Likely the lease agreement provides that the tenancy becomes month-to-month if no new writing is signed. It will pay you to read the agreement very carefully and determine which provisions continue to apply after the expiration of the first year and which do not.
If you want the money from the tenant, and he or she is not paying, then you will probably have to file suit on some theory, whether the written agreement is still valid or not. Depending on the circumstances, you may want to evict him or her as well.
If you sue, you can plead breach of written contract and breach of oral contract and other causes of action, perhaps waste or conversion or negligence--all depending on the facts--and recover what is due you under one theory or another. If the written agreement has an attorney fees clause, then you may be awarded attorney fees on top of other costs of suit when you win the judgment.
It would probably be worth it to get an hour's worth of time from an attorney to review the agreement and evidence of non-payment, damage to property, etc., and decide how to proceed from there.
Once a lease expires, it typically converts to a month to month tenancy. The general provisions of the lease will still apply, except that now either the tenant or the landlord can end the tenancy on 30 or 60 day notice.
If your tenants have not paid rent, then you can serve them with a three day notice and then an unlawful detainer lawsuit. The terms of the lease will be relevant and applicable in any dispute regarding the rent or tenancy.