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December 07, 2016 09:30 AM

Bradley Cooper‘s fatherly instincts are kicking in.

The actor, 41, and his girlfriend Irina Shayk are expecting their first child, and while “being pregnant hasn’t stopped [the model] from working hard in Europe,” a source says Cooper “seems more protective” of her.

“He has been spotted dropping her off at the airport with kisses as she is about to take off for another job,” says the source. “They are just a very sweet couple.”

Despite the time apart, the source says, “Bradley and Irina both seem very independent and supportive of each other’s careers.”

RELATED: Bradley Cooper and Irina Shayk are Expecting Their First Child Together

With their hectic careers, it always helps to have family close by. “They also are very close with their families,” says the source. “After Bradley’s dad passed away [in 2011], his mom Gloria mostly lives with Bradley in L.A.”

Cooper’s mom and Shayk, 30, already have a strong relationship and have been spotted hanging out together on multiple occasions.

“Irina was quickly introduced to Gloria and the two seem to get along great,” says the source. “They even spend time without Bradley. They take walks, go shopping or share long lunches.”

The Russian model is also “very close” with her mom, who the source says will “often travel with her on work trips.”

For more on Bradley Cooper and Irina Shayk, pick up the latest issue of PEOPLE, on newsstands Friday.

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