We have a JRE installed on our production environment, but not a JDK. The versions of the JRE and OS are below.
[me@mymachine ~]$ java -version
java version &
Built-in tools like jmap, jconsole, and jvisualvm are only available in a JDK. Another option is to add the VM argument -XX:+HeapDumpOnOutOfMemoryError which tells the JVM to automatically generate a heap dump when an OutOfMemoryError occurs, and the argument -XX:HeapDumpPath to specify the path for the heap dump.
If you cannot upgrade your JRE to use tools like the ones in the server JRE 7 (http://www.oracle.com/technetwork/java/javase/downloads/server-jre7-downloads-1931105.html), you may have to consider third-party profiling tools like JProfiler or ones list here.