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# Global analysis of a model of competition in the chemostat with internal inhibitor

• * Corresponding author: Mohamed Dellal

The authors are supported by the "Directorate General for Scientific Research and Technological Development, Ministry of Higher Education and Scientific Research, Algeria" and the Euro-Mediterranean research network TREASURE (http://www.inra.fr/treasure)

• A model of two microbial species in a chemostat competing for a single resource in the presence of an internal inhibitor is considered. The model is a four-dimensional system of ordinary differential equations. Using general growth rate functions of the species, we give a complete analysis for the existence and local stability of all steady states. We describe the behavior of the system with respect to the operating parameters represented by the dilution rate and the input concentrations of the substrate. The operating diagram has the operating parameters as its coordinates and the various regions defined in it correspond to qualitatively different asymptotic behavior: washout, competitive exclusion of one species, coexistence of the species, bistability, multiplicity of positive steady states. This bifurcation diagram which determines the effect of the operating parameters, is very useful to understand the model from both the mathematical and biological points of view, and is often constructed in the mathematical and biological literature.

Mathematics Subject Classification: Primary: 34C23, 34D20; Secondary: 92B05, 92D25.

 Citation: • • Figure 1.  Projection of equilibria $E_0$, $E_1$, $E_2$ and $E_c$ in the plane $(S,p)$ and existence and stability conditions of these points. Where stable equilibrium points are denoted by filled circles and unstable equilibrium points are denoted by empty circles

Figure 2.  The existence of multiple positive equilibria when the condition (20) $\rm(a)$: is not satisfied; $\rm(b)$: is satisfied

Figure 3.  Case $n = 3$: Local stability of $E_1$, $E_2$, $E_c^2$ and instability of $E_c^1$, $E_c^2$, $E_0$

Figure 5.  The operating diagram of the system (1) where $\mu_i$ are given by (2) and the curves $\Gamma_i^c$ do not intersect. $\rm(a)$: The occurrence of the bistability region $\mathcal{J}_7$, where the coexistence region $\mathcal{J}_6$ does not exist. $\rm(b)$: The occurrence of the coexistence region $\mathcal{J}_6$. The biological parameters used to construct Figs. Figs. 5(a, b) are exactly the same except that the values of $\alpha_i$ have been inverted

Figure 6.  The operating diagram of the system (1) where $\mu_i$ are given by (2) and the curves $\Gamma_i^c$ intersect. The pictures show the occurrence of the bistability region $\mathcal{J}_8$ of $E_2$ and $E_c^i$, The biological parameters used to construct Figs. Figs. 6(a, b) are exactly the same except that the values of $\alpha_i$ have been inverted

Figure 4.  Illustrative graph of $\Gamma_i^c$, $i = 1,2$ defined by (33), showing the relative positions of the roots $S_c^1(\!D\!)$ and $S_c^2(\!D\!)$ of (39) with respect to the root $S_i(\!D\!,\!S^0\!)$ of (38). Region Ⅰ: $S_i<S_c^1<S_c^2$; region Ⅱ: $S_c^1<S_c^2<S_i$; region Ⅲ : $S_c^1<S_i<S_c^2$

Figure 7.  The trajectories of the reduced model (28) where $(S^0,D)$ are chosen in regions of Fig. 6(a). (a): Bistability of $E_1$ and $E_2$ when $(S^0,D) = (0.1,0.9)\in\mathcal{J}_7^a$. (b): Bistability of $E_c^1$ and $E_2$ when $(S^0,D) = (0.05,1.15)\in\mathcal{J}_8^a$. (c): Global stability of $E_c^1$ when $(S^0,D) = (0.02,1.15)\in\mathcal{J}_6^a$

Table 1.  Existence and local asymptotic stability of equilibria of system (1)

 Equilibria Existence Local exponential stability $E_0$ Always $\lambda_1>S^0$ and $\lambda_2>S^0$ $E_1$ $\lambda_1F_2(S_1)$ $E_2$ $\lambda_2F_1(S_2)$ $E_{c}$ (23) has a solution $(\alpha_1\gamma_1-\alpha_2\gamma_2)[F'_2(S_c)-F'_1(S_c)]>0$

Table 2.  Parameter values used in Section 5 where $\mu_i$ are given by (2)

 Parameters $m_1$ $m_2$ $a_1$ $a_2$ $K_1$ $K_2$ $\alpha_1$ $\alpha_2$ Figures Units $h^{-1}$ $h^{-1}$ $gl^{-1}$ $gl^{-1}$ $gl^{-1}$ $gl^{-1}$ Case (a) 1.0 2.0 0.01 0.04 0.01 0.006 0.1 4.0 5(a) Case (b) 1.0 2.0 0.01 0.04 0.01 0.006 4.0 0.1 5(b) Case (c) 2.0 9.0 0.006 0.04 0.005 0.001 0.005 0.4 6(a), 7 Case (d) 2.0 9.0 0.006 0.04 0.005 0.001 0.4 0.005 6(b)

Table 3.  Existence and stability of equilibria in the regions of the operating diagrams of Fig. 5, when the curves $\Gamma_i^c$ do not intersect. The letter S (resp. U) means stable (resp. unstable) and empty if that equilibrium does not exist

 Region The relative positions of $S_i$ and $S_c^i$ $E_0$ $E_1$ $E_2$ $E_{c}^1$ $E_{c}^2$ $(S^0,D)\in\mathcal J_1$ $S_1$ and $S_2$ do not exist S $(S^0,D)\in\mathcal J_2$ $S_1$ does not exist U S $(S^0,D)\in\mathcal J_3$ $S_c^1 Table 4. Existence and stability of equilibria in the regions of the operating diagrams of Fig. 6, when the curves$ \Gamma_i^c $intersect  Region The relative positions of$ S_i $and$ S_c^i  E_0  E_1  E_2  E_{c}^1  E_{c}^2  (S^0,D)\in\mathcal J_1  S_1 $and$ S_2 $do not exist S$ (S^0,D)\in\mathcal J_2  S_1 $does not exist U S$ (S^0,D)\in\mathcal J_3  S_i

Table 5.  Parameter values of $S_i$ and $S_c^i$ used in Fig.7

 $(S^0,D)$ Regions $S_1$ $S_2$ $S_c^1$ $S_c^2$ Figures $(0.1,0.9)$ $\mathcal{J}_7^a$ 0.006 0.085 0.005 0.064 7(a) $(0.05,1.15)$ $\mathcal{J}_8^a$ 0.009 0.042 0.012 0.025 7(b) $(0.02,1.15)$ $\mathcal{J}_6^a$ 0.008 0.017 0.012 0.025 7(c)
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