Well Japan´s population grew with agricultural innovation, so I guess that without food importation the population can grow only until the carrying capacity of the land was reached. Korea was a food exporter, so why didn´t the population simply replenish itself? And France exported or imported of food before the 19th century?
The population
did replenish itself, although at a relatively slow rate. Reread what I wrote earlier.
Specifically, the Mongol invasions (1231-57/9, and an uprising from 1270-3) severely reduced Korea's population from 10-12 million to 4-6 million, suggesting that up to 2/3 of the populace may have effectively been wiped out, while up to a million were either transferred or immigrated to Manchuria (including Liaodong) in the following decades. Any population increases until the mid-14th century were negated by the late century due to severe wokou raids, the Red Turban Rebellion (which temporarily captured Gaegyeong, the capital), an invasion backed by the Yuan, and a temporary push into Liaodong to directly incorporate its Korean population, all of which severely displaced the populace.
After a coup ended Goryeo and founded Joseon in 1392, the population gradually increased from 6 to 8 million from the early 15th to late 16th centuries, although growth was again limited by frequent wokou raids across the coastline, in which 841 recorded incidents occurred from the late 14th to mid-16th centuries. The Imjin War (1592-8) and the two Jurchen/Manchu invasions (1627, 1636-7) then reduced the population by 1-2 million, while over 600,000 slaves were sold in Liaodong in 1637 after terminating hostilities.
As a result, it was not until the mid-17th century that the peninsular population growth began again in earnest due to cash crops, after which the figures swelled from around 6 to 15 million (some sources suggest up to 18 million) up to the late 19th century, essentially tripling in size. Hence why I had stated that population trends for Japan would have more closely resembled that for Korea had the former been invaded two or three times, potentially resulting in figures around 10-20 million during the 15th to 19th centuries.
I'm also unsure about what you mean by Korea's status as a "food exporter," as it essentially remained as a net importer for centuries due to China and Japan's larger populations and amounts of arable land, not to mention that the peninsula's cash crops through the Columbian Exchange were also imported directly from both neighbors. While Japan was diplomatically treated as a tributary by Korea in the aftermath of the Imjin War, the former continued to sustain higher volumes of trade since the 15th century.