The operating diagram for a model of competition in a chemostat with an external lethal inhibitor

Bachir Bar; Tewfik Sari

doi:10.3934/dcdsb.2019203

Article Contents

2020, Volume 25, Issue 6: 2093-2120. Doi: 10.3934/dcdsb.2019203

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The operating diagram for a model of competition in a chemostat with an external lethal inhibitor

Bachir Bar^1, and
Tewfik Sari^2, ,

1.
Université AbouBakr Belkaid, LSDA, Tlemcen, Algérie
2.
ITAP, Univ Montpellier, Irstea, Montpellier SupAgro, Montpellier, France

^* Corresponding author
^* Corresponding author

Received: December 31, 2018

Revised: April 30, 2019

Early access: September 2019

Published: June 2020

The authors are supported by the France-Algeria Partnership Tassili project 15MDU949 and the Euro-Mediterranean research network TREASURE (http://www.inra.fr/treasure)

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Abstract

The inhibition is an important phenomenon, which promotes the stable coexistence of species, in the chemostat. Here, we study a model of two microbial species in a chemostat competing for a single resource in the presence of an external lethal inhibitor. The model is a four-dimensional system of ordinary differential equations. We give a complete analysis for the existence and local stability of all steady states. We describe the bifurcation diagram which gives the behavior of the system with respect to the operating parameters represented by the dilution rate and the input concentrations of the substrate and the inhibitor. This diagram, is very useful to understand the model from both the mathematical and biological points of view.

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

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Figure 2. Illustrative operating diagrams for $ D $ fixed: The curves $ \Gamma_i $, $ i = 0\cdots 5 $ defined in the Table 3 divide the operating plane $ (p^0,S^0) $ into at most nine regions labeled $ \mathcal J_0 $, $ \mathcal J_1 $, $ \mathcal J_2 $, $ \mathcal J_3 $, $ \mathcal J_4 $, $ \mathcal J_5^S $, $ \mathcal J_5^U $, $ \mathcal J_6^S $ and $ \mathcal J_6^U $. Some of the regions may be empty. The existence and stability of the equilibrium points in the regions of these diagrams are shown in Table 4

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Figure 3. The biological parameters values are given in Table 5, Case 1. $ J_c = \varUpsilon_0 $: therefore $ E_c $ is LES whenever it exists. The operating diagram for $ D = 1 $ shows that $ \left(p^0 = 1,S^0 = 1\right)\in\mathcal{J}_6^S $. The existence and stability of the equilibrium points in the regions of this diagram are shown in Table 4

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Figure 5. The biological parameters values are given in Table 5, Case 2. (a): The operating diagram for $ D = 1 $ shows that $ p^0 = S^0 = 1 $ belongs to $ \mathcal{J}_6^S $. (b): The operating diagram for $ D = 2.2 $. The existence and stability of the equilibrium points in the regions of these diagrams are shown in Table 4

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Figure 7. The biological parameters values are given in Table 5, Case 3. (a): The operating diagram for $ D = 1 $. $ (b) $: A zoom showing the instability of $ E_c $ when $ p^0 = S^0 = D = 1 $. The existence and stability of the equilibrium points in the regions of these diagrams are shown in Table 4

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Figure 8. The biological parameters values are given in Table 5, Case 3. The operating diagram for $ D = 2.2 $. (a): $ \mathcal{J}_6^U $ is unbounded; (d): a zoom near the origin showing the regions $ \mathcal{J}_1 $, $ \mathcal{J}_2 $, $ \mathcal{J}_3 $ and $ \mathcal{J}_5^S $. The existence and stability of the equilibrium points in the regions of this diagram are shown in Table 4

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Figure 10. The biological parameters values are given in Table 5, Case 4. The operating diagram for (a): $ D = 0.01 $. (b): $ D = D_1 = 0.013 $. The existence and stability of the equilibrium points in the regions of this diagram are shown in Table 4

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Figure 12. The biological parameters values are given in Table 5, Case 5. The operating diagram for $ D = 1.26 $. (a): The full operating diagram. (b): A zoom showing that $ \mathcal{J}_4 $ is nonempty. The existence and stability of $ E_0 $, $ E_1 $, $ E_2 $ and $ E_c $ in the regions of these diagrams are shown in Table 4

