There are many. Some have been around for long enough to be accepted as natural and desirable in the US environment. These include earthworms, honeybees, pheasants. Others like Zebra Mussels are generally unknown to the public as they are not usually visible. However the Zebra mussels cost million of dollars a year in maintaining water treatment systems and environmental damage.

The USADA lists (for just animal invasive species:

Africanized Honeybee (Apis mellifera scutellata)
Asian Citrus Psyllid (Diaphorina citri)
Asian Long-Horned Beetle (Anoplophora glabripennis)
Asian Tiger Mosquito (Aedes albopictus)
Brown Marmorated Stink Bug (Halyomorpha halys)
Cactus Moth (Cactoblastis cactorum)
Chilli Thrips (Scirtothrips dorsalis)
Citrus Longhorned Beetle (Anoplophora chinensis)
Common Pine Shoot Beetle (Tomicus piniperda)
Emerald Ash Borer (Agrilus planipennis)
European Grapevine Moth (Lobesia botrana)
European Gypsy Moth (Lymantria dispar)
European Spruce Bark Beetle (Ips typographus)
Formosan Subterranean Termite (Coptotermes formosanus)
Glassy-Winged Sharpshooter (Homalodisca vitripennis)
Hemlock Woolly Adelgid (Adelges tsugae)
Khapra Beetle (Trogoderma granarium)
Kudzu Bug (Megacopta cribraria)
Light Brown Apple Moth (Epiphyas postvittana)
Mediterranean Fruit Fly (Ceratitis capitata)
Mexican Fruit Fly (Anastrepha ludens)
Oriental Fruit Fly (Bactrocera dorsalis)
Pink Bollworm (Pectinophora gossypiella)
Pink Hibiscus Mealybug (Maconellicoccus hirsutus)
Red Imported Fire Ant (Solenopsis invicta)
Russian Wheat Aphid (Diuraphis noxia)
Silverleaf Whitefly (Bemisia tabaci)
Sirex Woodwasp (Sirex noctilio)

Other Invertebrates

Giant African Snail (Lissachatina fulica)
Soybean Cyst Nematode (Heterodera glycines)

Vertebrate Invasive Species
Vertebrate species are animals with backbones or spinal columns. In some cases closely related species such as the hagfish which lack a spine but have a bony skull or cranium are included in the group. Example species include: bony fish; sharks; rays; amphibians; reptiles; mammals; and birds.

Brown Tree Snake (Boiga irregularis)
Burmese Python (Python molurus bivittatus)
European Starling (Sturnus vulgaris)
Wild Boar (Sus scrofa)

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