  CRISPR-Cas mediated immunity in bacteria allows bacterial populations to
protect themselves against pathogens. However, it also exposes them to the
dangers of auto-immunity by developing protection that targets its own genome.
Using a simple model of the coupled dynamics of phage and bacterial
populations, we explore how acquisition rates affect the survival rate of the
bacterial colony. We find that the optimal strategy depends on the initial
population sizes of both viruses and bacteria. Additionally, certain
combinations of acquisition and dynamical rates and initial population sizes
guarantee protection, due to a dynamical balance between the evolving
population sizes, without relying on acquisition of viral spacers. Outside this
regime, the high cost of auto-immunity limits the acquisition rate. We discuss
these optimal survival strategies in terms of recent experiments.
