Fig. 4: Eco-evolutionary feedbacks in a simple resource competition model.

a A simplified model of a microbiome: \({{{{{{{\mathcal{S}}}}}}}}\) resident species compete for \({{{{{{{\mathcal{R}}}}}}}}\) substitutable metabolites that are continuously supplied by the environment; r_μ,i denotes the uptake rate of metabolite i by species μ, while β_i denotes the input flux from the environment. In the absence of further mutations, growth and dilution lead to steady-state levels of resource concentrations (c_i) and species abundances (f_μ). b An example of an evolution-driven feedback: a mutation in a focal strain alters its resource uptake phenotype by an small amount \(\Delta \overrightarrow{r}\); if the mutation provides a fitness benefit (S_inv > 0), its descendants will proliferate in the community and cause a shift in the steady-state abundances of the resident species (f_μ → f_μ + Δf_μ). Colors denote the different species in (a). c Evolution-driven feedbacks in a simplified community with \({{{{{{{\mathcal{S}}}}}}}}={{{{{{{\mathcal{R}}}}}}}}=2\) (see Supplementary Note 1 for a mathematical derivation). Solid circles denote the resident strains, while dashed circles denote potential mutants. Shading indicates regions of phenotype space that are favored (S_inv > 0, white) or disfavored (S_inv < 0, shaded) to invade; when \({{{{{{{\mathcal{S}}}}}}}}={{{{{{{\mathcal{R}}}}}}}}=2\), these regions are completely determined by the resident strains, and are conditionally independent of the input fluxes from the external environment (β_i). The top two mutations lead to an increased abundance of the focal species when they invade, while the bottom example shows a beneficial mutation (S_inv > 0) that decreases the relative abundance of its focal species (\(\Delta {f}_{{\mu }^{*}} < \, 0\)).