First, we should note that the tendency of large animals to be concentrated in Africa is recent. The largest elephants that ever existed did not live in Africa. The largest birds ten thousand years ago were the elephant birds of Madagascar and the second largest were the moas of New Zealand. The largest primate who ever lived was a relative of the orangutan that lived in Asia.
Large animals frequently go extinct soon after people enter the environment. Humans clearly drove the elephant birds and the moas into extinction. But modern man evolved in Africa, so we are still left with the question of why the large animals survived in Africa.
One explanation is that the large animals had more time to adjust in Africa because man evolved in Africa. But the really large terrestrial mammals live in Africa and tropical Asia. They do not live in northern Europe and Asia. There were many large animals in northern Europe and Asia. These animals had plenty of time to learn how to coexist with humans, just as the African and tropical Asian animals did. But the large animals of northern Europe and Asia are all gone.
Why is that important? The more closely related two animals are the more communicable diseases they share in common. Humans and chimps share all their diseases. If humans get the disease so do chimps and vice versa. So apes and old world monkeys act as a reservoir of plagues that can infect and afflict humans thereby holding back rapid human population explosions that can drive other species into extinction.
Man's closest relatives, chimps and bonobos, and our second closest relatives, gorillas, live in Africa, along with many old world monkeys. In recent decades AIDS which came from chimps and/or monkeys hit Africa very hard. This provides a recent example of an on going problem for Africa.
So that is why Africa has so many large animals. Africa has more human diseases because man evolved there and that is where our closest relatives live. Those relatives carry diseases that have in the past constrained the African population and reduced the effects of Pleistocene and post Pleistocene overkill on large terrestrial animals. This is how the African elephants, rhinoceros, hippopotamus, giraffes, and ostriches escaped the fate of the woolly mammoth, woolly rhinoceros, ground sloths, elephant birds, moas, and many other large animals.
This is what biologists derisively call a just so story. Rudyard Kipling wrote a children's book full of very unscientific explinations for natural phenomena, with the title, Just So Stories If you want to read some you have the link. Nevertheless, let me close by saying, it happened just so.
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I have used this analysis based on the fact that close relatives share the same communicable diseases to explain many biological mysteries, from the huge size of whales, redwood trees, and late Cretaceous pterosaurs, to the success of dinosaurs. Check out this and much more on my Biology Index page.
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