# American Institute of Mathematical Sciences

February  2021, 26(2): 1129-1148. doi: 10.3934/dcdsb.2020156

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

 a. TAG_SUP Université Ibn Khaldoun, Tiaret 14000, Algérie b. TAG_SUP Ecole Normale Supérieure, Mostaganem 27000, Algérie c. TAG_SUP Université Djillali Liabès, LDM, Sidi Bel Abbès 22000, Algérie

* Corresponding author: Mohamed Dellal

Received  August 2019 Revised  January 2020 Published  February 2021 Early access  May 2020

Fund Project: 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.

Citation: Mohamed Dellal, Bachir Bar. Global analysis of a model of competition in the chemostat with internal inhibitor. Discrete and Continuous Dynamical Systems - B, 2021, 26 (2) : 1129-1148. doi: 10.3934/dcdsb.2020156
##### References:
 [1] N. Abdellatif, R. Fekih-Salem and T. Sari, Competition for a single resource and coexistence of several species in the chemostat, Mathematical Biosciences and Engineering, 13 (2016), 631-652.  doi: 10.3934/mbe.2016012. [2] B. Bar and T. Sari, The operating diagram for a model of competition in a chemostat with an external lethal inhibitor, Discrete and Continuous Dynamical Systems–B, in press. [3] J. P. Braselton and P. Waltman, A competition model with dynamically allocated inhibitor production, Mathematical Biosciences, 173 (2001), 55-84.  doi: 10.1016/S0025-5564(01)00078-5. [4] G. J. Butler and G. S. K. Wolkowicz, A mathematical model of the chemostat with a general class of functions describing nutrient uptake, SIAM Journal on Applied Mathematics, 45 (1985), 138-151.  doi: 10.1137/0145006. [5] M. J. De Freitas and A. G. Fredrickson, Inhibition as a factor in the maintenance of the diversity of microbial ecosystems, Journal of General Microbiology, 106 (1978), 307-320.  doi: 10.1099/00221287-106-2-307. [6] M. Dellal, M. Lakrib and T. Sari, The operating diagram of a model of two competitors in a chemostat with an external inhibitor, Mathematical Biosciences, 302 (2018), 27-45.  doi: 10.1016/j.mbs.2018.05.004. [7] P. De Leenheer, B. Li and H. L. Smith, Competition in the chemostat: Some remarks, Canadian Applied Mathematics Quarterly, 11 (2003), 229-248. [8] H. Fgaier, M. Kalmokoff, T. Ells and H. J. Eberl, An allelopathy based model for the Listeria overgrowth phenomenon, Mathematical Biosciences, 247 (2014), 13-26.  doi: 10.1016/j.mbs.2013.10.008. [9] S. R. Hansen and S. P. Hubbell, Single-nutrient microbial competition: Qualitative agreement between experimental and theoretically forecast outcomes, Science, 207 (1980), 1491-1493.  doi: 10.1126/science.6767274. [10] G. Hardin, The competitive exclusion principle, Science, 131 (1960), 1292-1297.  doi: 10.1126/science.131.3409.1292. [11] J. Harmand, C. Lobry, A. Rapaport and T. Sari, The Chemostat: Mathematical Theory of Microorganism Cultures, Wiley-ISTE, 2017. [12] P. A. Hoskisson and G. Hobbs, Continuous culture - making a comeback?, Microbiology, 151 (2005), 3153-3159.  doi: 10.1099/mic.0.27924-0. [13] S. B. Hsu, Limiting behaviour for competing species, SIAM Journal on Applied Mathematics, 34 (1978), 760-763.  