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Figure 13. The biological parameters values are given in Table 5, Case 3. The operating diagram in the $ (S^0,D) $-plane for $ p^0 = 1 $. (a): The region $ \mathcal{J}_6^U $ is unbounded. (b): A zoom showing the instability of $ E_c $ for $ S^0 = p^0 = D = 1 $. The existence and stability of the equilibrium points are shown in Table 4

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Figure 14. The biological parameters values are given in Table 5, Case 3. The operating diagram in the $ (p^0,D) $-plane for $ S^0 = 1 $, showing the instability of $ E_c $ for $ S^0 = p^0 = D = 1 $. The existence and stability of the equilibrium points in the regions of this diagram are shown in Table 4

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Figure 15. The biological parameters values are given in Table 5, Case 3, and $ \beta_1 = \beta_2 = 100 $. The operating diagram in the $ (p^0,D) $-plane for $ S^0 = 1 $. (a): The full diagram. (b): A zoom showing the regions near the $ D $ axis showing the regions $ \mathcal{J}_1 $, $ \mathcal{J}_3 $ and $ \mathcal{J}_5^S $. The existence and stability of the equilibrium points in the regions of this diagram are shown in Table 4

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Figure 1. (a): Definitions of $ \lambda_1 = \lambda_1(D) $, $ \lambda^- = \lambda^-(D,p^0,S^0) $, $ \lambda^+ = \lambda^+(D,p^0) $ and $ \lambda_2 = \lambda_2(D) $. (b): Definition of $ p^* = p^*(D,p^0,S^{0}) $ satisfying $ W(p^*,D,p^0) = \beta_2\left(S^0-\lambda_2(D)\right) $

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Figure 4. The biological parameters values are given in Table 1, Case 2. The regions $ \varUpsilon_0 $ and $ \varUpsilon_2 $ of $ J_c $, and the definitions of $ p_1(D) $ and $ p_2(D) $ for $ D_1<D<D_2 $: $ D_1\simeq 1.83 $, $ D_2\simeq 2.65 $, $ p_1(2.2)\simeq 0.65 $, $ p_2(2.2)\simeq 1.04 $

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Figure 6. The biological parameters values are given in Table 1, Case 3. The regions $ \varUpsilon_0 $, $ \varUpsilon_1 $ and $ \varUpsilon_2 $ and the definitions of $ p_1(D) $, $ p_2(D) $ for $ D_1<D<D_2 $, and $ p_3(D) $, $ p_4(D) $ for $ D_3<D<D_4 $, where $ D_1\simeq 0.63 $, $ D_2\simeq 1.27 $, $ D_3\simeq2.82 $ and $ D_4\simeq 3.59 $. The figure shows the values $ p_1(2.2)\approx 0.47 $, $ p_2(2.2)\approx 4.17 $, $ p_3(2.2)\approx 0.77 $, $ p_4(2.2)\approx 2.71 $

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Figure 9. The biological parameters values are given in Table 5, Case 4. In red curve of equation $ a_3 = 0 $, in blue, a component of the curve of equation $ \Delta = 0 $. In black, the curve $ p^0 = p_c(D) $ (a): The full regions $ \varUpsilon_0 $, $ \varUpsilon_1 $ and $ \varUpsilon_2 $. (b): a zoom showing the values $ p_1 = p_1(D) $, $ p_2 = p_2(D) $ $ p_3 = p_3(D) $ and $ p_4 = p_4(D) $ for $ D = 0.013 $. For the clarity of the figures, the points $ p_1(D) $ and $ p_2(D) $, for $ D = 0.01 $, are not depicted on the figure

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Figure 11. The biological parameters values are given in Table 5, Case 5. In red curve of equation $ a_3 = 0 $, in blue, a component of the curve of equation $ \Delta = 0 $. In black, the curve $ p^0 = p_c(D) $ (a): The full regions $ \varUpsilon_0 $, $ \varUpsilon_1 $ and $ \varUpsilon_2 $. (b): A zoom showing the values $ D_1 $ and $ D_2 $ and $ p_1 $ and $ p_2 $ corresponding to $ D = 1.26 $

