(CNN) California couple Rachel Nguyen and Joseph Orbeso went missing three months ago in Joshua Tree National Park . Searchers found them shot dead this week, locked in an embrace, officials say.

"Based on evidence located at the scene, detectives believe Orbeso shot Nguyen, then shot himself," the San Bernardino County Sheriff's Department said in a statement.

A gun found near the bodies was registered to Orbeso, officials said.

The owner of a bed-and-breakfast near the park reported the couple missing July 28 after they failed to check out of their room, according to authorities.

He told deputies that his guests possibly went hiking at Joshua Tree National Park.

Nguyen, 20, and Orbeso, 22, of Orange County were found dead this week in a rugged canyon with boulders in a remote area of the park. Police identified them Thursday.

'Desperate situation'

Water bottles and food wrappers led the group to the bodies, CNN affiliate KABC reported . Cindy Bachman, a spokeswoman for the San Bernardino County Sheriff's Department, told KABC that detectives believe the couple may have perceived they had no way out.

"They have nothing to suggest that Joseph had any intent to harm Rachel -- just that they probably found themselves in a very dire situation," Bachman told the TV station.

Investigators said the two used clothing to cover part of their bodies from the sun. It appears they had been rationing food and had no water, according to the sheriff's department.

The search

More than 250 searchers, including dog teams and helicopters, looked for the couple for days after authorities found their car near the Maze Loop Trailhead, a popular trail.

During the initial search, more than 10 people were injured and others dehydrated when temperatures reached 100 degrees, according to the sheriff's department.

A week later, authorities evaluated "the continuing risk to personnel vs the reward of finding the couple alive" and decided to scale back the search, the department statement said.

Orbeso's father then joined a volunteer rescue team that combed through miles of cliffs and boulders every weekend.

"I believed that I was going to find them. I didn't know when, but I had my answer today," Gilbert Orbeso told CNN affiliate KESQ earlier this week.

He said details show the deaths on the tough terrain may have been a result of desperation.

"Joseph is a protector, as well as a kind and caring person," he said. "We can't be judgmental right away until you get more facts."

He said he's mourning the loss of his son but felt a sense of relief he was able to bring them home.