doi: 10.1137/0134064. [14] S. B. Hsu, S. P. Hubbell and P. Waltman, A mathematical model for single nutrient competition in continuous cultures of micro-organisms, SIAM Journal on Applied Mathematics, 32 (1977), 366-383.  doi: 10.1137/0132030. [15] S. B. Hsu, Y. S. Li and P. Waltman, Competition in the Presence of a Lethal External Inhibitor, Mathematical Biosciences, 167 (2000), 177-199.  doi: 10.1016/S0025-5564(00)00030-4. [16] S. B. Hsu and P. Waltman, Analysis of a model of two competitors in a chemostat with an external inhibitor, SIAM Journal on Applied Mathematics, 52 (1992), 528-540.  doi: 10.1137/0152029. [17] S. B. Hsu and P. Waltman, Competition in the chemostat when one competitor produces a toxin, Japan Journal of Industrial and Applied Mathematics, 15 (1998), 471-490.  doi: 10.1007/BF03167323. [18] S. B. Hsu and P. Waltman, A survey of mathematical models of competition with an inhibitor, Mathematical Biosciences, 187 (2004), 53-91.  doi: 10.1016/j.mbs.2003.07.004. [19] R. E. Lenski and S. Hattingh, Coexistence of two competitors on one resource and one inhibitor: A chemostat model based on bacteria and antibiotics, Journal of Theoretical Biology, 122 (1986), 83-93.  doi: 10.1016/S0022-5193(86)80226-0. [20] B. Li, Global asymptotic behavior of the chemostat: General response functions and different removal rates, SIAM Journal on Applied Mathematics, 59 (1998), 411-422.  doi: 10.1137/S003613999631100X. [21] J. Monod, La technique de culture continue. Théorie et applications, Selected Papers in Molecular Biology by Jacques Monod, (1978), 184–204. doi: 10.1016/B978-0-12-460482-7.50023-3. [22] J. Monod, Recherches Sur la Croissance Des Cultures Bacteriennes, Hermann, Paris, 1958. [23] S. Pavlou, Computing operating diagrams of bioreactors, Journal of Biotechnology, 71 (1999), 7-16.  doi: 10.1016/S0168-1656(99)00011-5. [24] T. Sari, Competitive exclusion for chemostat equations with variable yields, Acta Applicandae Mathematicae, 123 (2013), 201-219.  doi: 10.1007/s10440-012-9761-8. [25] T. Sari and F. Mazenc, Global dynamics of the chemostat with different removal rates and variable yields, Mathematical Biosciences and Engineering, 8 (2011), 827-840.  doi: 10.3934/mbe.2011.8.827. [26] H. L. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition, Cambridge University Press, 1995.  doi: 10.1017/CBO9780511530043. [27] H. R. Thieme, Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations, Journal of Mathematical Biology, 30 (1992), 755-763.  doi: 10.1007/BF00173267. [28] M. J. Wade, J. Harmand, B. Benyahia, T. Bouchez, S. Chaillou, B. Cloez, J. Godon, B. Moussa Boudjemaa, A. Rapaport, T. Sari, R. Arditi and C. Lobry, Perspectives in mathematical modelling for microbial ecology, Ecological Modelling, 321 (2016), 64-74.  doi: 10.1016/j.ecolmodel.2015.11.002. [29] M. Weedermann, Analysis of a model for the effects of an external toxin on anaerobic digestion, Mathematical Biosciences and Engineering, 9 (2012), 445-459.  doi: 10.3934/mbe.2012.9.445. [30] G. S. K. Wolkowicz and Z. Lu, Global dynamics of a mathematical model of competition in the chemostat: general response functions and differential death rates, SIAM Journal on Applied Mathematics, 52 (1992), 222-233.  doi: 10.1137/0152012.

