"The kids are all OK. The man is OK," said Leelanau County Sheriff Mike Borkovich. They were found about 7:50 p.m. a rugged mile from where the man's car was found abandoned about 13 hours earlier. "It's a very strange event but it all ended well."
The 50-year-old man from the Detroit suburb of Orchard Lake was placed under arrest on the domestic violence charge and was being questioned. He will be jailed overnight and arraigned Tuesday in Emmet County, where the charge originated, Borkovich said.
The children told authorities they'd had nothing to eat or drink since Sunday morning. They arrived at the lakeshore later that day and spent the night in the woods, Borkovich said. He said they were "physically exhausted," very hungry and thirsty, and covered with mosquito bites.
The children are the man's 14- and 10-year-old daughters and his 9-year-old son.
The motive for the man's actions was unclear, and authorities said they hoped interviews with him and the children will provide answers.
"They didn't appear to have a lot of supplies or equipment with them," said Phil Akers, chief ranger at the national lakeshore. "Just kind of a spur-of-the-moment, 'let's take off into the woods and go camping' trip."
Borkovich characterized it as "a rogue camping event." He said the man's name would be withheld until he appeared in court.
The man was the subject of a massive search after an empty car he'd been driving was discovered near a museum at the lakeshore.
He got into a conflict with his wife Sunday during a vacation trip to Emmet County, Borkovich said. She filed a complaint against him, and the Emmet County Sheriff's Department issued a misdemeanor domestic violence warrant.
"It's not like it's a military guy and a survival expert. It's a dad with kids," Borkovich said before the four were found safe. "We don't see any signs of struggle, any signs of violence. We don't have any threats of him doing any harm to himself, to the kids or his spouse. We don't have any signs whatsoever to indicate violence. What we have is either lost people or people hiding."
About 7 a.m. Monday, a lakeshore park ranger spotted a car with a dented door parked on a road near the Coast Guard Maritime Heritage Museum. Borkovich said the ranger ran the license plate and learned there was an alert for the vehicle.
There was something about the car itself that caused authorities to be concerned, but Borkovich declined to be specific.
Borkovich said searchers walked the shoreline and combed wetlands and inland ridges. State police and Grand Traverse County Sheriff's Department K-9 units and Michigan Department of Natural Resources crews also took part.
Reinforcements from volunteer fire departments and other agencies arrived Monday night and began combing the area more systematically. The family was found about an hour later. The terrain in the park along the northern Lake Michigan coast is a mixture of towering sand dunes and swampy woodlands thick with trees and undergrowth.
Parts of the lakeshore, including the popular Dune Climb, were closed to the public during the search.