Toward the South (κατα μεσημβριαν). Old word from μεσος and ημερα, midday or noon as in Acts 22:16, the only other example in the N.T. That may be the idea here also, though "towards the South" gets support from the use of κατα λιβα in Acts 27:12.

The same is desert (αυτη εστιν ερημος). Probably a parenthetical remark by Luke to give an idea of the way. One of the ways actually goes through a desert. Gaza itself was a strong city that resisted Alexander the Great five months. It was destroyed by the Romans after war broke out with the Jews.