CRISPR (Cluster Regularly Interspaced Short Palindromic Repeats) is a newly discovered immune system of prokaryotes. It can protect prokaryotes from the infection of phage. Most bacteria and almost all archae possess CRISPR. The CRISPR system acquires short nucleotide sequences from viruses. These incorporated sequences, termed spacers, provide a historical record of the bacteria and phages coevolution. We simulate the coevolution of bacteria and phages with a mean field model. Phages can escape the recognition by CRISPR system via point mutation or recombination. We will discuss the different evolutionary consequences of point mutation or recombination on the coevolution of bacteria and phage. We will show that due to the arm race between phages and bacteria, the frequency of spacers and proto-spacers in a population can oscillate quite rapidly. We will also show that the substitution rate of phages is accelerated by the selection pressure imposed by CRISPR.
