I collect first and last month... one tenant texted me in wants to leave at the end of the month and has asked me to apply his last month despite... I'm ok with this but what if he doesn't leave attend of month?

Then you are stuck, did he pay deposit plus last month rent?
IF he does not leave, he will owe you the months rent, and if he does not pay, you will have to sue him in court.

First any question on law needs to include your general locale as laws vary by area. We need some more detail here. Is the tenancy month to month? How much notice did he give you? When is rent due? Did he pay a deposit as well?

I would tell him no. I would tell him that paying first and last protects you and you will refund the last after he vacates and you check the premises. Tell him if he doesn't pay the rent on time, you will serve him with a pay or quit notice.

You can't tell him no. You collected first and last rather than first and SECURITY. Last month's rent is exactly what it sounds like - he gets to not pay the last month.
All you can do is hope for the best.

...one tenant ... has asked me to apply his last month despite...

Despite what?

It appears to me that, as joypulv wrote, the agreement as to the pre-paid rent suggests that the last month is already paid; so you can't treat it as a deposit. But be sure to give him a written notice to quit (or a writing from him agreeing to be out at the termination date).