IkuKasahara's Links
It took Jun a long time to narrow down who exactly was responsible for his death, but once he had a name and face, he exhausted every resource to hunt him down. Once Jasper was no longer tied to the organization he had worked for it became much harder to track him, which just frustrated Jun more. Jun made a very negative image of Jasper in his mind, but the more he learnt about him (through various interactions they have during the hunt) the more... "human" his perception of Jasper became. By the time Jun got the chance to finally end Jasper, he instead gave him a chance to redeem himself. All he really wanted was some indication that Jasper regretted his actions.
Unfortunately he never gets that, and even after Jasper is dead, the relationship they had still sticks with Jun for a very long time.
Jasper was barely aware of Jun's existence until the end of the organization's existence. Once he was on the run, it wasn't hard to recognize that Jun was the one doing the hunting. Jasper eventually figures out why that is, and at that point develops what he thinks of as a playful rivalry. Because of his luck, Jasper doesn't see Jun as a legitimate threat, simply as the cat to his mouse, and enjoys foiling his plans.