Unlike Glacier or Yellowstone National Park where bear spray is a requirement, you'll likely be just fine going without while you're hiking in Big Bend.
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NOTE: Bear spray is not allowed in every national park.Check with your national park to see if bear pepper spray is recommended or allowed for the activities you have planned. Having bear spray isn't useful if you don't know how to use it!
Remember, always carry bear spray when hiking in Grand Teton, practice with inert cans often, and be bear aware! Stay safe out there and enjoy yourself while recreating responsibly in Grand Teton.
However, there have been only six reported bear attacks in the park since 1994 - none of which were fatal. Noise is your best friend when it comes to avoiding bears while out on the trail. Bears will usually move out of the way if they hear people approaching.