Well, on the plus side you seem to have used one of the very heavy-duty UV-stabilized silver tarps (yes?) and you seem to be in a location pretty well sheltered from wind. Those things will help.
On the minus side, that is a really flat pitch and your supports are pretty far apart, and yes you CAN rip the grommets out of those heavy silver tarps (or bend the support posts over unless they are ultra well set). That's not a bungee I see attaching the tarp to the closest post, is it?
At the very least, I guarantee that unless you are out there with a roof rake every hour during every single snowfall, you will (probably frequently) get enough snow on there to sag down and dent your run top way down.
Realistically I will bet you all the Timbits you can eat in one sitting that it will go "splat" in a heavy snowfall before Groundhog day, either rupturing the tarp or its attachments or just sagging to the point where your run is half its original height.
If there were a way to increase the steepness of the pitch and run a RIGID ridge (like a 2x6 on edge) to support the highest part, your odds would be improved. Also, STRONGLY recommend reconfiguring that "strapping" on the run top by using 2x4s (at least) ON EDGE, not on the flat as they currently are. Is the treetrunk rigidly attached to the strapping so it can't slip off one side or the other, or fall over?
Good luck, you will need it
,
Pat