Washington (CNN) Neither President Donald Trump nor his front-running Democratic foil could seem to believe it.

"He said my name so many times that people couldn't stand it anymore," Trump exclaimed at an ethanol plant in Council Bluffs, Iowa, referring to his latest bête noire, former Vice President Joe Biden

"Apparently he had my speech on Air Force One," Biden said around the same time, 250 miles away in Mount Pleasant, Iowa. "I guess he is really fascinated with me. I find it fascinating."

It was a political version of the putdown favored by passive-aggressive teenagers everywhere -- why are you so obsessed with me? -- rendered by two septuagenarians eager to occupy the other's head.

In the end, it was Biden who singled out Trump more often by name -- 35 times, according to CNN's count -- compared to 15 mentions of Biden by the President.

Perhaps ceding to advisers who have warned against elevating the Democratic front-runner , or maybe hoping to dispel the notion that Biden is causing him electoral anxiety, Trump avoided any mention of his rival during an evening speech to a fundraiser in Des Moines.

Still, for both men -- who brushed against each other in Iowa Tuesday but never crossed paths -- the reasons for obsession are plenty and obvious.

Trump has singled out Biden as the most formidable threat to the blue-collar voters who propelled him to the White House in 2016 -- and polls (both public and internal) have shown Trump lagging behind the former vice president in states that were central to his victory.

He's also threatening the iron grip Trump once held on the news cycle. Cable networks all carried some or all of Biden's midday speech in Iowa, a fact that did not go unnoticed aboard Air Force One as the President was jetting to the state.

That's on top of Trump's lingering fixation with his predecessor, President Barack Obama, and the link to that administration Biden represents.

Biden, meanwhile, has framed his entire campaign on the grave necessity of defeating the President. Unlike some of his Democratic rivals -- who were left on the sidelines Tuesday as the center-ring candidates sparred -- Biden has decided to focus squarely on Trump, even as he talks sanguinely of not being drawn into a a mudslinging match.

A preoccupation with the incumbent has lent his campaign a general election sensibility, all in keeping with the image of a front-runner he's eager to maintain.

Blow for blow

With those motivations in mind, each tore into the other on Tuesday. On their first day trading attacks in an early voting state, a picture emerged of a potential general election match-up colored by personal insults, open hostility -- and attacks striking in their similarity.

Over the course of three events in the eastern part of the state, Biden deemed the President an "existential threat" to the country, questioning his intelligence on matters of policy and casting him as clueless to the struggles of working Americans.

"I don't think the President really gets the basics," Biden said during an evening speech in Davenport, which was so heavy on Trump insults that his campaign sought to program the whole day around it, releasing prepared text of the entire thing at 6 o'clock in the morning.

Lampooning Trump's onetime declaration that "I alone can fix" the country's problems, Biden scoffed: "Fix yourself first, Donald Trump."

He decided to nix a line from the prepared text declaring cashiers at Target "know more about economics than Trump."

The President, too, went after Biden's smarts, albeit in far cruder terms.

"Joe Biden is a dummy," he declared on the White House South Lawn before departing for Iowa, adding later: "He looks different than he used to, he acts different than he used to, he's even slower than he used to be. So I don't know."

Trump, of course, is no stranger to allegations of mental decline; he was so eager to disprove claims of senility that he asked his doctor to administer a mental acuity test and release the results publicly in 2017. Like many of Trump's insults, the accusation didn't come with many specifics.

Distractions and politics

Asked about the allegations later in the day, Biden was not inclined to shrug them off. Instead, he turned the insult into a contest of lucidity.

"Look at him and look at me and answer the question," Biden said. "It's self-evident, you know it's a ridiculous assertion on his part."

"All I can say is watch me," he said. "Just watch me."

With Democrats largely quiet in their criticism of the front-runner -- and with his own strategy against Biden still coming into focus -- Trump sought to fill the void using material dating to his last campaign.

"He makes his stance in Iowa once every two weeks and then he mentions my name 74 times in one speech," he said during his event at the ethanol facility, which was ostensibly a taxpayer-funded official engagement, but nonetheless included several political jabs.

"That reminds me of Crooked Hillary," he added. "She did the same thing. And then, when it came time to vote, they all said, 'You know, she doesn't like Trump very much, but what else does she stand for?' The same thing is happening with Sleepy Joe."