Log filled with CRM_Utils_System::permissionDenied
It appears that the error simply is reporting that someone is trying to access a CiviCRM page and they do not have sufficient permission to access it.

This is either due to an authorized person trying to access the page, but not having sufficient permission set, or to some unauthorized person (robot?) attempting to access the page.

I would probably try to contrast the error messages with the web server access log to try to identify what page is being attempted to be accessed, and its IP address.  If it's a robot you may be able to detect that as well. (It could simply be a search engine robot that knows of the page and is regularly attempting to access it without the proper credentials.)