How large/heavy the rug is makes a difference. Both the tacks and staples ideas above are good ways to go. . . . especially for black bear. It certainly works to get the rug "flush" against the wall.
I personally like to have some "tougher" hardware involved if it is a large black bear or a brown bear. You can use the kind of hardware with wire loops/metal rings attached, and then use nails/bolts to adhere to wall. You usually screw that kind of hardware into the underneath of the head and/or paw regions for one to five contact points. There are different types of screw/loop/ring for this, but the gist is the same. You get stronger metal involved for weightbearing with heavier parts of the bear. You can then tack down the edges with similar color tack as felt of rug. That way the weightbearing is done from underneath. The tacks or staples that keep the rug flush are not bearing all the weight, and, therefore, are not tearing or putting stress on the felt part of the rug.
I have a black bear rug that I hung with one screw/loop under the chin then attached to the wall with a nail. I then tacked down the felt edges in order to aid in weightbearing and to make flush with the wall. I have a brown bear rug that has five contact points (head and four paws) that are all loop/rings screwed into the underneath of the rug and then nailed into the wall (into studs on this one). Because they are "heavy duty" and under tension in all directions I do not have any of the edges tacked/stapled. It appears flush with the wall.
There a variety of ways, and it depends what kind of wall it's attached to, how heavy the rug/bear is, and how comfortable you are with its permanence as hung on the wall etc. I tend towards overkill I think.
As a side note: When hanging on a wall and unsure of exactly how/where to hang it for angle/location/size etc. . . . lay the rug on the floor and cover it with an old bed sheet. Then cut out the bed sheet like you are tracing the rug. You can then "hang" the bed sheet on the wall to see exactly how you want the rug to look and establish where on wall you want your contact points. It's a pretty slick idea, and one that I have to give my wife all the credit for. Not only does she enjoy the taxidermy theme . . . she wants input as to where it goes and how it all fits together.
Good luck figuring out what works for you. BTW That looks to be a nice bear rug. Congratulations