Humberside Police have named the victim as Shane Gilmer, 30, from Southburn, a village near Driffield, following the Friday evening attack.
According to reports, a crossbow was used in the attack and police have cordoned off the rural village and have been searching local woods.
The force has issued an appeal to find the suspect, 55-year-old Anthony Lawrence, and officers have found his car abandoned.
What the police say
Detective Chief Inspector Stewart Miller said: "We are able to confirm that the victim killed in the incident was 30-year-old Shane Gilmer.
"We are supporting Mr Gilmer's family during this difficult and sad time.
"The investigation to find the person responsible is ongoing."
Police received reports at around 9.20pm on Friday that a couple had been seriously injured in a disturbance at their home.
Mr Gilmer died at Hull Royal Infirmary on Saturday, while the woman, who is said to be pregnant, remains in a stable condition.
Police thanked the public for help in tracing Mr Lawrence's blue Vauxhall Meriva following an appeal.
Chief Inspector Andy Parsons said: "We are still appealing for anyone who believes they have seen Mr Lawrence to call us immediately on 999.
"While we still believe this was an isolated incident and there is no threat to the wider public, please do not approach him yourself."
Neighbours have spoken of their horror at hearing of the tragedy in this normally peaceful village.
"I've never known a quieter village for something like this to happen in," said Chris King, 25.
"You have a lot of fields and hiding places in and around the village so the police are going to have a tough time finding him.
"I can see that the police have now gone into the woods and they are searching there."
East Riding councillor for the Southburn area, Coun Felicity Temple said: "It is a terrible shame and I am very sorry it has happened.
"Obviously, my deep heart-felt sympathies go out to the family of those involved."
The community has spoken of its shock in this normally quiet village, with a Gofundme page set up to support the family of Mr Gilmer, who is said to be a father.
The page's creator Amanda Merrey said: "I'm hoping our whole community will stand beside me in helping to give Shane the send off he truly deserves and hopefully raise enough to help his family and children through this devastating time :( rip Shane you will be so missed."
Click here to go to the page.Description of Anthony Lawrence
Mr Lawrence is white, around 5ft 10in tall, with short dark hair and large build.
Anyone who sees him is asked to call 999.