The Time Safari company chooses dinosaurs that are about to die, and that can be shot with their guns, and marks them with red paint.
Time Safari is a company that takes people back in a time machine to kill a dinosaur. It is the ultimate thrill. They explain to Eckles that there is a special antigravity path that they must stay on, and they must use the special guns to kill the dinosaurs.
Some dinosaurs have two brains, one in the head, another far down the spinal column. We stay away from those. That's stretching luck.
The animals they choose are marked with red paint. They determine which ones are about to die, and mate the least. Those are the ones they can kill without affecting anything else.
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.
In this way, they don’t affect the future. Unfortunately, they can’t fully avoid the stupidity of their customers. Eckels steps on a butterfly and changes things so much that the spelling on the sign is different and there is a different president. One small change can have a big impact.