Bacterial defense and phage counter-defense lead to coexistence in a modeled ecosystem

Bacteria have evolved many defenses against invading viruses (phage). Typically, each bacterium carries several defense systems, while each phage may carry multiple counter-defense systems. Despite the many bacterial defenses and phage counter-defenses, in most environments, bacteria and phage coexist, with neither driving the other to extinction. How is coexistence realized in the context of the bacteria/phage arms race, and how are the sizes of the bacterial immune and phage counter-immune repertoires determined in conditions of coexistence? Here we develop a simple mathematical model to consider the evolutionary and ecological dynamics of competing bacteria and phage with different immune/counter-immune repertoires. An analysis of our model reveals an ecologically stable fixed point exhibiting coexistence. This fixed point agrees with the experimental observation that each individual bacterium typically carries multiple defense systems, though fewer than the maximum number possible. However, in simulations, the populations typically remain dynamic, exhibiting chaotic fluctuations around this fixed point. We obtain quantitative predictions for the mean, amplitude, and timescale of these dynamics. Our results provide a framework for understanding the evolutionary and ecological dynamics of the bacteria/phage arms race, and demonstrate how bacteria/phage coexistence can stably arise from the coevolution of bacterial defense systems and phage counter-defense systems.