Taking students would be taking business although this is not something that could be trademarked or patented (aikido is aikido). Perhaps a lawyer could respond to this one.
Yes, the students are not trademarks or trade secrets, but they may be "goodwill." Clientel acquired through the running of a business is called "goodwill" in law. This applies in torts and covenants not to compete. What happens if someone breaches their contract not to compete and takes the "goodwill" from their former employer, they might to have to pay for those profits that the former would have earned from those clients (students). But of course to recover for this there must have been a convenant not to compete to begin with.
Anne Marie "recent law grad but not an attorney yet" Giri