Gowan’s shipyard in Berwick started operating in Berwick in 1751 and in 1867 Arthur took over the yard. He upgraded the yard in 1874 and, in 1877 formed partnership with John Wilson and converted the yard to construct iron ships. However the shipbuilding industry in Berwick was starting to decline (probably due to the expansion of the railways) and in March 1879, the company went into liquidation and had to attempt to sell its assets to pay its debts.  Link 

In 1894 Elizabeth Margaret Gowan (29yo spinster of 47 Huskisson St Liverpool, daughter of Arthur Byram Gowan, shipbuilder, deceased) married James Brand McArthur (30yo analytical chemist) and that the marriage was witnessed by Arthur Byram Gowan.

In The Edinburgh Gazette of 4 July 1871, Queen Victoria appointed Arthur Byram Gowan as vice-consul for Berwick-on-Tweed, Eyemouth, North Sunderland and Holy Island.    Link 

Argentinian immigration records show A B Gowan (born Berwick) as arriving there on the ship Arlanza having embarked at Southampton. This may be the son born in 1862. Perhaps one son went to Argentina, one to Australia after the shipyard folded.