Likely going you have to pay a tresspass fee to shoot them on any private land.
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There are public land areas but in SD most all of them will what are termed walk in areas. You have to park on the road and walk in, no driving. You would also have to figure out which of those areas contained dogs. Game wardens west of the MO river may be able to help you with that? Also, there is a season on them on all public land in SD, not sure when it is but would be listed on the GF&P website. I know it is closed in the spring when they have little ones.
I think most trespass fees run in the $100-$150 day range.
Likely going you have to pay a tresspass fee to shoot them on any private land.
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OnX map has a prarie dog layer you can purchase for $10. Overlay that with public land...