Definition of the variables and parameters of the chemostat system Eq (1) and of the growth rates Eq (2).

<p>Definition of the variables and parameters of the chemostat system <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0197462#pone.0197462.e001" target="_blank">Eq (1)</a> and of the growth rates <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0197462#pone.0197462.e013" target="_blank">Eq (2)</a>.</p

Phase plane (<i>X</i>₁, <i>X</i>₂) with nullclines for Eq (4) for increasing growth rates: <i>r</i> = [0.02, 0.02] (a), [0.027, 0.027] (b), [0.05, 0.02] (c), [0.05, 0.05](<i>d</i>).

<p>Different steady state configurations are found at the intersections of the nullclines: both species become extinct <i>E</i>, both species survive <i>L</i>₁₂, only one species survives <i>L</i>₁, <i>L</i>₂. Stable solutions are indicated by the solid circle, while unstable saddle solutions are shown by the open circle.</p

Zoom of Fig 3(b), using the same notation.

<p>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.</p

Different dynamical regimes for the mutualist-competitive system exist, depending on the values of the parameters.

<p>We define these regimes as follows: R1: Extinction. R2: Competitive exclusion, only species <i>X</i>₁ survives. R3: Competitive exclusion, only species <i>X</i>₂ survives. R4: bistability between extinction and coexistence. R5: bistability between survival of <i>X</i>₁ and coexistence. R6: bistability between survival of <i>X</i>₂ and coexistence. R7: coexistence of <i>X</i>₁ and <i>X</i>₂. (a) Influence of the growth rates <b><i>μ</i></b> in the chemostat system (b) Influence of the growth rates <b><i>μ</i></b> in the extended LV model (c) Influence of the flow rate Φ and the inflow in the chemostat for <b><i>μ</i></b> = [1600, 800]. (d) Influence of the flow rate Φ and the inflow in the extended LV model for <b><i>r</i></b> = [0.04, 0.02].</p

Time simulations for different values of the growth rates <i>μ</i> illustrate the different behavioral regimes of the system.

<p>For each panel 10 simulations are shown using initial densities varied between 0 and 20. (a) Extinction of the species <i>X</i>₁ and <i>X</i>₂ for all initial densities (<b><i>μ</i></b> = [800, 800]). (b) Bistability: depending on the initial densities the species will either survive (state 1) or become extinct (state 2) (<b><i>μ</i></b> = [1600, 1600]). (c) Bistability: There are two final states possible: coexistence of the species, <i>X</i>₁ ≠ 0 and <i>X</i>₂ ≠ 0 (state 1) or extinction of species <i>X</i>₂ (state 2) (<b><i>μ</i></b> = [2400, 1200]). (d) Survival of <i>X</i>₁ and <i>X</i>₂ for all initial densities (<b><i>μ</i></b> = [2400, 2400]).</p

Schemes of the two-species system for the chemostat model (a) and the simplified model (b).

<p>Microbial species are represented by variables <i>X</i> (blue), nutrients by <i>S</i> (red), arrows represent the consumption or production of a nutrient by a species. In this system <i>X</i>₁ and <i>X</i>₂ compete for the consumption of <i>S</i>₀ and they are mutualistic due to the cross-feeding through nutrients <i>S</i>₁ and <i>S</i>₂. Self-inhibition in the simplified system is determined via the parameter <i>b</i>_<i>ii</i> for species <i>X</i>_<i>i</i> (<i>i</i> = 1, 2), <i>b</i>_<i>ij</i> (<i>i</i> ≠ <i>j</i>) quantifies the strength of mutualism and <i>c</i>_<i>i</i> the strength of competition, <i>d</i>_<i>i</i> is the death rate.</p