Extended Data Fig. 1: Ecological theory of strain replacement varying toxin potency instead of toxin investment, and varying supplementation of private nutrients for both strains instead of just the invader.
From: Strain displacement in microbiomes via ecological competition
Additional scenarios related to Fig. 1 that capture three key aspects of the natural ecology of bacterial competition: ecological strain invasions where an invadNumerical solutions of four scenarioser (red) invades the niche of a resident (blue), nutrient competition over shared nutrient (grey), and interference competition (red stars for toxins produced by the invader). A-E) Varying toxin potency instead of toxin investment. F-J) Varying toxin investment, but now varying supplementation of private nutrients for both strains instead of just the invader. K-O) Varying toxin potency as well as varying supplementation of private nutrients for both strains. A,F,K) Invasion success as a result of varying toxin investment (z; abbreviation=invest.) or potency (p) on the y-axis, or the supplementation of a private nutrient for the invader (mI) or both the invader and the resident (mI=mR) on the x-axis. Invasion fails (white region) if private nutrients are not sufficiently available. If an invader has sufficient access to a private nutrient, it can co-exist with a resident strain (light grey region), but if it invests sufficiently into a toxin or has a sufficiently potent toxin, it can displace the resident strain (dark grey region; abbreviation=disp.). The invasion boundary is analytically determined (Eq. S12; see Supplementary text) and the displacement boundary is plotted numerically (see Methods), with the dashed line delimiting the analytically derived bound for the displacement boundary (Eq. S17; see Supplementary text). B-E,G-J,L-O) Numerical solutions of four scenarios panels A,F,K, indicated as black points. Solid lines indicate the abundance of strains (resident in blue, invader in red), the dotted red line indicates the abundance of the invader toxin, and the dashed lines indicate the abundance of nutrients (shared nutrient in grey, private nutrient for the resident in blue, private nutrient for the invader in red). B-E) The invader has a non-potent toxin and no private nutrient (panel B; mI=0, p=0), a non-potent toxin but an abundant private nutrient (panel C; mI=1, p=0), a potent toxin but no private nutrient (panel D; mI=0, p=1), and a potent toxin and an abundant private nutrient (panel E; mI=1, p=1). Parameter values used for simulating the invasion dynamics from Eq. 3 (Methods): \(m={m}_{R}=1,\delta =D=d=0.15,{R}_{R}={r}_{R}={R}_{I}=\)\({r}_{I}=1,{C}_{R}={c}_{R}={C}_{I}={c}_{I}=1,s=1,{k}_{R}={k}_{I}=\)\(1,{K}_{R}={K}_{I}=K=10,z=0.5,g=1.\) G-J) The invader has not invested in the production of a toxin and there are no private nutrients (panel G; mI=mR =0, z=0), the invader has not invested in the production of a toxin but there are private nutrients for both strains (panel H; mI=mR =1, z=0), the invader invested in a toxin but there are no private nutrients (panel I; mI=mR=0, z=0.5), and the invader has both invested in a toxin and there are private nutrients (panel J; mI=mR =1, z=0.5). Parameter values used for simulating the invasion dynamics from Eq. 3(Methods): \(m=1,\delta =D=d=0.15,{R}_{R}={r}_{R}={R}_{I}={r}_{I}=\)\(1,{C}_{R}={c}_{R}={C}_{I}={c}_{I}=1,s=1,{k}_{R}={k}_{I}=\)\(1,{K}_{R}={K}_{I}=K=10,g=1,p=0.7.\) L-O) The invader does not have a potent toxin and there are no private nutrients for either strain (panel L; mI=mR =0, p=0), the invader does not have a potent toxin but there are private nutrients for both strains (panel M; mI=mR =1, p=0), the invader has a potent toxin but there are no private nutrients for either strain (panel N; mI=mR =0, p=1), the invader has a potent toxin and there are private nutrients for both strains (panel O; mI=mR =1, p=0). Parameter values used for simulating the invasion dynamics from Eq. 3(Methods): \(m=1,\delta =D=d=0.15,{R}_{R}={r}_{R}={R}_{I}={r}_{I}=\)\(1,{C}_{R}={c}_{R}={C}_{I}={c}_{I}=1,s=1,{k}_{R}={k}_{I}=\)\(1,{K}_{R}={K}_{I}=K=10,z=0.5,g=1\).
