The engineering school at M.I.T., for example, often gets 400 applicants for every open assistant professor job, says Richard Larson, an operations research professor there. Many, he adds, are “superstellar.”

One way to see what is happening is to look at a measure, called R0, used in demography to show how a population is growing. If every baby girl in a population grows up to have one baby girl on average, R0 is one, and the population size will remain constant. If R0 is significantly greater than one, the population can explode.

Dr. Larson and his colleagues calculated R0s for various science fields in academia. There, R0 is the average number of Ph.D.s that a tenure-track professor will graduate over the course of his or her career, with an R0 of one meaning each professor is replaced by one new Ph.D. The highest R0 is in environmental engineering, at 19.0. It is lower — 6.3 — in biological and medical sciences combined, but that still means that for every new Ph.D. who gets a tenure-track academic job, 5.3 will be shut out. In other words, Dr. Larson said, 84 percent of new Ph.D.s in biomedicine “should be pursuing other opportunities” — jobs in industry or elsewhere, for example, that are not meant to lead to a professorship.

Biomedical sciences have been among the hardest hit. The field had an 83 percent increase in Ph.D.s between 1993 and 2013, to about 192,000 from 105,000. But although most got jobs somewhere, only about half got jobs in academia and only a quarter got tenure-track positions, which, for many, is what all that training was preparing them for.

“It used to be that the majority who got a Ph.D. in the biological sciences would go into an academic career,” said Dr. Michael Lauer, deputy director for extramural research at the National Institutes of Health. “Now,” he says, “that is very much the minority.”

Many spend years in a holding pattern as postdocs, which are temporary positions, working for a professor and being paid from the professor’s research grant. The average pay in 2016 for a beginning postdoc in the biomedical sciences is around $44,000, a figure that, adjusted for inflation, has not changed since 1998.

Why would any smart person work for so little? The goal for postdocs is to get grants of their own eventually, but the success rate for those applying has plunged.