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30 April 2012, 18:45 | Updated: 30 March 2016, 13:50
North Yorkshire Police say searches will continue around Whitby this week and have asked people to check for any discarded clothes or signs that anyone has been living rough.
Detective Chief Inspector Steve Smith, who is leading the North Yorkshire Police investigation, said: "As we continue the inquiry into the murder of Julie Davison in Whitby, we are appealing to members of the public in the area to check their bins, gardens and outbuildings for any discarded personal items including footwear and clothing, and for signs of anyone living rough."
Cleveland Police launched a murder investigation last week after 81-year-old Colin Dunford was found dead at his terraced house in Leven Street in Middlesbrough last Monday.
His death was then linked to the death of Julie Davison, whose body was found at her home in Whitby on Wednesday.
Statement from the family of Julie Davison
"We are devastated by the sad loss of our lovely Julie.
"She was one of the kindest people you could ever wish to know and went out of her way to help others.
"She did not deserve to die in such terrible circumstances.
"We are so very sad that her hopes of building a new life back in her home town of Whitby ended so tragically.
"We would like to thank everyone who has sent us messages of support during this traumatic time.
"We now ask that the media respect our privacy and allow us to grieve in peace while we try to come to terms with what has happened."
36-year-old James Allen was arrested on Crown Point Road in Leeds on Sunday morning and is being questioned in connection with the two deaths.