Oh just realized Tokyo Jungle is on ps4 too.
There's a story mode but there's also just a straight up Roguelike where you try to survive as long as possible.
You basically roam around a post apocalyptic Tokyo as some type of animal trying to survive as long as possible as you hunt prey (or seek plants) to keep your hunger in check, avoid predators (and a toxic mist that roams around) and reproduce before you die of old age (and then play as your offspring until they too grow old, and so on).
It has a nice progression in that at first you can only choose stuff like small dogs or a deer but as you play you'll unlock new animals who are higher up on the food chain (there's even some dinos towards the end of the unlocks).
It's also a blast in local co-op