They come out of hiding at night and scuttle across the house in search of starchy crumbs on the floor, a sticky sugary stain on the counter and sometimes even a nibble of toothpaste or soap.

Cockroaches.

Yet out of the 4,500 species of cockroaches, the German cockroach is probably the primary source of your frustration. It has overpowered other cockroach species and is considered the world’s most prevalent indoor pest. How exactly this wild insect became our personal problem — so well-adapted to living in places with humans it’s barely found in nature — has eluded scientists for some time.

A new study describes the scavenger’s origin story, and reveals the genetic variations that make the insect “different from other cockroaches,” said Qian Tang, an evolutionary biologist now at Harvard University and an author of the study, published Monday in the Proceedings of the National Academy of Sciences. “And then that helps us find a solution to control them.”

The German cockroach was given its name in the late 1700s in Central Europe. Scientists later concluded that the species, Blattella germanica, originated in northeastern Africa.

But there is another species, the Asian cockroach, or Blattella asahinai, that scientists saw as a good candidate for B. germanica’s ancestor. It looks almost identical to the German cockroach, although it has distinctive traits, like an attraction to light, an ability to live outdoors and the power of flight. As technology advanced, genetic analyses made the case that B. germanica shared more with B. asahinai than just looks.

Dr. Tang was itching to get to the bottom of B. germanica’s genealogical tree, so his team sourced DNA from 281 German cockroaches from 17 countries to study their genetic differences. Then they traced the pest’s journey across the planet, from where it first evolved until it crawled into your kitchen.

This is a “landmark study,” said Chow-Yang Lee, an urban entomologist from the University of California, Riverside who has studied German cockroaches for 30 years but was not involved in this research.

The data confirmed that B. germanica evolved from the Asian cockroach, somewhere in India or Myanmar around 2,100 years ago as human settlements bloomed. Some Asian cockroaches lived near human settlements or plantations, and they probably switched to eating crops planted by humans, Dr. Tang speculates. Then, because human dwellings had similar food sources, they moved indoors and eventually became household pests.

“That’s approximately when the Asian cockroach started to become the German cockroach,” Dr. Tang said.

The insects moved westward in two waves. They first hitched rides in soldiers’ bread baskets to the Middle East 1,200 years ago — much earlier than previously thought, Dr. Tang said. They reached Europe, where they would get their name, only 270 years ago, probably aboard European colonial ships.

Global trade in the 19th and 20th centuries allowed the scavengers to infiltrate most of the world’s nooks and crannies, and indoor plumbing and heating enticed them to stay.

“It makes perfect sense,” said Dini Miller, a professor of urban pest management at Virginia Tech who was not involved in the study. “We provided them with food, moisture and warmth. And they have just been with us basically ever since.”

She works on cockroach management projects across the U.S. and often finds 700 cockroaches in traps left in infested buildings for one night. “They’re pretty prolific,” Dr. Miller said, and they’ve evolved resistance to almost all of the pesticides they’ve been exposed to in the past 60 years.

To understand what makes the German roach such a formidable invader of urban spaces, scientists must unravel the insect’s ancient genetic history, said Erich Bornberg-Bauer, a professor of molecular evolution and bioinformatics at the University of Muenster in Germany who was not involved in the study.

“Then you can reconstruct the path of adaptation,” Dr. Bornberg-Bauer said, and see what genes have been lying dormant throughout history, waiting to come into action with each new challenge.

His own research found that the German cockroach has genes for many receptors for smell and a high number of proteins to help them resist toxic substances. Those are most likely the genes that make them so cunning at sensing new food sources and quickly developing resistance to insecticides.

“They have a very high number of genes, so they have a high, high potential for adaptation,” Dr. Bornberg-Bauer said. “To rapidly evolve into anything further.”