Introduced species are no laughing matter. Burmese pythons are a major threat to not only the wildlife of the everglades but people, pets, and everything else. Introduced pigs in Hawaii are rapidly eradicating hundreds of native species. Ask an Australian about rabbits sometime, for that matter.
While it's all well and good to be an animal lover, those that are not where they should be, and causing massive destruction in the process need to be removed, and trapping is almost never a viable option. Humans put them there, humans are responsible for removing them. In remote areas, removal of the carcasses is usually impossible, but nature generally has ways of dealing with the remains. Kill a goat, grow some bushes.
Burmese Pythons are a great example. Thanks to Hurricanes or the very irresponsible pet owner, they have escaped and made a very unpleasant home in the Everglades. Irresponsible pet owners are a small junk surprisingly, the larger majority are destroyed Zoos, Breeding Facilities, and the likes of that. But, compared to other things (I'll list them below), the Burmese Pythons are as destructive as a native species.
Here are said pest:
-Household Cats are a big threat to native animals. They kill birds in mass quantities along with snakes, lizards, and small rodents. They kill nest with eggs or young in them, and can wipe out entire populations of animals (Which they have done).
-Dogs of non-native breeds will eat up any little small creature they can, along with eating animal eggs.
-Domesticated Pigs along with Non-native Dog breeds, Household Cats, and Wild Boars are the big four. They dig up bird/lizard/snake/crocodile eggs, and they eat up native plants.
-Wild Boar are same as Domesticated Pigs.
-Domesticated Cattle they are, of course, large animals, and thus need A LOT of vegetation to stay happy. So they eat tons of native plantlife.
-Wild Cattle same as domesticated cattle.
-Domesticated Goats eat up plants like lawnmowers.
-Wild Goats are the same as domesticated ones.
-Brown Tree Snakes are a problem on the island of Guam. Once rich with birds, Guam is now infested with Brown Tree Snakes that have eaten every last bird up.
-Foxes are a problem in Southern Australia were the Endangered Little Penguin resides, they have been shown to kill and eat them and thus are killed by trained professionals.
- Snakeheads - This fish is invading every state in the US. It's highly adaptable, living in any kind of water sustainable for life. It also has proto-lungs, so it can breathe out of water and last out of water for at least 3 days. They're able to crawl out of water to find a new water source, were they will wipe out any animal they can find that will fit in there mouth. Oh, and they're great parents. Both parents will protect the babies and will violently attack anything nearby.
I would also like to note that most invasive species that change the land forever are mammals, snakes, and fish. Pretty strange, ain't it?