Where was Bertha's mother living in 1930? Is her full name Anna Lee or are they two given names? If not in Moline, then it is another locality to look in for James Pershing Davis. If you have a link on familysearch or ancestry, that would help.
Here is a the link to the record on Ancestry.
https://www.ancestry.co.uk/search/collections/6224/records/94993710?tid=192997417&pid=312533858083&_phsrc=YhV13&_phstart=default&usePUBJs=true
Thank you. So they are in Tecumseh, Pottawatomie County, Oklahoma. The brother Robert married in that county in 1931.
https://www.familysearch.org/ark:/61903/1:1:Q29M-HW2G?lang=enfrom Bertha's obituary, 1934
surviving are the widower, her parents Mr and Mrs Robert L Davis of Moline, two sisters, Mrs Emelie Holtgrave, Whittier, Calif.; Miss Lula Mae Davis, Moline, six brothers, Albert, Robert and Willard Davis, Long View, Texas; Francis of Yukon, Okla., etc.
So what I think happened is that Robert E Davis left most of the children behind in Oklahoma when he went back to Illinois. He left Lula May with her grandmother to keep house for her. He took Bertha with him to keep house for him. The sons might have been left with neighbors or who knows where. James is back with his father by 1934. One of the brothers married in Iowa in the early 1940's. And I found a few of the brothers in Oklahoma or Texas on later records. (The familysearch tree has the records.) But I didn't find any of them in 1930.
So my guess is James Pershing Davis is somewhere in Oklahoma or Texas in 1930.
If it's any consolation, no one else seems to be finding any of them in 1930 either. The tree at familysearch doesn't have any of them and ancestry's hints didn't point to anyone that matched in 1930.