Yes, if you've seen some you like and don't want to spend much the ideal would be to arrange to dig up some of the kind you have seen already growing there. In spring, just before the new shoots come up.

Hopefully you are not talking about Japanese knotweed. Don't plant any of that.

And if you do accidently plant japanese knotweed, you can accidently send it here.

If you wait awhile it may show up on its own.

What steps would be involved in it showing up on its own?

A pest species in North America. Wait long enough and it may appear on its own.

But there would have to be a way to spread. I haven't heard that j weed has spread much by seed. I only hear of people being its primary way it gains foothold. Then people blame the plant of being of the devil. But if looking at history, the plant is doing what humans do to the environment: alter it to benefit itself with no regard to other species.

"Before 1850, common Japanese Knotweed ( Polygonum cuspidatum = Reynoutria or Fallopia japonica ) was introduced from E Asia to North America as an ornamental plant called JAPANESE FLEECEFLOWER or MEXICAN BAMBOO. It spread like mad in this country. This tough, bold perennial reseeded as well as increased by rootstock. It has proved as persistent as any plant, mightily resisting manual eradication or doses of poison. And it can grow anywhere. And its hybrid is way more common in the Seattle area. Virtually all books covering plants of the Northwest describe and illustrate the hybrid under the name JAPANESE KNOTWEED. But purebred JAPANESE KNOTWEED is relatively rare and small; its hybrid is larger, since the other parent is GIANT KNOTWEED*."

--Arthur Lee Jacobson, Wild Plants of Greater Seattle

* Polygonum sachalinense

The only saving grace with Japanese Knotweed in our area is that all the clumps appear to be a clone of a single orginating plant. As a dioecious species, we are currently seeing it's spread through division only, not seeding. This obviously helping slow it's spread in non-cultivated or wild areas.

Simon

Maybe they're the hybrid.

There is a super pretty variegated version of the japanese knotweed that is a great perennial plant. It's called fallipia japonica variegata and it's real pretty. They call it Mountain fleece or something like that, but it's knotweed.
Yellow groove bamboo should grow great in Michigan where you live.

Here is a link that might be useful: variegated japanese knotweed