Extended Data Fig. 2: Invasibility conditions.

Invader strain denoted by red and resident strain denoted by blue. A) Impact of toxin investment z on invasion success. Invasion success decreases with investment into toxin production (dashed lines) and increases with the supplementation of private nutrients to the invader. Dashed lines correspond to the growth rate of the invader in the spent medium of the resident (Eq. S12) and \(\delta =0.5\) denotes the dilution rate (\(p=0.7\), (1): \({m}_{I}=0.1\), (2): \({m}_{I}=1.5\)). B) Impact of toxin potencyp on invasion success. Invasion success is independent of toxin potency and increases with the supplementation of private nutrients to the invader. Dashed lines correspond to the growth rate of the invader in the spent medium of the resident (Eq. S12) and \(\delta =0.5\) denotes the dilution rate (\(z=0.5\), (1): \({m}_{I}=0.4\), (2): \({m}_{I}=4\)). C) Impact of toxin investment z on displacement. The possibility of displacement increases with the supplementation of private nutrients and a suitable toxin investment, as long as the metabolic cost of toxin production does not prevent the success of invasion (red cross). Dashed lines correspond to the growth rate of the resident in the spent medium of the invader (Eq. S17) and \(\delta =0.5\) denotes the dilution rate (\(p=0.7\), (1): \({m}_{I}=0.5\), (2): \({m}_{I}=2\), (3): \({m}_{I}=4\)). D) Impact of toxin potency p on displacement. The possibility of displacement increases with toxin potency and supplementation of private nutrients. Dashed lines correspond to the growth rate of the resident in the spent medium of the invader (Eq. S17) and \(\delta =0.5\) denotes the dilution rate (\(z=0.5\), (1): \({m}_{I}=1.5\), (2): \({m}_{I}=2\), (3): \({m}_{I}=4\)). Parameter values used for plotting this figure: \(m={m}_{R}=1,\delta =0.5,D=d=0.15,{R}_{R}=\)\({R}_{I}=2,{r}_{R}={r}_{I}=1,{C}_{R}={c}_{R}={C}_{I}=\)\({c}_{I}=1,s=1,{k}_{R}={k}_{I}=1,{K}_{R}={K}_{I}=K=\)\(10,z=0.5,g=1,p=0.7\).