Figure 1

Bistability in mutualistic systems creates a survival threshold. (A) The microbial system we study consists of two species, characterized by their densities \({\rho }_{1}\) and \({\rho }_{2}\), consumption of substrates (concentrations \({S}_{1}\) and \({S}_{2}\)) and mutualistic cross-feeding via the production of nutrients (concentrations \({P}_{1}\) and \({P}_{2}\)). We theoretically investigate the growth of these species in a chemostat reactor: a well-mixed growth vessel with an inflow of nutrients with concentrations \({\tilde{S}}_{i}\), \({\tilde{P}}_{i}\) (\(i=1,2\)) and an outflow of the suspension, occuring at an equal rate: the dilution rate \(\,d\). (B) The behavior of the system is simulated with Eq. (1) using the growth rate Eq. (2). The dilution is gradually increased from \(\,d=1\) to \(\,d=0.24\) with steps of \(0.08\) for consecutive time frames of \(t=100\), using a logarithmic scale for the species density for clarity. The equilibrium densities of the species decrease with the dilution. For \(\,d=0.24\), a threshold is reached and the species will be washed-out. Decreasing the dilution to its previous rate does not lead to the recovery of the initial abundances of the species. The dilution needs to be further decreased to \(\,d=0.08\) the make the population growing again. This is a phenomenon called hysteresis: the system has memory of the previous state. It is a consequence of bistability between survival and extinction at intermediate dilution, so that a density threshold for survival exists. Initial conditions: \({\rho }_{1}(0)=0.4\), \({\rho }_{2}(0)=0.7\), \({S}_{1}(0)=0.3\), \({S}_{2}(0)=0.9\), \({P}_{1}(0)=1.0\), \({P}_{2}(0)=0.9\). Parameters values: \({\mu }_{1}=2\), \({\mu }_{2}=2\), \({K}_{s1}=2\), \({K}_{s2}=2\), \({K}_{p1}=1\), \({K}_{p2}=1\), \({\tilde{S}}_{1}=1.1\), \({\tilde{S}}_{2}=0.9\), \({\tilde{P}}_{1}=0.2\), \({\tilde{P}}_{2}=0.2\), \({\nu }_{s1}=1\), \({\nu }_{s2}=1\), \({\nu }_{p1}=1\), \({\nu }_{p2}=1\), \({a}_{1}=2\), \({a}_{2}=2\) (arbitrary units).