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Figure 16. The plots of curves $ p^0 = p_c(D) $, in black, $ \Delta = 0 $, in blue, and $ a_1a_2 = a_0a_3 $ in magenta. The biological parameters values are given in Table 5, Case 2

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Figure 17. The curve $ p^0 = p_c(D) $ is plotted in black. (a): The plots of curves $ \Delta = 0 $ ($ \mathcal{C}_1\cup\mathcal{A}_1\cup\mathcal{A}_2 $, in blue), $ a_3 = 0 $ ($ \mathcal{C}_2 $, in red). (b): The plots of curves $ a_1a_2 = a_0a_3 $ ($ \mathcal{C}_3\cup\mathcal{C}_4 $, in magenta) and $ a_2 = 0 $ ($ \mathcal{C}_5 $, in cyan). The biological parameters are given in Table 5, Case 3

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Figure 18. The curve $ p^0 = p_c(D) $ is plotted in black. (a): The plots of curves $ \Delta = 0 $ ($ \mathcal{C}_1\cup\mathcal{A}_1\cup\mathcal{A}_2 $, in blue), $ a_3 = 0 $ ($ \mathcal{C}_2 $, in red), $ a_1a_2 = a_0a_3 $ ($ \mathcal{C}_3\cup\mathcal{C}_4 $, in magenta) and $ a_2 = 0 $ ($ \mathcal{C}_5 $, in cyan). (b): A zoom of the strip $ 0<p^0<1 $. The biological parameters are given in Table 5, Case 3

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Table 6. Meanings and units of the variables and parameters of (1)

	Meanings	Units
$ S $, $ x $, $ y $, $ p $	Concentrations of substrate, species and inhibitor	mass/volume
$ S^0 $, $ p^0 $	Input concentrations of substrate and inhibitor	mass/volume
$ D $	Dilution rate	1/time
$ m_1 $, $ m_2 $	Maximal growth rates of the competitors	1/time
$ K_1 $, $ K_2 $	Half saturation constants of the competitors	mass/volume
$ \delta $	Maximal growth rate of detoxification	1/time
$ K $	Half saturation constant of detoxification	mass/volume
$ \gamma $	Lethal effect of $ p $ on $ x $	volume/mass
$ \beta_1 $, $ \beta_2 $	Growth yield coefficients	dimensionless

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Table 3. Boundaries of the regions in the operating diagram. The color code is used in Figs. 2, 3, 5, 7, 8, 10, 12, 13, 14, 15

Boundary	Color	Equation in $ \left(p^0, S^0\right) $-plane, with $ D\in I_c $ fixed
$ \Gamma_1 $	blue	Graph of $ S^0 =f_1^{-1}(D+\gamma p^0) $
$ \Gamma_2 $	black	Horizontal line $ S^0 = \lambda_2(D) $
$ \Gamma_3 $	red	Vertical line $ p^0 = p_c(D) $ and $ S^0> \lambda_2(D) $
$ \Gamma_4 $	cyan	Oblique line $ S^0=\frac{D\left(p^0-p_c(D)\right)}{\beta_2g(p_c(D))}+\lambda_2(D) $ and $ p^0>p_c(D) $
$ \Gamma_5 $	green	Curve of equation $ F_3(D,p^0,S^0)=0 $

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Table 4. Existence and stability of equilibrium points in the regions of the operating diagram, shown in Figs. 2, 3, 5, 7, 8, 10, 12, 13, 14, 15

Regions	$ \mathcal J_0 $	$ \mathcal J_1 $	$ \mathcal J_2 $	$ \mathcal J_3 $	$ \mathcal J_4 $	$ \mathcal J_{5}^S $	$ \mathcal J_{5}^U $	$ \mathcal J_{6}^S $	$ \mathcal J_{6}^U $
$ E_0 $	S	U	U	U	U	U	U	U	U
$ E_1 $		S		S	U	U	U
$ E_2 $			S	U	S	U	U	U	U
$ E_{c} $						S	U	S	U

