GTA III Era
Grand Theft Auto: Vice City featured sharks as non-solid, non-interactive objects in the water. Grand Theft Auto: San Andreas was originally set to have land animals, but were removed later on in development. It's possible that Rockstar could not properly program animal A.I., which they would later do in Red Dead Redemption. The game did include birds and sea creatures, however, with dolphins, turtles, fish and jellyfish present in the water and birds including hawks in the sky. These animals are also non-interactive.
GTA IV Era
Grand Theft Auto IV featured 200 pigeons hidden around Liberty City as collectables. In game they are referred to as "flying rats". Killing all 200 flying rats adds 2.5% to the game completed percentage and also unlocks an Annihilator helicopter on a helipad near Star Junction in Algonquin.
GTA IV also introduced many other types of birds. Many of them don't physically appear but their chirps can be heard in the ambient background.
In the episodes The Lost and Damned and The Ballad of Gay Tony, the flying rats had been replaced with 50 seagulls in each game. In TLAD collecting all seagulls will reward you an Innovation outside your second safe house. In TBoGT the reward for collecting all of the seagulls is an APC at Yusuf Amir's construction site.
In all three games the pigeons/flying rats and seagulls are only collectables and do not exhibit animal behaviour.
Grand Theft Auto V
Many animals, both wild and domesticated, appear in Grand Theft Auto V. Dogs are frequently seen in the trailer, and in fact Chop, a rottweiler dog, is a usable animal in at least one mission. Dan Houser has stated in an interview that the countryside would be lifeless without animals, and Rockstar decided to add animals in the game. A related game on RAGE, Red Dead Redemption, had a variety of wildlife as well.