Figure 5

Bistability requires a sufficient production of cross-feeding nutrients. (A) Different regimes are distinguished as a function of the parameter values, using Eq. (4). For \(b < 1\), the growth is limited by the cross-feeding nutrient, leading to monostable dynamics similar to logistic growth. There is survival for small dilution (\(\,d < {r}_{a}{C}_{a}a\)) and extinction for large dilution. For \(b > 1\), the growth only becomes limited by the substrate, allowing for bistability. There is a weak Allee effect, corresponding to monostable survival, for small dilution, a strong Allee effect for intermediate dilution \((r{C}_{s}{C}_{p} < \,d < 0.25r(b-1){\left({C}_{s}+\frac{{C}_{p}}{b-1}\right)}^{2})\) and monostable extinction for large dilution. (B) For \(b < 1\), the growth is limited by the cross-feeding nutrients (Eq. (9)), so that the physical growth region is small and does not allow bistability. For small dilution the dynamics is similar to logistic growth as the population monotonically grows towards its equilibrium. (C) For \(b > 1\), high population densities are possible, whereby the growth is limited by the substrate \(S\) (Eq. (8)). Increased values of \(b\) lead to a larger growth region, allowing for bistability when \(b > 1\). Increasing the dilution rate \(\,d\) causes the nullclines to bend into hyperbola with the linear functions at \(\,d=1\) as asymptotes. Bistability is obtained when the hyperbolic nullclines intersect twice parameter values as listed in Table S6).