I meant to post this ages ago but forgot, sorry. I had a look through the 1890 surnames of Ireland database it shows the surname Durkan being mainly in the province of Connaught - in particular counties Mayo and Sligo and Ellicot/Elliott mainly in province of Ulster - Fermanagh, Antrim and Donegal, province of Leinster - Dublin and also in Connaught - Mayo, Sligo, Roscommon and Galway
As both names are in Mayo and Sligo these maybe a good place to start.