Observant aliens visiting Earth and studying its civilizations would probably be pretty obsessed with wheat. They couldn’t fail to note how staggeringly many people we feed with the crop on this planet.

“Wheat is one of the main staple crops in the world and provides 20% of daily protein and calories,” notes the Wheat Initiative, a project launched by G20 agricultural ministers. “With a world population of 9 billion in 2050, wheat demand is expected to increase by 60%. To meet the demand, annual wheat yield increases must grow from the current level of below 1% to at least 1.6%.”

That’s why the punchline of a new study in the Proceedings of the National Academy of Sciences is pretty troubling. A warming climate, it suggests, could drive wheat yields in the opposite direction – down — in the United States and, possibly, elsewhere.

“The net effect of warming on yields is negative,” write Jesse Tack of the agricultural economics department of Mississippi State University and two colleagues, “even after accounting for the benefits of reduced exposure to freezing temperatures.”

That’s no small matter, the study notes, in that wheat is “the largest source of vegetable protein in low-income countries.”

The study compared results from nearly 30 years of winter wheat trials across Kansas — a state that produced $2.8 billion worth of wheat crop in 2013 — with data on weather and precipitation. Winter wheat grows from September to May and faces two major temperature-related threats during this cycle — extreme winter cold, and extreme spring heat.

Global warming ought to cut down on the freezing temperatures, but also amp up really hot ones. The study found, however, that on balance, the effect is more negative than positive, with a roughly 15 percent decline in wheat yields under a 2 degrees Celsius warming scenario, rising to around 40 percent with 4 degrees (C) of warming.

As for whether the Kansas-based research can easily be extrapolated to other regions where wheat is grown around the world, that depends highly on the local climate, says lead author Tack. So long as warming creates a situation in which temperatures in a given place more frequently reach 34 degrees Celsius (or 92.3 degrees Fahrenheit) during the growing season, then it could be bad for wheat, based on his study.

“The tipping point is 34 degrees Celsius,” says Tack. “In terms of the estimated warming impacts, it’s largely going to be a matter of whether the new climate has increased exposure over 34 degrees.”

That means that if you were in a cooler and more northerly region, one where 34 degrees isn’t very common, but where global warming actually cut down on the number of freezing days in the fall, wheat could actually benefit. And thus, perhaps one way of adapting the world to warming temperatures may be to shift wheat farming more toward the poles.

The study also found that the wheat varieties that breeders have been creating of late are less temperature-resistant than older varieties. Hence, the study says, there seems to be to a “tradeoff between average (mean) yield and ability to resist extreme heat across varieties.”

“One of the thing we found in this study is that the heat resistance of newer seed varieties is lower than historical ones. From a breeding perspective, that raises questions about the current path of wheat breeding efforts: Are they too focused on average yield?” says Tack.

There’s one other factor, too, that’s critical to the wheat equation — water, which can, to some extent, offset hot temperatures, the study finds. If there’s more rainfall or better irrigation of wheat fields, then hot temperatures won’t take as much of a toll.

But, of course, water itself is a resource that can be threatened by climate change.

“The converse of that is that if things get drier and there’s no irrigation, that’s going to further enhance the negative effects of warming temperatures,” says Tack.