A stochastic, agent-based mathematical model of the coevolution of the archaeal and bacterial adaptive immunity system, CRISPR-Cas, and lytic viruses shows that CRISPR-Cas immunity can stabilize the virus-host coexistence rather than leading to the extinction of the virus. In the model, CRISPR-Cas immunity does not specifically promote viral diversity, presumably because the selection pressure on each single proto-spacer is too weak. However, the overall virus diversity in the presence of CRISPR-Cas grows due to the increase of the host and, accordingly, the virus population size. Above a threshold value of total viral diversity, which is proportional to the viral mutation rate and population size, the CRISPR-Cas system becomes ineffective and is lost due to the associated fitness cost. Our previous modeling study has suggested that the ubiquity of CRISPR-Cas in hyperthermophiles, which contrasts its comparative low prevalence in mesophiles, is due to lower rates of mutation fixation in thermal habitats. The present findings offer a complementary, simpler perspective on this contrast through the larger population sizes of mesophiles compared to hyperthermophiles, because of which CRISPR-Cas can become ineffective in mesophiles. The efficacy of CRISPR-Cas sharply increases with the number of proto-spacers per viral genome, potentially explaining the low information content of the proto-spacer-associated motif (PAM) that is required for spacer acquisition by CRISPR-Cas because a higher specificity would restrict the number of spacers available to CRISPR-Cas, thus hampering immunity. The very existence of the PAM might reflect the tradeoff between the requirement of diverse spacers for efficient immunity and avoidance of autoimmunity.