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More details of issues of using Mesquite under Mac OS X are described in the Support page.
For more complete instructions about starting Mesquite and what can go wrong, see the file Mesquite_Starter_INSTRUCTIONS-macOS.txt. The different versions of Starters also provide different amounts of memory. Which Mesquite_Starter should you use? We offer several different versions because the diversity of macOS versions and Java versions make it too challenging to provide you a single solution.
If you don't have Java installed, you may be asked to install it. Double click one of the Mesquite_Starter icons to start Mesquite.Its contents should be approximately as follows: Drag this Mesquite_Folder to your hard drive (e.g., your Applications folder). If the downloaded file doesn't mount into a disk image automatically, double click it to mount.Download the latest version of Mesquite from the Mesquite Github repository.
First, however, you may want to take from Mesquite_Folder/mesquite any third party packages (e.g., PDAP, TreeSetViz) before deleting the directory.
If you have an older version of Mesquite, you may want to delete it (by deleting Mesquite_Folder), so as to avoid getting confused as to what version of Mesquite is starting up. (Java 1.8 is also called Java 8.) Mesquite may or may not run on older systems. System requirements: Mac OS X 10.10 or higher, with Java 1.8 or higher required.