Optimal trading strategy for raising silkworms

Jingles mentioned that when she was a child, she used to trade silkworms for Mulberry leaves because otherwise her worms would starve. Here is the question: what is the optimal trading strategy to have the most living silkworms by day T?

Some basic assumptions:
1, trade is one way only.
2, each silkworm consumes 1 unit of leaf every day.
3, leaves have a half life of k days (2 leaves from k days ago is equivalent to 1 leaf from today).
4, the price formula (# of leaves traded for one worm) is P(t) on day t, for starters, assume P(t) is a constant function.
5, silkworm will starve to death if not fed for one day, however, one can choose which worms to feed (so if leaves are insufficient, at least some of the worms can survive).