified voter of the city for at least one year next before his appointment and shall continue to reside therein during his tenure in office. Each member appointed by the chief executive officer of the county shall have been a qualified voter of the county for at least one year next before his appointment and shall continue to reside therein during his tenure in office. The legislative body of the constitutional charter city upon the petition of the chief executive officer of such city may remove any member appointed by said chief executive officer of that board for misconduct or neglect of duties and the legislative body of the county may upon petition of the chief executive officer of the county may remove any member appointed by him of the board for misconduct or neglect of duties. No member so removed shall be eligible for appointment to said district board thereafter. The board shall elect its own chairman, secretary and such other officers that it deems necessary and expedient and it may make such rules, regulations and bylaws to effectuate its purposes it deems necessary.