Figure 1.

Model behavior under variations in the rates of autoimmunity (α) and CRISPR-Cas system loss (μ) Equilibria (Supplementary Table 1) derived from Equations 1–4 are shown in (a) where orange indicates a stable equilibrium with all populations coexisting and defended host dominating phage populations, green indicates that all populations coexist but phages dominate, and blue indicates that defended bacteria have gone extinct but phages and undefended bacteria coexist. In (b) we find numerical solutions to the model at 80 days using realistic initial conditions more specific to the experimental setup (R(0)=350, D(0)=10⁶, U(0)=100, P(0)=10⁶). In this case orange indicates coexistence at 80 days with defended host at higher density than phages, green indicates a phage-dominated coexistence at 80 days, and blue indicates that coexistence did not occur. Numerical error is apparent as noise near the orange-blue boundary. We neglect coevolution and innate immunity in this analysis (). (c-e) Phase diagrams with perturbed starting conditions. Numerical simulations with starting conditions (X(0)=[R(0),D(0),U(0),P(0)]) were perturbed by a proportion of the equilibrium condition where Y∼U[0, 1] and signifies an equilibrium value to explore how robust the equilibria are to starting conditions. A single simulation was run for each parameter combination.