Well-mixed competition of a bacterial population consisting of a CDI+ and a CDI- species. (A) Phase diagram of the competing population. Grey region: competition of the population may lead to the extinction of one of the two species, depending on the initial conditions of the system. Upper white region: the CDI+ species (v) always goes extinct regardless of initial conditions. Lower white region: the CDI- species (u) always goes extinct regardless of initial conditions. (B) Plot of the Lyapunov function  of the competing population in the regime when extinction is possible for either species. The parameter set here corresponds to the blue cross in the grey region of Panel A. Notice that the minima for the system lie on the respective axes, showing two possible extinctions. The dark circle is the saddle point of the system. C-E. Sample time course trajectories of the two competing species. (C) Species u (CDI-) always dominates when growth advantage outweighs inhibition. Here, α / β = 2 and c = 0.5, corresponding to the cross in the upper white region of Panel A. (D) Species v (CDI+) always take over the population when it has a growth advantage and exerts inhibition. Here, α / β = 0.5 and c = 2.0, corresponding to the cross in the lower white region of Panel A. (E) Both of the species may dominate depending on their initial conditions. Here, c = 2.0 and α / β = 2.0, corresponding to the cross in the grey region of Panel A. In each panel (C-E), 100 pairs of trajectories (orange: CDI+, blue: CDI-) with random initial conditions are plotted.