If parents perceive the public school is not providing suitable services, and the private school is, then the public school finds reason to correct the deficien

If parents perceive the public school is not providing suitable services, and the private school is, then the public school finds reason to correct the deficiency - competition works. And if the private school can't handle the inundation, then there's incentive to either expand capacity and/or another school adapt/open to share the load.