Figure 4
From: Mutualistic cross-feeding in microbial systems generates bistability via an Allee effect
Original chemostat Eq. (1) confirm findings in the reduced Eq. (5). (A) The first-order Taylor approximation of the growth rate (Eq. (2)) causes the wash-out limit to be reached for lower dilution rates. (B) By constructing the bifurcation diagram as a function of \(\,d\), using chemostat Eq. (1) and the Monod-like growth rates, we can estimate the error by considering the point at which the saddle-node bifurcation (SN) occurs (T: transcritical bifurcation). A fold change \(F\) of the Monod constants (\({K}_{s{\prime} }=F\cdot {K}_{s}\) & \({K}_{p{\prime} }=F\cdot {K}_{p}\)) is used to quantify the deviation from the simplified bifurcation curve of Eq. (5). At \(F=1\) the error is large as \({K}_{s}\) and \({K}_{p}\) are of the order of \(S\) and \(P\) (Table S5), the error is significantly reduced for \(F=3.3\) and almost negligible for \(F=10\). (C) Using \(F=1\), The calculated regions of survival, bistability and extinction in the reduced system (dashed curves) map on the simulated regions of the chemostat equations (solid curves) so that the qualitative behavior is conserved. (D) The error of the simplification is visualized by mapping the critical dilution rate at which the saddle node bifurcation (SN) occurs. For \(F=1\) the critical value of the dilution (\({d}^{\ast }\)) is about 50% lower than the estimated value using the simplified model (Parameter values listed in Table S5).
