When police approached 22-year-old Xiangyu Zhang at a gas station near his home in La Marque, Texas, last July, they found him sitting in his vehicle with two loaded rifles, including an AM-15 semiautomatic. Zhang, an undocumented immigrant from China, had threatened in an online chatroom for troubled military veterans to shoot schoolchildren, and in December he pleaded guilty to possessing a firearm while in the country without papers.

Zhang lived undocumented in the United States for two years, but he didn’t arrive illegally by walking across the border from Mexico. He entered legally, holding a temporary visa, and when the visa expired, Zhang stayed, becoming one of hundreds of thousands of foreign nationals who in recent years have overstayed visas and are now living in the country illegally.

President Donald Trump has argued that building a wall on the U.S.-Mexico border is the only way to stop illegal immigration. Without a wall, he said during a Jan. 10 visit to McAllen, Texas, there would be “death. A lot of death.”

But the fact is that more illegal immigration stems from visa overstays than illegal border crossings. In fiscal 2017, for instance, the most recent year for which both metrics are available, the Border Patrol apprehended just over 310,000 undocumented border crossers. But more than 700,000 foreigners overstayed visas (out of a possible 52.7 million people issued temporary visas for tourism, business, education, etc.), according to the Homeland Security Department, and more than 85 percent of them were thought still in the United States at the year’s close.

This hasn’t always been the case. For instance, the government is catching far fewer immigrants at the border today than throughout the 1990s and early 2000s, when apprehension totals regularly eclipsed 1 million. But a 2017 report by the nonpartisan Center for Migration Studies estimates the overstay population has outnumbered border-crossers every year since 2007.

About 4.5 million of the 11 million undocumented immigrants in the United States, or 42 percent of the total undocumented population, overstayed visas, according to the report.

“That percentage will continue to increase as long as overstaying continues to be the predominant mode of arrival into the undocumented population,” the report said.

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Donald M. Kerwin, the center’s executive director and author of the report, said the pattern of overstays outnumbering border-crossers “seems to be the new normal.”

“And the most obvious point to make is that a wall doesn’t help capture those people,” he said.

The trend toward visa overstays comprising the bulk of the undocumented population not only raises questions about the need for a wall, Kerwin wrote in his report, “but about the allocation of immigration enforcement resources and funding levels for border enforcement compared to other strategies that might reduce new arrivals into the undocumented population.”

But stopping travelers from overstaying their visas isn’t a simple fix. One possibility is to issue fewer visas to people from countries with high overstay rates, but it’s not very practical given the sheer scope of international travel and the importance of the tourism industry.

Robert Warren, a senior visiting fellow at the Center who led the Immigration and Naturalization Service’s statistics division for nine years, said another option is to crack down on companies that hire undocumented workers, which would remove the financial incentive for staying in the country illegally.

“People will overstay if they can come and work here,” he said. “The number of overstays has always been there. It’s not a new phenomenon.”

Even as Trump remains laser-focused on the quickest way to build his wall, other immigration hawks are working on the overstay issue. Former Rep. Lou Barletta, a Pennsylvania Republican who was one of Trump’s earliest 2016 endorsers, has pushed legislation that would impose fines and prison sentences on those who overstay a visa and ban them from ever being issued another one.

Rep. Steve King of Iowa, who has recently been criticized even by fellow Republicans for endorsing white supremacist views, introduced separate legislation last year that would require some travelers to post a bond up to $10,000 upon receiving a U.S. visa. If they do not leave before their visa expires, the money is forfeited and handed over to the Homeland Security Department to help enforce immigration laws.

King, who estimates his bill could have pulled in more than $6.2 billion in fiscal 2016, has said the strategy could even help pay for Trump’s wall. “I know the construction industry,” he said when the bill was introduced, “and $6.2 billion would pay for a lot of the wall we need on our border with Mexico.”