You mean the wind that is still going on?
I have a kennel I made of field fencing for our little chicken killer, one end is covered in black tarp. The cage is 12ft by 4ft by 4ft, 5ft end is covered down to 2ft from the bottom on 3 sides. The wind kept flipping the WHOLE thing like it was nothing. I tried small wire, then thick rope. Now I am using thick fencing wire to keep the kennel attached to the tree.
We made the kennel with ends sticking out, so it pierced the plastic on 3 sides and then I cut a long U on each side[front and back] to have strips to use of the plastic to tie itself to the kennel.
If you do cut plastic, do not leave a jagged or unclean edge. Cut smooth so the plastic won't have a way of ripping itself apart.
I'd go with using the plastic itself to tie it. If you make any holes, the wind could easily pull it and tear it. Unless you make plenty of holes/staples only a few inches apart and as smooth to the run as possible.
BTW, it's winter now, what will you do for a ton of rain or 2-3ft snow like last year??
Is it A shaped, so water easily comes off? Snow will be very hard to get off unless the A is very steep.
The kennel, the tree above the 'window/door' is where it's tied to now. The tarp is from the middle of the window, up and over and the short end.