Keep them all (native wildlife), but that doesn't mean getting rid of all non-natives. Personally I think getting rid of a small population of Mt.Goats up in Washingtons Olympic Penninsula because we
speculate they are causing a small plant to decline for intance, is a poor excuse for trying to do something useful. Lower the numbers and use the money made from tags to help control/manage a species that we know have a much more profound negative affect on the landscape such as horses.
Notice I said manage, not eliminate. I feel the same way about predators; manage them to a point where we have huntable species.
I find it interesting when all of these scenarios don't include humans as part of the equation, as far as, being a consumptive user.
Yes we have to put controls on the amount of game we can take, because we are too efficient and there are too many of us. But when it comes to large predators, I am for managing to a point that we have a peice of the pie also.
Like it or not we have altered the wintering grounds, migration corridors and the landscape in general with towns, cities and highways. We can do what we can to fix our mistakes and mitigate with habitat projects, but it isn't going back to pre civilization or settlement!
Since obviously we can't control human population we put limits on take, but that doesn't mean we can't manage predators or non-natives to a level that allows for sustainable native wildlife, while still allowing for non-natives that are a benefit to native wildlife and large predators, including humans.
The huge benefit to including humans, is the funding that is gained to put into habitat projects and scientific studies. Although we have contributed to major loss of habitat and made many mistakes through the uncontrolled market hunting in the past. The fact is we as humans are the only portion of all the aforementioned groups/species that contributes funds to the programs to repair those mistakes that benefit the others.
Repair the habitat as much as possible while using some common sense about all the factors that are depressing some native wildlife, not just eliminate all non-natives to later find out they were just one minor factor/piece of a puzzle, and now we don't have any of them. If we control non-natives to reasonable levels and repair the available habitat, we can enjoy native wildlife, non-natives and predators.
Just my opinion, but you can get rid of all non-natives and stop all hunting, and all you will end up with, is less wildlife, a disconnected user group and a huge loss of income to do anything to benefit the habitat that is left or implement studies and projects to boost struggling or depressed native wildlife.
We can't go back to some of the population numbers of game animals in the heydays of no predators; the habitat can't withstand it. We can't go back to the numbers of large predators to populations of pre-bounty or pre modern civilization levels either because the prey levels can't sustain it either.
We can benefit from some non-native species for consumptive uses (humans and other predators) and enjoyment (=money to benefit other species) as long as we use common sense and science to help us stay in the constraints of the habitat that is currently available also!
For instance if we find we are able to rid the landscape of cheat grass etc. and find that chukars are taking up habitat that native gamebirds need, due to population number recoveries, then at that point reduce chukar numbers to a level that is exceptble in that location. (For the record I don't think this will happen, as the landscape is changed/changing and the wildlife is adapting to it as it changes. If we restore the habitat to the best level possible, I believe the native wildlife/non-native wildlife/predators will adjust back to levels that the particular habitat will support.
This doesn't include invasive or noxious wildlife, plants or insects; I am talking wildlife that has been integrated into our landscape and with native wildlife for many years. Hopefully this statement will stop far reach rebuttles

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All this depends on us to assist with population levels to help eliminate the huge population swings created by uncontrolled predator/prey population swings made unsustainable or at the very least undesirable due to limited habitat, especially lack of winter range habitat created by us!
I know all this kind of strays off the main OP question, but it all seemed somewhat relevant and is only my opinion