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Table 5. Biological parameters values used in the numerical computations shown in the figures. The yields are $ \beta_1 = \beta_2 = 1 $, excepted for Fig. 15 in which $ \beta_1 = \beta_2 = 100 $. The last column of the table shows the value of $ \overline{D} $ such that $ E_c $ exists for $ D\in(0,\overline{D}) $

Case	$ m_1 $	$ m_2 $	$ K_1 $	$ K_2 $	$ \delta $	$ K $	$ \gamma $	Figures	$ \overline{D} $
1	4.0	5.0	0.3	1.0	3.0	0.3	4.0	24	3.54
2	4.0	5.0	0.06	1.0	5.0	1.3	4.0	25, 5, 16	3.94
3	4.0	5.0	0.03	1.0	5.0	1.3	4.0	26, 7, 8, 13, 14, 15, 17, 18	3.97
4	1.7	2	0.4	0.9	15	0.03	0.025	9, 10	1.46
5	4.0	5.0	0.03	1.0	0.5	1.3	4.0	11, 12	3.97

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Table 1. Existence and stability of equilibrium points $ E_0 $, $ E_1 $, $ E_2 $ and $ E_c $ of (3), given in Prop. 1. Here, $ \lambda_2 $ and $ \lambda^+ $ are given by (5), $ \lambda^- $ is given by (7) and $ A_1 $, $ A_2 $, $ A_3 $ and $ A_4 $ are given by (17)

	Existence	Local exponential stability
$ E_0 $	Always	$ \min(\lambda^+,\lambda_2)>S^0 $
$ E_1 $	$ \lambda^+<S^0 $	$ \lambda^+<\lambda_2 $
$ E_2 $	$ \lambda_2<S^0 $	$ \lambda_2<\lambda^- $
$ E_{c} $	$ \lambda^-<\lambda_2<\min(\lambda^+,S^0) $	$ A_3(A_1A_2-A_3)> A_1^2A_4 $

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Table 2. Existence and stability of equilibrium points of (3), with respect to the operating parameters $ D $, $ S^0 $ and $ p^0 $. The functions $ F_1 $, $ F_2 $, $ F_3 $ are defined by (19), (20), (21), respectively

	Existence	Local exponential stability
$ E_0 $	Always	$ D>\max(f_1(S^0)-\gamma p^0,f_2(S^0)) $
$ E_1 $	$ D<f_1(S^0)-\gamma p^0 $	$ D<F_1(D,p^0) $
$ E_2 $	$ D<f_2(S^0) $	$ S^0<F_2(D,p^0) $
$ E_{c} $	$ D>F_1(D,p^0) $ & $ S^0>F_2(D,p^0) $	$ F_3(D,p^0,S^0)>0 $

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Table 7. The signs of functions $ a_2 $, $ a_3 $, $ a_1a_2-a_0a_3 $ and $ \Delta $. Here $ \mathcal{A} = R_1\cup\mathcal{A}_1\cup R_2\cup\mathcal{A}_2 $

Function	$<0 $	$ =0 $	$>0 $
$ \Delta $	$ Ext\left(\mathcal{C}_1\right)\setminus \mathcal{A} $	$ \mathcal{C}_1\cup\mathcal{A}_1\cup\mathcal{A}_2 $	$ Int\left(\mathcal{C}_1\right)\cup R_1\cup R_2 $
$ a_3 $	$ Int\left(\mathcal{C}_2\right) $	$ \mathcal{C}_2 $	$ Ext\left(\mathcal{C}_2\right) $
$ a_1a_2-a_0a_3 $	$ Int\left(\mathcal{C}_3\right)\cap Ext\left(\mathcal{C}_4\right) $	$ \mathcal{C}_3\cup\mathcal{C}_4 $	$ Ext\left(\mathcal{C}_3\right)\cup Int\left(\mathcal{C}_4\right) $
$ a_2 $	$ Int\left(\mathcal{C}_5\right) $	$ \mathcal{C}_5 $	$ Ext\left(\mathcal{C}_5\right) $

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