From the Model Mayhem FAQ:

How come I keep getting logged out?
When you are logging in, to stay logged in, you simply need to click to the check box next to the "Remember Me" box. It should look like this:

Without that check, the site will automatically log you out and cause you to have to login frequently.

If you are checking "Remember Me" and are still getting logged out, you will need to check your browser's settings and ensure your browser is set to allow Cookies from Model Mayhem.