Does an Animal Companion's Support Benefit double on a crit?
Title. I have a party member picking up an animal companion, Bird specifically, who's support benefit says,
>Support Benefit The bird pecks at your foes’ eyes when you create an opening. Until the start of your next turn, your Strikes that damage a creature that your bird threatens also deal 1d4 persistent bleed damage, and the target is dazzled until it removes the bleed damage. If your bird is nimble or savage, the persistent bleed damage increases to 2d4.
Compared to something like the Bear,
>Support Benefit Your bear mauls your enemies when you create an opening. Until the start of your next turn, each time you hit a creature in the bear’s reach with a Strike, the creature takes 1d8 slashing damage from the bear. If your bear is nimble or savage, the slashing damage increases to 2d8.
Since the Bird's benefit says the persistent damage is dealt by the PC, while the Bears says that it's dealt by the Bear, does the Bird's benefit double on a crit or not? Thank you.