I have written to my hosting company about this and they replied:
"It's also possible that there is a script (php or otherwise) which is not terminating properly which could be taking up resources. Other than that you can try optimizing your databases."
So, is it possible that there is a php script within moodle that is not 'terminating properly'? I have a feeling that it is a server problem and not a moodle problem, but would like to see if anyone out there can suggest any moodle php files that may be possible causes of the problem.
I have not made any changes to php files, apart from some of the theme files.
I believe that the following directives in the php.ini can have a significant effect on the perfromance of moodle - they have been set as follows:
and in the config.php file:
$CFG->dbpersist = true
I've also tried 'false' for this last one, but no change.
Any advice or suggestions will be greatly appreciated.