I'm growing lots of oyster mushrooms, and now I have lots of black flies. I don't know what species they are.
How do I tell if they are fungus gnats or fruit flies ? How do I control them ?
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There are many things you can try to control fungus gnats including
and best is to prevent them getting into your grow room in the first place.
So, your question has two parts:
How do I tell if they are fungus gnats or fruit flies ?
You could always take a picture and put it on StackExchange! Or, you can go to the very useful https://bugguide.net and upload a picture, then wait for an expert to look at it. You'll need to set up an account, but I've had very good results and you'll be notified by email when someone responds.
But, that's for bug identification in general. For your specific case, here's a handy chart from the page https://dengarden.com/gardening/Fungus-Gnats-All-you-need-to-know:
What's the Difference Between a Fungus Gnat and a Fruit Fly? Fungus Gnat smaller than a fruit fly tiny black body attracted to moisture and decaying organic material feeds on moist, decaying organic material found in houseplants' soil Fruit Fly small brown or tan body attracted to fruits and vegetables feeds mainly on produce
How do I control them ?
The link above, https://dengarden.com/gardening/Fungus-Gnats-All-you-need-to-know, has a lot of helpful-looking tips, and you also seem to have found a good link on your own at https://www.smallfootprintfamily.com/how-to-control-fungus-gnats-organically.