Hm, right now the stamina only decreases when jumping if holding the sprint key. In fact, that's the only time stamina is ever depleted. It would be interesting to decrease stamina when jumping. (Actually, the stamina is depleted down to zero if starved.)
Another idea I'm thinking about is implementing a cooldown timeout, such that if the player totally runs out of stamina, they have to wait for it to fully recharge before it can be used again. I should also lessen the effect of the "stamina fill rate when moving" multiplier.
Okay, I've created three issues on the GitHub issue tracker. Thank you googol, it's a good suggestion and one that touches on a bigger aspect: what else besides sprinting does stamina relate to?
Update: The player is now silent while sneaking, making no footstep sounds. Secondly, there's now a crunch sound that plays at various pitches when eating. Update #2: Players are now physical, unless they're dead, then they're not until they respawn.