International applicants for a prestigious Dartmouth College engineering program dropped 30 percent this year, startling the dean of a school long accustomed to growth in demand from overseas. His colleagues at other top schools told him they were seeing danger signs as well, as they anxiously monitor whether the sharp turn in U.S. immigration and travel policies under President Trump will keep foreign scholars away.

“I totally agree that we need to vet people, we need to be careful. It makes sense to take a close look at people applying to come to the U.S.,” said Terrie Fox Wetle, dean of public health at Brown University. “But when you begin to send a message that you don’t welcome good people … it has a chilling effect.”

Wetle said she had just attended a conference at which other deans were talking about international students who retracted their acceptance of offers of admission. She herself has heard a lot of anxiety from international students. “They’re afraid to come.”

Except for a short pause after the 9/11 terrorist attacks, the number of international students in the United States has been climbing for generations, as U.S. universities have drawn brilliant scholars from around the world. But this year many university officials are on edge, fearful of immediate and long-term impact on higher education as Trump seeks to tighten immigration enforcement and restrict travel from certain countries.

There are a host of unknowns, with fast-moving executive orders and legal challenges, and concerns are largely anecdotal at this point. But the possible effects of last year’s campaign rhetoric and this year’s policy changes are wide-ranging: the risk of losing elite researchers to other countries, a potential decline in international applications and enrollment, and new obstacles to collaborations that had been speeding technological breakthroughs. There are even worries that thousands of doctors from overseas matched with residency programs last week won’t be able to get visas soon enough to prevent a disruption in patient care in the United States.

When Trump announced his first executive order on immigration in January, temporarily suspending the refugee program and banning entry of people from seven majority-Muslim countries, supporters praised it as an essential step to protect the country from the threat of terrorism. “We want to make sure we are not admitting into our country the very threats our soldiers are fighting overseas,” Trump said at the time.

A federal appellate court blocked that order.

University leaders were some of the most ardent critics of that ban — and of Trump’s second executive order, issued this month, which has also been frozen by a federal judge — saying it violates core principles of democracy and higher education.

Along with their contributions to campus culture, international students provide colleges with crucial revenue, often paying full tuition. With 1 million international students contributing an estimated $36 billion a year to the U.S. economy, according to the Institute of International Education, a drop in enrollment could have a huge impact.

For elite universities, these students represent something arguably more valuable: America’s ability to attract the brightest minds from around the world.

“Things are looking pretty grim,” said Amr Elnashai, the dean of engineering at Penn State.

His biggest short-term worry is the 40 faculty searches under way at the engineering school. Half of the school’s faculty is international, and many who are applying are worried about unknowns, such as whether it might become more difficult to get a visa. Candidates have asked him: ” ‘How much will you defend us if we are attacked by the government?’

“We are really in very, very peculiar times,” he said.

He said overseas applications to the college are down 4 percent this year. “I heard much larger reductions from other institutions, but our deadlines are earlier,” he said. “It is inconceivable that this will not have an impact next year.”

Joseph Helble, dean of engineering at Dartmouth, said colleagues from 19 other schools told him recently of unusual drops in overseas applications — a surprising development for a field that ordinarily draws intense global demand. The declines for various types of programs were mostly in the range of 10 percent to 30 percent, akin to the 30 percent decline in international applications for Dartmouth’s program in engineering management.

Helble said he and his colleagues are worried about slowing technological innovation. “I’m a sports fan,” he said. “When you field a sports team, you want to put the best possible team on the field and see how they compete against the best possible competition. We want them all to have the opportunity to collaborate with the best and the brightest; that’s how you move the best technology from the lab to the marketplace.”

Anxious that new policies and the political climate might hinder recruiting, higher education associations surveyed schools about international applicants. Initial results released this month were mixed: 26 percent reported no change in international applicants, and 35 percent reported an increase. But 39 percent reported a drop — a finding that suggested a significant contrast with trends of recent years.

The biggest decline, the survey found, was from students in the Middle East.

Similarly, some top universities told The Washington Post they had a 3 or 4 percent increase in applications from students abroad, and some noted a drop. Some declined to answer. And many cautioned that they won’t truly know whether there is an impact for weeks or months, when they know how many students have accepted offers of admission.

“We have accepted more international students than ever before,” said Michael Klag, dean of public health at Johns Hopkins University. “The key question is: Will they come?”

