What's the Landlord's position when a tenant dies?

Other

We have just learned that a tenant of over 9 years died on 18 January. Does that mean the tenancy automatically ends?
There is a daughter of about 21 who has been at college taking courses leading to a career in hotel and hospitality and a younger son serving with the Army and they have asked if they can stay on maybe for as much as a year. Their father, the tenant's ex husband, apparently will help with rent.

Clearly we don't want to add to their grief but would appreciate advice on what our legal position now is. The DPS has advised on how the deposit is handled depending on whether there is a will or not but that's for the future.
Meanwhile, we don't have a tenancy agreement with the son or daughter - they were at school when the original tenancy started.
After leaving it for a period while they sort themselves out we will have to take some sort of action. What would you suggest?