What state are you in ? Here in central ky invasive bush honeysuckle has taken over the area I hunt It has green deciduous leaves and small red berries.
IMO it's having a drastic impact on the landscape turning semi open areas into thickets
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We have honeysuckle and haven't ever had a issue with it taking over a place. Our problem with the Chinese Privet came from there being an old house place on the property. Once the canopy got taken away those ornamental shrubs spread like wild fire. Like so...
From your description of the problem bushes it almost sounds like our Yaupon which happens to get used as ornamental shrubs in other parts of the country. The dang song birds carry and poop out the seeds from those red berries and it grows quickly and thick. The more open the area the faster it grows, if it takes root under a canopy of trees it'll likely look more like what's in the pic below at maturity. That whole mass is coming from that twisted base to the left of the deer.(Those like that always seem to have scrapes under them) starting out in an open area they'll look like that wall of brush in the top pic, only with red berries.
I'm in west central Wisconsin. -12 where I live and -25 on the way into work this morning!
34 and flooding this morning.