Figure 1

(Color online) Dependence of the surviving strain as a function of the lattice size $N$. (a) Survival probability of $R$ (triangles) and $S$ (squares) for global neighborhood. Probabilities are averages of 1000 independent simulations on lattices with $N\times N$ sites, starting from random initial conditions. For each set of initial conditions, the fraction of survival was determined by following the time evolution until just one strain remains in the lattice. The $K$ strain survives just for very small lattices (smaller than $25\times 25$ sites), being omitted from the graph. (b) Evolution of the crossover lattice size, $N_c$, as a function of $\nu$, the mixing frequency introduced by Károlyi et al. [10]. To determine $N_c$, for each value of $\nu$ we computed a graph like the one in Fig. 1a and located the crossing. Each value of $N_c$ represents an average over 200 independent sets of initial conditions. Error bars reflect the fact that crossovers occur within a finite interval, not in well-defined points.Reuse & Permissions

Figure 2

(Color online) Identification of the quasiextinction period. Temporal evolution of each strain of E. coli according to the mean-field model (thin lines) and according to simulations on a lattice with $200\times 200$ sites (thick lines). In both cases, during the interval $25\le t\le 150$ the $S$ strain comes very close to zero, characterizing a period of quasiextinction.Reuse & Permissions

Figure 3

(Color online) Effect of quasiextinction period of $S$ on the crossover lattice size $N_c$. (a) $N_c$ as a function of ${\Delta }_K$, the death rate of killer $K$. Variation of ${\Delta }_K$ affects strongly the crossover position in Fig. 1a since to decrease ${\Delta }_K$ implies an increase in the resistant phase $R$. (b) $N_c$ as a function of the quasiextinction period, calculated for the same eleven values of ${\Delta }_K$ shown in Fig. 3a. The behavior is well approximated by an exponential: $f\left(x\right)\simeq -1.1+4.0\text{exp}\left(0.011x\right)$. (c) $N_c$ computed similarly as in Fig. 3b but for nine values of ${\Delta }_{S,0}$. The behavior is well approximated by an exponential: $g\left(x\right)\simeq -72.9+31.0\text{exp}\left(0.017x\right)$. (d) $N_c$ computed as before for ten values of ${\Delta }_R$. The behavior is well approximated by an exponential: $h\left(x\right)\simeq -111.8+8.8\text{exp}\left(0.015x\right)$. The metric similarity of the three exponentials suggests that the growth of $N_c$ is a general feature that does not depend of the parameter changed.Reuse & Permissions