Sounds like a classic memory leak. If you were ever a BlackBerry user, you should be familiar with that concept.
I am not aware of any tools for finding the cause or for viewing individual processes in iOS devices.
A memory leak, in computer science (or leakage, in this context), occurs when a computer program consumes memory but is unable to release it back to the operating system. In object-oriented programming, a memory leak happens when an object is stored in memory but cannot be accessed by the running code. A memory leak has symptoms similar to a number of other problems (see below) and generally can only be diagnosed by a programmer with access to the program source code; however, many people refer to any unwanted increase in memory usage as a memory leak, though this is not strictly accurate from a technical perspective.
Because they can exhaust available system memory as an application runs, memory leaks are often the cause or a contributing factor of software aging.