(CNN) As farmers and ranchers face a crisis of livestock overcrowding, shuttered meat-packing plants and plummeting profits, industry experts are warning that the Trump administration's efforts to help pork and beef producers are not nearly enough to save the farms that supply America's meat.

Their pleas for more relief come amid concerns about potential meat shortages and desperation from pork producers who are running out of options -- and space for their pigs.

Agriculture Secretary Sonny Perdue announced last week that his department would use stimulus funding to offer farmers and ranchers $16 billion of direct payments in an effort to cushion their losses due to Covid-19 . USDA will also purchase $3 billion of fresh meat, dairy and produce to distribute at food banks and other community groups.

But some meat producers say the payment program contains too many restrictions to reach those most responsible for keeping up the nation's meat supply.

"It is woefully short, it's more of a political gesture," said Stephen Mayer, an economist at Kearns & Associates working with the pork industry.

The Trump administration and Congress have struggled to keep up with the unprecedented demands created by Covid-19 and the economic fallout it has sparked. Congress late last month passed the largest stimulus bill in US history -- and has already had to grapple with the need for yet another, as well as the depletion of a $350 billion small business loan program. USDA received $9.5 billion in that stimulus bill and will tap into other funding sources for its agriculture relief effort.

The USDA has set aside $5.1 billion for payments to beef producers and $1.6 billion for payments to pork producers. Those payments are capped at $250,000 per farmer or entity and $125,000 per commodity -- meaning the larger farms that supply vast quantities of meat won't come close to recouping their losses.

"The $1.6 billion in payments doesn't really scratch the surface of the losses incurred by producers," Jim Monroe, spokesman for the National Pork Producers Council, told CNN. "We're going to lose a lot of producers through this situation."

"Specifically, the $250,000 cap per individual and the $150,000 cap per commodity leaves many of our producers -- those who have made major investments in hogs -- behind and will do long-term damage to a pork production system that is the envy of the world if not addressed soon," Monroe added.

The warnings come as more meat processing plants across the country shut their doors each week due to illness among workers.

"I think pork supply is going to get pretty tight beginning probably next week," Mayer said. "Even though food service is slow, you're going to see that supply get stretched awfully thin trying to keep retail stores open."

Mayer said roughly 29% of the industry's processing capacity is currently down -- and the worst could be yet to come, with Covid-19 cases not yet peaking in the parts of the country where some of the largest plants are located.

Tyson Foods closed its largest pork plant on Wednesday after the Waterloo, Iowa, facility was linked to 182 cases of coronavirus. The company later announced it would also temporarily shutter its Logansport, Indiana, facility -- which processes 3 million pounds of pork per day -- by the end of the week.

A USDA spokesperson said the agency will continue to update its relief program with more details down the road, and noted its officials would "evaluate impacts and work with Congress" if more funding is needed.

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"USDA worked to build up this program from the ground up, evaluating the impacts of COVID-19 throughout the agricultural and food sector and then reviewed the resources we had available to do the best we can with what we have," the spokesperson told CNN. "The topline numbers released are estimates based upon that process and clearly do not account for all the possible losses that have occurred. This is intended to be an inclusive program and more information will be provided in the rulemaking."

Dermot Hayes, an economics professor at Iowa State University, said the Trump administration's direct payment plan leaves out so much of the pork industry because it is dominated by major producers.

"The top 40 sow farms produce and own two-thirds of the pigs in the US," Hayes said. "The smallest of the top 40 producers, they're going to lose about $18 million this year."

"$125,000 is just a drop in the bucket, and it makes no economic sense," he added.

With so many processing plants offline, some pork producers are struggling to care for and house the growing number of pigs they can no longer get to market.

Monroe said he has not learned of specific cases where pork producers are being forced to put down their livestock due to the shutdown of major processing plants -- but he said the desperate conditions facing farmers could soon force them to confront "very tragic animal welfare decisions."

"The last thing a farmer wants to do is to euthanize animals. It's a last resort," he said. "But if you can't send pigs to market, barns start to get overcrowded, you start to have issues with individual animals' access to food and water, and those are welfare challenges. And the last thing you want is for any animals to suffer."

"This is not milk that can be spilled or crops that can be dug under, these are live animals, and there's very few options if they can't get them to market," he said.

"Hog values have dropped to virtually zero and you've got farmers in financial crisis," Monroe added.

The beef industry has struggled as well.

The National Cattlemen's Beef Association wrote to Perdue on Thursday, warning in a letter that the current structure of the administration's agricultural relief program "would result in many producers deriving almost no real relief from these funds."

Colin Woodall, CEO of the National Cattlemen's Beef Association, said industry leaders are still waiting for details from the Trump administration about how the payments will work -- but said the current outline suggests the administration is not offering as much relief to agricultural producers as they have to other parts of the economy.

"There is a concern that farmers and ranchers are being treated differently here, and we need to figure out how to fix that," Woodall told CNN. "It's obvious that we have to go back to Congress and ask for more funding."