After the collapse of the Ur III Dynasty in the Middle East around 2000 BCE the region fragmented into several different rival states which fought amongst themselves trying to establish overall political control. This lasted throughout the Middle Bronze Age and the Late Bronze Age, until the Assyrian Empire rose to control the whole region in the late 8th Century BCE. This chapter of the book is split into three sections, and this blog post is only really about the first of these which covers the earlier and more southern & eastern states in the region.
2nd Millennium BCE
The last part of the Mesopotamia chapter in this book covers the Third Dynasty of Ur, which was a Sumerian empire that arose a short time after the fall of the Akkadian Empire. The book doesn't give dates for the empire - having looked at the wikipedia page I think that's because there's a high degree of uncertainty about when the dates were. Two different possibilities are 2112-2004 BCE or 2055-1940 BCE.