How to get rid of cheat grass?

Our 40 acres in high desert New Mexico is covered with cheat grass. The surrounding national forest and neighbors properties are also covered in cheat grass. We do not burn ----- too dangerous with unpredictable winds and forest. We do not use chemicals/poisons at all --- ever.
We had the property brush hogged last year when the seeds were setting..........but now I learn the seed heads will mature even when cut.
I am thinking of using my lawn tractor with bagger to cut, bag and dispose of in garbage sacks. 40 Acres will take forever. If that is the only way though.......I will do it.

How do we get rid of this invasive grass?Nothing eats it.

Thank you in advance.

Well, what do plants need? Sunlight, proper nutrition, water. You could cover it up with paper, cardboard, or mulch but 40 acres is a lot for that. Maybe you could mow it over then plant something else that will outcompete it and rob it of nutrients.

Where I live it there isn't a lot of it so I can just pull it up or leave it, most other grasses outcompete it. Definitely tricky since the seed heads get everywhere when cutting.

I guess just do your best to kill it back and introduce another grass that is easier to manage. Maybe vetch if it grows in down there.

Thank you T Simpson! I have a bagger on order. Mow and seed with a high desert grass that will compete with it.........and is drought and cold resistant. We have no ground water or wells. Harvest rainwater or haul. So no irrigating here. I am also contacting local soil and water district to see if they can help.

Cheat grass is an annual so if you can prevent it from going to seed then it will disappear, depending on if it stores seed in the seed bank for very long. Most annual grasses can be terminated by a severe grazing or a mowing close to the ground at just the right stage. This is usually during or immediately after the grass has flowered and is in what's called the "dough stage". This is the same stage in the life cycle of the cover crop cereal rye when it is best to terminate it through the method of crimping.  While most will probably be killed with this method some of the individual cheat grass plants may have just enough energy left over to resprout and create a seed. And if this happens you will just need to wait again for these plants to reach the dough stage and mow or heavily graze them a second time during that season.

A potentially even more important step here is to replant the area with plants you do want that way when eventually more cheatgrass seed comes onto your property from the neighbors you will already have other plants established that will prevent the cheatgrass from establishing.  

Also what do you want growing in these areas? Pasture for livestock?  Food forest? Lawn? Wildlife habitat?  It's always more important in the long run to manage for what you want instead  of managing for what you don't want.  However knowing the lifecycle of the "weeds" you don't want can help you manage for what you do want.

Changing the ecosystem will change what plants thrive in it...

Pay attention to where the cheat grass is not growing. What is growing in those places? Plant more of those things, or modify the soil to match areas where cheat grass is not growing.

Any chance very high stocking rates and grazing pressure can supress it?  Even the cattle don't eat something, hoof damage can really set back undesirables...

Rent some goats. They will eat cheat grass.
Goats will eat....
thistles.
Blackberry bushes
Canada thistle
Cheat grass
Curly dock (Indian tobacco)
Dalmation toadflax
Dyers woad
Hoary cress (Small white top)
Himalayan knotweed
Ivey
Junipers
Leafy spurge
Perennial pepperwood (Tall white top) It is like candy to them
Poison oak
Quack grass
sedge
Scotch thistle
Spotted knapweed
Yellow toadflax



Common tansey
Diffuse knapweed
Gorse
Japenese knotweed
Kochia
Mountain mahagony
Medusha head
Musk thistle
Poison ivey
Nettle
Purple star thistle
Russian knapweed
Russian olive
Scotch broom
Spanish broom
Wild rose
Willows
Yellow star thistle