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My only encounter with a wild 'predator' was with bears in New Jersey. Not quite as scary as a cat but still shat me up pretty good.
My gf and I went out with some friends to one of their families' houses in the sticks. They had a large-ish plot of land that was mostly wooded but they had a go-kart buggy thing with a rollcage. It only had two seats so my mate drives me around the woods to show me where to go while our missus' stay around the house. When we get back, I hop in the driving seat, the gf gets in and we set off. when we get further in to the woods, suddenly to our right we see a mother bear about 6 meters away and then two large Cubs which got half way up some trees in the two seconds it took to pass them.
We both give each other the "did you fucking see that" look and carry the fuck on.
30 seconds later, the trail dips down in to a shallow but muddy ravine that I had earlier crossed with my mate and he had said "maybe don't go this way when it's your turn"....The buggy looses power completely half way up the other side, as if the clutch wasn't working properly or something. It had big knobbly tyres and it just made it the previous run.
I tell my gf she has to get out and push while I try and drive it up, which she was not too pleased about considering second earlier, we had just startled a family of fucking bears.
The buggy won't budge and it's super heavy so we reluctantly abandon it and walk the rest of the way.
We follow the trail on the other side of the ravine and spot a really tiny bear cub on a tree trunk on the other side, 30 feet away. We can't see any of the other bears which immediately turned the "nope" levels right up. I start thinking about that bit in Grizzly Man when Herzog is describing the audio of the moment when Timothy Treadwell and his girlfriend were eaten in front of each other and start to wonder if one of us is going to have to see the other die.
We get back to the house in one piece where our friends are starting to freak out that we might have had an accident and we explain what happened. The guy says me and him will have to go and retrieve the buggy and I explained again that we just pissed off a fucking bear.
He goes: "Oh, those things are fine, my 12 year old niece bumps in to them all the time, they're vegetarians".
We manage to retrieve the buggy.Three hours later we're in a pub in Manhattan. That felt surreal.
Edit: it was one of these:

Well_is_it
Yeah, was whistler, it was doing the rounds on my mates facebook. it's why it's good to have a (big) dog when you go hiking. normally enough to scare them away.
There were always sightings around some of the cross country ski tracks so I never went. The last thing i wanted was to be strapped into some skis on flat ground with one of those coming at me!