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CRISPR is a newly discovered prokaryotic immune system. Bacteria and archaeawith this system incorporate genetic material from invading viruses into theirgenomes, providing protection against future infection by similar viruses. Theconditions for coexistence of prokaryots and viruses is an interesting problemin evolutionary biology. In this work, we show an intriguing phase diagram ofthe virus extinction probability, which is more complex than that of theclassical predator-prey model. As the CRISPR incorporates genetic material,viruses are under pressure to evolve to escape the recognition by CRISPR. Whenbacteria have a small rate of deleting spacers, a new parameter region in whichbacteria and viruses can coexist arises, and it leads to a more complexcoexistence patten for bacteria and viruses. For example, when the virusmutation rate is low, the virus extinction probability changes non-montonicallywith the bacterial exposure rate. The virus and bacteria co-evolution not onlyalters the virus extinction probability, but also changes the bacterialpopulation structure. Additionally, we show that recombination is a successfulstrategy for viruses to escape from CRISPR recognition when viruses havemultiple proto-spacers, providing support for a recombination-mediated escapemechanism suggested experimentally. Finally, we suggest that the reentrantphase diagram, in which phages can progress through three phases of extinctionand two phases of abundance at low spacer deletion rates as a function ofexposure rate to bacteria, is an experimentally testable phenomenon.

Nonclassical phase diagram for virus bacterial co-evolution mediated by CRISPR