Lysogens are bacterial cells that have survived after genomically incorporating the DNA of temperate bacteriophages infecting them. If an infection results in lysogeny, the lysogen continues to grow and divide normally, seemingly unaffected by the integrated viral genome known as a prophage. However, the prophage can still have an impact on the host's phenotype and overall fitness in certain environments. Additionally, the prophage within the lysogen can activate the lytic pathway via spontaneous prophage induction (SPI), killing the lysogen and releasing new progeny phages. These new phages can then lyse or lysogenize other susceptible nonlysogens, thereby impacting the competition between lysogens and nonlysogens. In a scenario with differing growth rates, it is not clear whether SPI would be beneficial or detrimental to the lysogens since it kills the host cell but also attacks nonlysogenic competitors, either lysing or lysogenizing them. Here we study the evolutionary dynamics of a mixture of lysogens and nonlysogens and derive general conditions on SPI rates for lysogens to displace nonlysogens. We show that there exists an optimal SPI rate for bacteriophage λ and explain why it is so low. We also investigate the impact of stochasticity and conclude that even at low cell numbers SPI can still provide an advantage to the lysogens. These results corroborate recent experimental studies showing that lower SPI rates are advantageous for phage-phage competition, and establish theoretical bounds on the SPI rate in terms of ecological and environmental variables associated with lysogens having a competitive advantage over their nonlysogenic counterparts.
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