NEW YORK — (CNN) First they ignored Donald Trump, but he wouldn’t go away. Then they mocked him as a reality show star on a self-promotion binge, but The Donald refused to take the hint.

So now the real estate mogul’s rivals for the Republican nomination, spooked by his poll surge and fearing his potential to damage both the party and their own White House hopes, are taking it up a notch, significantly hardening their rhetoric.

Trump, they say, is not just a threat to the GOP’s hopes of winning back the presidency by alienating key constituencies — like Hispanic voters and veterans — he’s a threat to the nation itself. The brash businessman is likely to provoke further vitriol on Thursday by touring the Texas border.

On Wednesday, former Texas Gov. Rick Perry lashed out at his rival for the Republican nomination for what he described as “Trumpism.” He charged that its author was a dangerous extremist who had whipped up a “toxic mix of demagoguery, mean-spiritedness and nonsense” and was spreading a”cancer on conservatism.”

His criticism was matched by another GOP candidate, South Carolina Sen. Lindsey Graham, who called the “Celebrity Apprentice” star a “car wreck” during an interview with CNN’s “The Situation Room” Tuesday.

“People who say the things he said will never lead a great nation in my opinion,” he said.

The attacks were just the latest efforts by those in the Republican Party to try to manage the Trump boom, which has hijacked the party’s hopes of a smooth primary process and threatens to turn the first party debate early next month into a riot.

For sure, Trump does not play the game like a regular politician. And with his tirade against Mexican immigrants and allegations that some are rapists and criminals, he seems ready to exploit fears and prejudices to boost his personal political ambition.

“I’m doing this for the good of the country,” Trump told CNN’s Anderson Cooper in an interview on Wednesday. “Somebody’s got to do it. The politicians are never going to turn this country around.”

But by using the “D-word” — demagoguery — Perry appeared to be putting Trump among more dubious political company than he has hitherto been, a long list of rogues and firebrands that speckle American history.

Its members include the race-baiting former Alabama Gov. George Wallace and the power-hungry former Louisiana Gov. Huey Long, who greeted prominent guests in green pajamas and built a personal fiefdom in his state in the late 1920s and early 1930s.

And Trump’s mastery of driving controversy in the media also has parallels with the bewitching talents of Father Charles Coughlin, a charismatic radio priest who used the wireless to harness opposition to Franklin Delano Roosevelt’s New Deal and rail against Jews.

“These candidates try to appeal to constituencies who feel that they have no other option,” said Julian Zelizer, a professor of history at Princeton University.

“They play on the worst feelings of the electorate, economic and racial … or they step into where there is a vacuum where neither party is dealing with it.”

But while Trump certainly is cranking up the populist rhetoric and doesn’t shy from whipping up fears over illegal immigration, he is tough to classify as a classic demagogue.

“There is a difference between demagogues and demagogic behavior,” said Michael Signer, an author and lawyer who has made a 15-year study of political demagogues and wrote a book on what he calls an endemic threat to democracy.

“Everything he has been doing has been at the level of him benefiting from controversy. There’s a propagandistic element to it. He knows how to outrage people for his own benefit, and he’s done that his whole career,” he said.

But he said that doesn’t mean Trump can be classified as a genuine demagogue — even if his opponents think he’s heading that way.

Signer has established four tests to identify genuine demagogues, and so far Trump, for all his bluster, seems well short of membership in this notorious political club.

Classic demagogues — like Venezuela’s late dictator Hugo Chavez — tend to present themselves as men of the people, a rank that Trump seems to have little desire to join, considering he rarely tires of reminding voters that he is stinking rich.

And under Signer’s test, true demagogues strike an overpowering connection with the masses that they then use to power their own political development.

Trump, for all his outsize personality, can hardly claim to have captured the public imagination at this point: His percentage in an ABC/Washington Post poll this week, though the best in the GOP field, was only 24%.

Given that he also lacks a detailed program of political priorities and policies, he seems to fall short here, too. Many in the political world also doubt whether he will go to the trouble of building a full political operation that could get him on the ballot countrywide and power a genuine bid to lead the GOP into the general election.

Finally, and most importantly, demagogues threaten to break established rules of governance and to challenge the power of political institutions — a feat that Trump’s distance from power seems to render an unlikely prospect.

“The question now is whether he’s really going to set himself up to actually mirror the common people,” said Signer. “I don’t see him seriously developing either a political persona or a program that’s about regular people and their problems.”

So if Trump is not a demagogue in the true sense of the word, where can he be placed on the political spectrum?

With his background in business and refusal to rule out a independent bid for the presidency should he fail to win the nomination, Trump gives some Republicans nightmare flashbacks to Ross Perot — who helped defeat President George H.W. Bush in 1992 and handed the White House to Democrat Bill Clinton.

Other observers see similarities to Pat Buchanan, the culture warrior who so softened up Bush in the New Hampshire primary that year that it helped seed the ground for Perot’s challenge.

And if Trump’s campaign turns out to be a media comet that burns out and leaves no discernable impact on the Republican field, he may end up in the same category of busted politicians as Sarah Palin, Michele Bachmann and Herman Cain, who experienced stratospheric political ascents but couldn’t keep up the momentum.

Trump’s rise and political persona also seem to bear some comparisons to Italian media mogul Silvio Berlusconi’s talent for shaping media narratives. Berlusconi successfully navigated Italy’s fractured political climate to grab genuine power as prime minister.

Trump’s use of public antipathy over immigration, meanwhile, is similar to that of flamboyant British politician Nigel Farage of the UK Independence Party — who once threatened to rout Prime Minister David Cameron’s Conservatives but didn’t even win his own seat in Parliament in this May’s election.

Whatever his political fate, Trump clearly is determined to arrive at it his own way.

Conventional politicians build coalitions. Trump takes delight in splintering them — as he alienates Hispanics, veterans and evangelicals and flings insults at GOP rivals he deems “stiff” or stupid and generic politicians who “don’t know what they are doing.”

Most candidates with their eye on office avoid sacred cows. Trump slaughters them. Feel-your-pain politicians like Bill Clinton tell voters it’s all about them. Trump wants you to know it’s all about him — that, in fact, appears to be the central selling point of his campaign.

Even a few months ago, no one thought he would risk his business interests and mammoth earnings and actually jump into the GOP race — let alone that he would lead the field, albeit it more than six months before the first nominating contests.

But jump in he did. And despite suspicions he’s on a mass publicity tour, he insists his motives are pure.

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