Successfully complete the game in Regular Survival mode to unlock Serious Survival mode. In this mode, you must survive as long as possible without a partner.
Go far into the jungle to find a grassy place where you can make another house.
You can ask your partner for food and firewood. When you have 40% or less, tell her or him to get food or firewood.
You can only go to sleep if you do not have 0% for water and food. However, that only applies to your character. The game will still allow you and your partner to go to sleep if you have at least 1% food and water, even if your partner is at 0%. You can both rest to fill up the stamina meter as long as the character you are playing as has food/water. Hold the partner's hand to rest.