Option 1: Convert everything to dates and use the usual date comparison functions.
Option 2: Convert all the dates to a 4-character string (MMDD) (making sure you put leading zeros in the day or month <10) and do a simple string comparison.
In both cases, just test if between April 10 and Sept. 20. If so, it's season 1. Otherwise, it must be season 2.
(I don't know PHP, so I can't give you the specific functions or code to accomplish it, but this will hopefully point you in the right direction.)