Running new iOS 13 on older mac

IOS 13 is an operating system for Apple iPhones and iPads, not for the Mac.

The maximum macOS for your 2011 iMac is macOS 10.13 High Sierra.

Your 2011 iMac CANNOT run macOS Mojave OR the up and coming macOS Catalina.

JAG3510 wrote:

Can a mid 2011 Mac be upgraded to run new iOS 13

No. It can't run ANY version of iOS. iOS only runs on iPhones, iPods, an iPads.