Prices go up on the food most people prefer to eat, first. People who cannot afford the foods they're accustomed to eating will switch to lower-priced foods they would never have considered otherwise.

That's largely how tomatoes, potatoes, and more came into use... Desperate, hungry peasants.

When food suppliers see the increase in price, they will work hard to squeeze more yield out of their existing supplies, and as soon as possible will invest in producing more (typically for next year). Others who are capable of producing food, but normally produce something else due to economics, will see the equation change in favor of food and switch their output.

Famines are not caused or encouraged by supply/demand economics. It is vastly more common for communist governments to cause them.