The dogs were brought along strictly for the purpose of carting gear/food around. Providing things went according to plan, they would have been required throughout the entire journey. Once the Endurance was no longer seaworthy and it became more and more obvious they were going to be trekking/sailing in life rafts to safety rather than completing their mission, they started downsizing their dog numbers. Dogs eat a LOT of food. A massive part of the reason the members were able to survive is because they were able to ration their food appropriately, something that would have been tough feeding the 80+ dogs they started with. Once they had determined the dogs were essentially dead-weight, the decision was made to kill the last team. I can't recall off the top of my head but I think it might have been Worsely's team killed last, and even he lamented he saw no use for them moving forward in their efforts. I mean, all the crew found themselves on three 22ft by 6.5ft life boats. Imagine trying to put dogs on there as well? Especially considering they were large breed dogs, who were not typically well socialized. In a survival situation, killing the dogs was a no brainer. That isn't even mentioning that there were times when food was scarce and eating the dogs was a very logical choice to make.