CRISPRs offer adaptive immunity in prokaryotes by acquiring genomic fragments from infecting phage and subsequently exploiting them for phage restriction via an RNAi-like mechanism. Here, we develop and analyze a dynamical model of CRISPR-mediated prokaryote-phage coevolution that incorporates classical CRISPR kinetics along with the recently discovered infection-induced activation and autoimmunity side effects. Our analyses reveal two striking characteristics of the CRISPR defense strategy: that both restriction and abortive infections operate during coevolution with phages, driving phages to much lower densities than possible with restriction alone, and that CRISPR maintenance is determined by a key dimensionless combination of parameters, which upper bounds the activation level of CRISPRs in uninfected populations. We contrast these qualitative observations with experimental data on CRISPR kinetics, which offer insight into the spacer deletion mechanism and the observed low CRISPR prevalence in clinical isolates. More generally, we exploit numerical simulations to delineate four regimes of CRISPR dynamics in terms of its host, kinetic, and regulatory parameters. 