So are you suggesting that we give beavers more rights than humans? Where would you draw the line then? Should we not build foundations for our houses at the risk of chopping the heads off of some worms? If your house were to get infested by roaches would you welcome them with open arms, tu casa es mi casa? No, why not, is it because they aren't as cute as beavers or deer? Oh no you'd have to, you'd be an outright bigot if you didn't

. No I don't think that killing a beaver to stop if from cutting down trees is killing the rainforest. I really hope you are kidding, because if you aren't I feel sorry for you.
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On the contrary - I'm pretty serious, but in a serious way. Why KILL the beaver? Why not set up a trap, and then transfer them to their natural habitat somewhere down/up the river? Or even to other state that doesn't have beavers? Or any other solution, which is NOT involving killing. Because there are many. Ask european zoos, and (although I'm not an expert) they will should be happy to have one. BUT. This involves more money/effort, and doesn't have the killing thrill, does it?
And yes, I will exterminate cockroaches, etc. Because they are NOT the same as beavers. I have plenty of (i think even different types) of mice that live around, and half of them (I think, from the sounds

) are moving into my attic for the winter. They move away in spring, though.
I have moles in my garden, and I don't hunt them - somehow my apple trees are still there. I don't know, maybe I'm lucky, but I simply believe that there are other possibilites.
I just see it through onther lens. Lens, in which the only "rodents" or "vermint", whatever, that is abundant are stray cats or dogs, or mice or rats. And yes, you can still hunt here - BUT you must be a registered hunter and there is a strict limit of how many of what you can kill in the forest. So some will survive.
Just look at so many other species that were considered super-abundant, that were nearly extint. Buffalo? Sea Otter? There even was some kind of pidgeon that used to migrate in bilions over US, and now they are ALL dead....
And i stress it - I'm NOT an eco-freak. I just want to give a break to the nature. I'm going to use a vehicle. But it will never have bigger engine than 1500 cm3, and most probably less. And there are many other ways to ease the disaster we are inflicting on our planet WITHOUT going back to the stone caves. That's all I'm saying.