Q. Is the Srimad Bhagavatam written by Srila Vyasadeva only for this Kali-yuga ? Or will it be written again in the next cycle of the four yugas, as the four cycles repeat?
A. The message of Srimad Bhagavatam and Srimad Bhagavad Gita is eternal.
For example Srimad Bhagavatam has references to events in Satya, Treta, Dvapara and Kali yugas . Srila Vyasadeva compiled the scriptures at the beginning of this Kali yuga and wrote them down because the people of Kali yuga have very poor memory and cannot retain the knowledge just by hearing, which was the process in the previous ages.
In the Bhagavad-Gita also Krishna explains to Arjun that the knowledge He is about to give is eternal. Only when the chain of knowledge breaks does Krishna descends to reestablish the principles once again.
The message in all the scriptures is to know God and to love God. The validity of this message is there in all the ages. The spiritual master and other authorized devotees may speak the same message after taking into consideration the desha, kaala and patra (time, place and circumstance).
It shall be again written in next Kali-yuga, since at the beginning of next cycle there shall be need of re-establishing the principles of religion as Lord Krishna does in every millennium.