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##### References:
 [1] N. Abdellatif, R. Fekih-Salem and T. Sari, Competition for a single resource and coexistence of several species in the chemostat, Mathematical Biosciences and Engineering, 13 (2016), 631-652.  doi: 10.3934/mbe.2016012. [2] B. Bar and T. Sari, The operating diagram for a model of competition in a chemostat with an external lethal inhibitor, Discrete and Continuous Dynamical Systems–B, in press. [3] J. P. Braselton and P. Waltman, A competition model with dynamically allocated inhibitor production, Mathematical Biosciences, 173 (2001), 55-84.  doi: 10.1016/S0025-5564(01)00078-5. [4] G. J. Butler and G. S. K. Wolkowicz, A mathematical model of the chemostat with a general class of functions describing nutrient uptake, SIAM Journal on Applied Mathematics, 45 (1985), 138-151.  doi: 10.1137/0145006. [5] M. J. De Freitas and A. G. Fredrickson, Inhibition as a factor in the maintenance of the diversity of microbial ecosystems, Journal of General Microbiology, 106 (1978), 307-320.  doi: 10.1099/00221287-106-2-307. [6] M. Dellal, M. Lakrib and T. Sari, The operating diagram of a model of two competitors in a chemostat with an external inhibitor, Mathematical Biosciences, 302 (2018), 27-45.  doi: 10.1016/j.mbs.2018.05.004. [7] P. De Leenheer, B. Li and H. L. Smith, Competition in the chemostat: Some remarks, Canadian Applied Mathematics Quarterly, 11 (2003), 229-248. [8] H. Fgaier, M. Kalmokoff, T. Ells and H. J. Eberl, An allelopathy based model for the Listeria overgrowth phenomenon, Mathematical Biosciences, 247 (2014), 13-26.  doi: 10.1016/j.mbs.2013.10.008. [9] S. R. Hansen and S. P. Hubbell, Single-nutrient microbial competition: Qualitative agreement between experimental and theoretically forecast outcomes, Science, 207 (1980), 1491-1493.  doi: 10.1126/science.6767274. [10] G. Hardin, The competitive exclusion principle, Science, 131 (1960), 1292-1297.  doi: 10.1126/science.131.3409.1292. [11] J. Harmand, C. Lobry, A. Rapaport and T. Sari, The Chemostat: Mathematical Theory of Microorganism Cultures, Wiley-ISTE, 2017. [12] P. A. Hoskisson and G. Hobbs, Continuous culture - making a comeback?, Microbiology, 151 (2005), 3153-3159.  doi: 10.1099/mic.0.27924-0. [13] S. B. Hsu, Limiting behaviour for competing species, SIAM Journal on Applied Mathematics, 34 (1978), 760-763.  doi: 10.1137/0134064. [14] S. B. Hsu, S. P. Hubbell and P. Waltman, A mathematical model for single nutrient competition in continuous cultures of micro-organisms, SIAM Journal on Applied Mathematics, 32 (1977), 366-383.  doi: 10.1137/0132030. [15] S. B. Hsu, Y. S. Li and P. Waltman, Competition in the Presence of a Lethal External Inhibitor, Mathematical Biosciences, 167 (2000), 177-199.  doi: 10.1016/S0025-5564(00)00030-4. [16] S. B. Hsu and P. Waltman, Analysis of a model of two competitors in a chemostat with an external inhibitor, SIAM Journal on Applied Mathematics, 52 (1992), 528-540.  doi: 10.1137/0152029. [17] S. B. Hsu and P. Waltman, Competition in the chemostat when one competitor produces a toxin, Japan Journal of Industrial and Applied Mathematics, 15 (1998), 471-490.  doi: 10.1007/BF03167323. [18] S. B. Hsu and P. Waltman, A survey of mathematical models of competition with an inhibitor, Mathematical Biosciences, 187 (2004), 53-91.  doi: 10.1016/j.mbs.2003.07.004. [19] R. E. Lenski and S. Hattingh, Coexistence of two competitors on one resource and one inhibitor: A chemostat model based on bacteria and antibiotics, Journal of Theoretical Biology, 122 (1986), 83-93.  doi: 10.1016/S0022-5193(86)80226-0. [20] B. Li, Global asymptotic behavior of the chemostat: General response functions and different removal rates, SIAM Journal on Applied Mathematics, 59 (1998), 411-422.  doi: 10.1137/S003613999631100X. [21] J. Monod, La technique de culture continue. Théorie et applications, Selected Papers in Molecular Biology by Jacques Monod, (1978), 184–204. doi: 10.1016/B978-0-12-460482-7.50023-3. [22] J. Monod, Recherches Sur la Croissance Des Cultures Bacteriennes, Hermann, Paris, 1958. [23] S. Pavlou, Computing operating diagrams of bioreactors, Journal of Biotechnology, 71 (1999), 7-16.  doi: 10.1016/S0168-1656(99)00011-5. [24] T. Sari, Competitive exclusion for chemostat equations with variable yields, Acta Applicandae Mathematicae, 123 (2013), 201-219.  doi: 10.1007/s10440-012-9761-8. [25] T. Sari and F. Mazenc, Global dynamics of the chemostat with different removal rates and variable yields, Mathematical Biosciences and Engineering, 8 (2011), 827-840.  doi: 10.3934/mbe.2011.8.827. [26] H. L. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition, Cambridge University Press, 1995.  doi: 10.1017/CBO9780511530043. [27] H. R. Thieme, Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations, Journal of Mathematical Biology, 30 (1992), 755-763.  doi: 10.1007/BF00173267. [28] M. J. Wade, J. Harmand, B. Benyahia, T. Bouchez, S. Chaillou, B. Cloez, J. Godon, B. Moussa Boudjemaa, A. Rapaport, T. Sari, R. Arditi and C. Lobry, