Actually, in first world, not even an homeless gain less than 2$ per day.
So we can safely leave all the expenses for fuel and cookware out of the math.
Some don't get $2, not even $1. Some are lucky to get $0 instead of losing money every day.
Sure, some homeless people get assistance from public or private sources, but many others are too mentally ill, depressed, or self-destructive to seek, find, or even accept aid (when aid is available, which it not always is).
And yes, these people do die of starvation, exposure, malnutition, illness, suicide, drug overdose, etc..
They also rob each other when they walk out of the recycling depot with cash in their pocket. Since you live in an alley you don't have security
To be clear, I'm not saying that is an easy life, I'm only saying that is difficult die of starving in a modern first world nation