James said a man who lived at the property was also transported to hospital and underwent surgery on Monday.
He said the victims were related and lived together at the house.
“Police are not seeking anyone else in relation to the incident, however no charges have been laid at this time,” James said.
James said close to 30 police officers were working on the homicide investigation.
Police forensic experts remained at the scene on Tuesday morning doing analysis, with cordons still in place.
Oranga Tamariki regional commissioner east coast Julie Tangaere told Hawke’s Bay Today the agency had “no involvement with the tamariki at this address”.
Avenue Rd East in Hastings remained closed on Tuesday morning. Photo / Jack Riddell
“Our thoughts are with all those affected by the incident,” Tangaere said.
On Monday, Coroner Bruce Hesketh granted an interim non-publication order in relation to the names of the mother, her two children and the man.
Coroner Hesketh’s ruling said the mother and children died of stab wounds.
The coroner said police were yet to speak to the man, who also suffered stab wounds.
“He is currently unconscious and in Wellington Hospital.
“Police wish to speak to [the man] so he can assist with their inquiries. He is considered a person of interest”.
Flowers laid at the scene of a triple-homicide investigation in Hastings. Photo / Jack Riddell
Police also wanted to speak to extended family members, including both the man and woman’s former partners, Hesketh’s ruling said.
Coroner Hesketh made the orders until May 4.
Flowers were being laid on Monday by the community as a team of officers worked on an ongoing scene examination around and in the duplex property.
A nearby resident, who did not want to be named, said she had come down with a bouquet because the situation had “really hit home” when she learned it involved children.
A car is towed from a family home at the centre of a triple-homicide investigation in Hastings. Photo / Jack Riddell
A police tent had been erected at the scene on Monday morning, and a blue hatchback in the driveway of the property was removed.
Neighbours spoken to by Hawke’s Bay Today said they heard no noise from the home on Saturday night or Sunday morning, and the first they had heard of the homicides was when news articles began to appear late on Sunday morning.
They described the area as a generally quiet piece of suburban New Zealand.
Hastings Mayor Wendy Schollum said on Monday losing two children, alongside their mother, was a “grief no family should ever have to carry”.
“Many people across Hastings will be feeling deeply shaken by this today, especially those closest to the family.
“When something like this happens, our community wraps around each other.
“That support will matter deeply in the days ahead.
“I know this will be deeply unsettling for people right across our community, especially those living nearby.
“Police have been clear that this was an isolated incident and that there is no wider risk to the public, and that reassurance will matter for many today.”
Jack Riddell is a multimedia journalist with Hawke’s Bay Today and has worked in radio and media in the UK, Germany, and New Zealand.