Population of bacteria and phages with time. We show results for constant and density-dependent phage growth rates. The parameters are c = 0.15, r = 0.05, β = 2 × 10−5, and γ = 0.1. The mutation rate per sequence per replication is μ = 0.01. The maximum population of phage is vmax = 6000, and the maximum population of bacteria is xmax = 12000. The maximum number of spacers in a CRISPR is 30. When the number of spacers in the CRISPR array is over 30, the oldest spacer is deleted from the leader-distal end. There are 149 phage strains with a logarithmic initial population distribution.