I spent the morning chatting with a PI on another continent that was once my Ph.D. student. This former student of mine is now setting up an aDNA facility, and is planning for future work as well as for ongoing projects. All of these plans are naturally organised to support the new team, but I was lucky to get invited to participate in some of them. Those that have a clear overlap with what I am interested in. When you train a Ph.D. student, you are obviously best at teaching what you know. Thus, if this student is successful, he or she will either become a competitor, or a collaborator in the future. As he or she has been trained in the field where you are active, by the one who knows everything that you know, that is you. And this can be a fantastic resource. If you manage to train your Ph.D. students in a way so you separate on good terms when they graduate, then you will eventually have a network of scientists out there who enjoys collaborating with you and your team. And I would strongly advise to take this approach, it is so much better and more fun than the alternative. That is, to lay the groundwork for a set of emerging scientists who do not like you at all and are prepared to throw a glove your way on any research-topic.
Randomly selected really old thesis from Stockholm University