Fig 1.
Schemes of the two-species system for the chemostat model (a) and the simplified model (b).
Microbial species are represented by variables X (blue), nutrients by S (red), arrows represent the consumption or production of a nutrient by a species. In this system X1 and X2 compete for the consumption of S0 and they are mutualistic due to the cross-feeding through nutrients S1 and S2. Self-inhibition in the simplified system is determined via the parameter bii for species Xi (i = 1, 2), bij (i ≠ j) quantifies the strength of mutualism and ci the strength of competition, di is the death rate.
Table 1.
Definition of the variables and parameters of the chemostat system Eq (1) and of the growth rates Eq (2).
Fig 2.
Time simulations for different values of the growth rates μ illustrate the different behavioral regimes of the system.
For each panel 10 simulations are shown using initial densities varied between 0 and 20. (a) Extinction of the species X1 and X2 for all initial densities (μ = [800, 800]). (b) Bistability: depending on the initial densities the species will either survive (state 1) or become extinct (state 2) (μ = [1600, 1600]). (c) Bistability: There are two final states possible: coexistence of the species, X1 ≠ 0 and X2 ≠ 0 (state 1) or extinction of species X2 (state 2) (μ = [2400, 1200]). (d) Survival of X1 and X2 for all initial densities (μ = [2400, 2400]).
Fig 3.
Phase plane (X1, X2) with nullclines for Eq (4) for increasing growth rates: r = [0.02, 0.02] (a), [0.027, 0.027] (b), [0.05, 0.02] (c), [0.05, 0.05](d).
Different steady state configurations are found at the intersections of the nullclines: both species become extinct E, both species survive L12, only one species survives L1, L2. Stable solutions are indicated by the solid circle, while unstable saddle solutions are shown by the open circle.
Fig 4.
Zoom of Fig 3(b), using the same notation.
The blue dashed lines are the linear approximations of the nullclines and need to intersect in order to have bistability. The regions (1)-(5) are discussed in the text. The blue and red areas correspond to the basins of attraction of each steady state.
Fig 5.
Different dynamical regimes for the mutualist-competitive system exist, depending on the values of the parameters.
We define these regimes as follows: R1: Extinction. R2: Competitive exclusion, only species X1 survives. R3: Competitive exclusion, only species X2 survives. R4: bistability between extinction and coexistence. R5: bistability between survival of X1 and coexistence. R6: bistability between survival of X2 and coexistence. R7: coexistence of X1 and X2. (a) Influence of the growth rates μ in the chemostat system (b) Influence of the growth rates μ in the extended LV model (c) Influence of the flow rate Φ and the inflow in the chemostat for μ = [1600, 800]. (d) Influence of the flow rate Φ and the inflow
in the extended LV model for r = [0.04, 0.02].