For students not depending on financial aid, it could be long after the April 15 commitment deadline before results are known. “We won’t know until the academic year starts whether the executive order and the anti-immigrant, anti-Muslim dialogue will affect students’ willingness to come to our school,” Klag said.

Klag said international students, especially those from majority-Muslim countries, were “incredibly disturbed” by the first executive order. (The revised order, including a temporary ban on the issuance of new visas for people from six countries and a 120-day suspension of the refugee program, was frozen by a federal judge in Hawaii.) “They see it as an attack on them.”

More than three-quarters of the schools surveyed reported concern about potential impact.

“Obviously we’re all really concerned about the order,” said Mark Wallace, dean of the graduate school at Vanderbilt University. “It raises concerns about our ability to recruit the best and brightest from international locations.” The first order came after their application deadline, and Wallace said the numbers of international applicants were stable, with a few exceptions, such as engineering. As for next fall, he said, “We all worry about that.”

“The biggest concern is beyond the six countries,” listed in the ban, the president of George Washington University, Steven Knapp, said. GW has 72 students and recent alumni directly affected if the order is upheld, Knapp said, “but the broader impact we’re really concerned about is the deterrent effect.”

GW’s international applicants were up this year, he said, but many applied before the Trump orders, and he doesn’t know how many who were accepted will choose to come. He said GW is trying to send a strong message that its doors are open. The school recently hung posters all over campus, welcoming students in multiple languages.

Frederick Hess, director of education policy studies at the American Enterprise Institute, said some of Trump’s campaign talk, such as calling for a ban on Muslims entering the country, was destructive and reckless. But Hess said he thought the higher education community was also deterring international students with “some of the remarkably theatrical responses and some of the purple rhetoric.”

The survey was led by the American Association of Collegiate Registrars and Admission Officers, working with several other groups. It found that college recruiting specialists reported deep concerns from students and families. Their worries included perceptions that more student visas were being denied; that international students are no longer welcome in the United States; that Trump’s executive orders could extend travel bans to other countries, or be made permanent; and that there might be other complications in the visa process, making travel in and out of the country risky.

Amirhossein Sharifzadeh, a junior at Dartmouth from Iran — one of the countries targeted in Trump’s executive orders — is majoring in computer science and physics. He created a crowd-funding page when he realized he was unlikely to get a visa to work this summer as he had planned, and if the March order is upheld, would not be allowed to return to the United States if he went home to Iran for the break. He applied for a summer program at the University of California at Berkeley to study quantum computing, hoping to go on to earn a doctorate and work for NASA one day. “Due to our president’s executive order both Dartmouth and UC Berkeley cannot let me work to earn and pay my own way,” he wrote on the site. People donated more than his $5,000 goal within one day.

Asked about the visa concerns of international students, a State Department official said in an email: “Studying abroad is a personal decision. As every student has in the past, they will have to look at what options are available to them and make the decision that best fits their needs and career goals.”

Gillian Christensen, a spokeswoman for the Department of Homeland Security, said in a written statement that Trump administration actions aim to strengthen a U.S. immigration system that “has been repeatedly exploited by terrorists and other malicious actors who seek to do us harm.” But she said the department would “treat all of those we encounter humanely and with professionalism.”

With policy in flux, those options aren’t always clear at this point. William Pinsky, who leads the Educational Commission for Foreign Medical Graduates, said he is not sure that the more than 4,000 international medical graduates who need visas will be able to get them before residencies begin July 1, which would disrupt not only their educations but staffing at the hospitals.

One of every four doctors in the United States went to medical schools outside of this country and Canada, he said, so those international graduates are critical. “It’s not that we don’t want the best security for the country,” he said, but he is worried the policies may unintentionally harm health security.

Their visas require them to stay in the country to work with under-served populations after they complete the training, said Joann Porter of the Creighton University School of Medicine, and they’re far more likely than U.S. graduates to go into primary care. “I think it’ll impact health care in the U.S. by limiting our international graduate population.”

After the executive order was announced, Creighton officials didn’t want to discriminate against good candidates based on country of origin, and decided not to despite the rigid training schedules. “If there’s any delay, it can really affect the trainee,” Porter said. “They can get behind. And it can affect the program.” The residents and fellows from the six countries listed in the order have told her they are frightened, she said; one didn’t go home for a mother’s funeral because of the risk of not being allowed back into the country. “We risk losing some really qualified candidates.”