The back and forth laid bare the rising political stakes for each man, even with the election about 17 months away. Trump has zeroed in on Biden as a potential threat to his reelection chances and is testing themes to beat him back. Biden, meanwhile, is campaigning as a front-runner, relishing the one-on-one fight with Trump while making sure he doesn't ignore the demands of the Democratic primary.
"People don't respect him," Trump said after touring a renewable energy facility in Council Bluffs. "Even the people that he's running against, they're saying: 'Where is he? What happened?'"
With a dose of exaggeration and name-dropping his 2016 rival, Hillary Clinton, the president added: "He makes his stance in Iowa once every two weeks and then he mentions my name 74 times in one speech. I don't know. That reminds me of Crooked Hillary. She did the same thing."
At almost the same moment in Mount Pleasant, Biden noted the attention he has been getting from Trump. Biden's appearances from earlier in the day were playing on TV screens when Air Force One landed in Iowa.
"I guess he's really fascinated by me," said Biden, who mentioned Trump by name about a dozen times during his first two events in Iowa. "I find it fascinating." He started to say more but then stopped himself, quipping: "My mother would say: 'Joey, focus. Don't descend. Stay up.'"
Pressed later by reporters about his repeated assurances that he wouldn't openly criticize Trump while campaigning, Biden said, "By not talking about him personally - talking about where I disagree with him on the issues, why he's doing such damage to the country - that's totally different than attacking his character or lack thereof."
Speaking Tuesday night in Davenport, Biden said voters must stop the president's attempts to elevate his office beyond its traditional limits of power. Trump is "breaking down the barriers that constrain his power," Biden said, and acting like he has "complete power."
"No, you don't, Donald Trump!" Biden thundered into the microphone, sparking cheers from a crowd of around 500 at the Mississippi Valley Fairgrounds.
Trump, meanwhile, insisted that Biden was the candidate he hoped to face off against in 2020.
"I'd rather run against Biden than anybody," Trump told reporters on the White House lawn before flying to Iowa. "I think he's the weakest mentally, and I like running against people that are weak mentally."
Biden began the day in Ottumwa, the heart of Wapello County, a meat-packing and agricultural manufacturing center that Trump was the first Republican to carry since Dwight D. Eisenhower. It's part of Biden's dual track approach: campaigning for the caucuses while projecting himself as someone who can win in territory Trump snatched from Democrats in 2016, such as Michigan, Pennsylvania and Wisconsin.
There, the former vice president hit Trump on the economy - an issue the president often promotes as his chief strength in a time of low unemployment.
"I hope his presence here will be a clarifying event because Iowa farmers have been crushed by his tariffs toward China," Biden said. "It's really easy to be tough when someone else absorbs the pain, farmers and manufacturers."
Biden added that Trump "backed off his threat of tariffs to Mexico basically because he realized he was likely to lose" in manufacturing states such as Michigan and Ohio. He broadly branded Trump "an existential threat to this country" and said his behavior is beneath the office of the presidency.
For Trump, the power of incumbency was on display in this state dominated by agriculture interests. As the president toured a plant that produces and sells the corn-based fuel additive ethanol, he signed an executive order to "promote agricultural biotechnology." He said the measure would help farmers by speeding up the scientific review of their innovative products and help get them to market faster.
"I fought very hard for ethanol, but you proved me right," Trump said, adding that he fought "for the American farmer like no president has fought before."
But he then again mocked Biden.
"He was some place in Iowa today," the president said, "and he said my name so many times that people couldn't stand it."
Later addressing an evening fundraiser in West Des Moines, Trump refrained from mentioning Biden by name but took a veiled swipe at the former vice president's recent reversal on the Hyde Amendment, a ban on using federal funds to pay for abortions. Biden now says he opposes the ban.
"They go for one thing for a lifetime, and then they flip, and they go to something totally different," Trump said. "It depends on which race they're running."
Superville reported from Council Bluffs and West Des Moines. Weissert reported from Washington. Associated Press writers Alexandra Jaffe in Mount Pleasant, Iowa, and Zeke Miller in Washington contributed to this report.
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