To avoid the sixth mass extinction we will probably have to push harder for conservation. Endangered species may need to be moved to help them deal with a changing climate. Think re-wilding: reintroducing species like wolves or beavers that were once in a certain ecosystem but have since disappeared. Aggressive conservation might also mean killing off newcomer species to preserve or make room for local plants and animals; in New Zealand, killing off local rats have helped kakapo parrots survive.
In the most extreme case new animals could be brought in. They could fill the role of animals that have gone extinct. For example, European sailors ate their way through the Indian Ocean islands of Mauritius, killing off the dodo bird and the local tortoise species. But tortoises from the neighboring Seychelles archipelago have been imported recently. They have helped restore the island ecosystem, including bringing back endangered local trees.
There is even some hope of bringing back entirely extinct species in the future using advances in genetics and synthetic biology. Bringing back extinct species, or replacing extinct animals with similar ones to restore ecosystems, could pose problems. Right now, new invasive species are hurting ecosystems, like the Asian carp invading lakes in the U.S. We'll have to be careful.
But we are not doomed to cause a sixth mass extinction, at least not yet. Based on an estimate published in Nature in 2011, we have a century or two before our actions assure a mass extinction. Unlike an asteroid, we could choose to change course.