Getting Jasper neutered will definitely help. At six months old he's certainly ready to be neutered, so make an appointment for this ASAP. Remember that even after he's neutered, it will take at least a month for all those male hormones to dissipate!
Jasper and your other cats are at very different activity stages in their lives, so there are bound to be some "clashes".
Are you positive that Jasper isn't just trying to play with your older cats? Kittens tend to roughhouse and sometimes rough play can be mistaken for an "attack". If Jasper is actually attacking, you should separate him from your other cats when he does. Give him a "time out" in another room for about 5-10 minutes. Be consistent about this. I had the same problem introducing a 5 month old kitten to two older (6 and 12) girls. At first the kitten just wanted to play, but became more aggressive when he realized he was intimidating the bigger cats. He soon learned that going after the other cats got him time in "solitary confinement". It didn't take long for him to change his behavior.
Many of the members here have also used Feliway plug-ins with success. They give off a scent that is calming to cats.
It's wonderful that you've rescued Jasper, and I'd hate to see him lose a great home because of what is likely only a temporary behavior problem. I hope some of these suggestions help.