Persia would have had no interest in Arabia NO ONE did. The desert had no wealth or anything but misery to offer so it was left alone to produce tribes of wild barbarians. Mohammed harnessed their ferocity and cupidity to unify them and make them capable of raiding throughout the world. Their GREED and poverty propelled Islam's spread. Religion had little to do with it as Mo was making it up as he went along leading his bands of criminals and paying them off by stealing from their neighbors: Jews, Christians and animists.
When attacked by would be conquerers the Arabs of old merely withdrew further into the deserts from which they could not be attacked and to which they were acclimated. After the invader grew tired of chasing them he would withdraw until the next tribal raid and the same futile procedure would start anew.
I believe the Tuareg and Berbers were in Arabia - is this correct?