How does Lesperance select the animals that can be hunted?

Lesperance and Travis work for a time-traveling safari company.  They offer customers the chance to travel back in time to hunt exotic beasts like dinosaurs.  As time-traveling employees, they are acutely aware of the dangers of time travel.  They understand that changes made in the past could have serious repercussions in the present and future.  That is why the safari company goes to such great lengths to ensure that they do not make changes in the past.  They even provide a walkway that floats above the surface of the ground.  This ensures that nobody accidentally steps on a blade of grass or a butterfly. 

“And that,” he said, “is the Path, laid by Time Safari for your use. It floats six inches above the earth. Doesn’t touch so much as one grass blade, flower, or tree. It’s an anti-gravity metal. Its purpose is to keep you from touching this world of the Past in any way. Stay on the Path. Don’t go off it. I repeat. Don’t go off. For any reason! If you fall off, there’s a penalty. And don’t shoot any animal we don’t okay.”

Eckels does not understand why they have to be so careful, so Travis gives a lengthy explanation about how one animal being killed could quite possibly cause the death of billions of future organisms.  Eckels then asks how the safari company decides which animal they are going to hunt in the first place if any animal death is such a big deal.  Lesperance explains that he goes back and tracks an animal that is about to die from natural causes. By doing this, he knows that no animal is depending on it for food, and he knows that it will not mate again. Because the animal is going to die within minutes anyway, its death from hunting will not affect the time stream. 

When I find one that’s going to die when a tree falls on him, or one that drowns in a tar pit, I note the exact hour, minute, and second. I shoot a paint bomb. It leaves a red patch on his side. We can’t miss it. Then I correlate our arrival in the Past so that we meet the Monster not more than two minutes before he would have died anyway. This way, we kill only animals with no future, that are never going to mate again. You see how careful we are?