When does this new tenancy offer take effect?...... If at all!

Tenancy Types and Management

In 2010 I issued a written 13week (summer only ) contact to a student. The agreement was that this would expired on 30th Aug 2010. As the 30th approached the tenant expressed an interest in having a new longer term contract. I prepared and sent a new payment schedule and contract that started on the 1st September 2010. This was for a 38week period at a higher rent. Tenant made last payment in August 2010, he received the new higher payment details but has not yet paid any of the payments due. Questions:- On the 1st September (i) did the 13week contract actually default into a periodic tenancy at the same rent as the 13week contract (which can only be ended by issuing a S21A) ,or (ii) by staying in the property after receiving the new rent offer, has the tenant, by default, actually acepted the new higher payment?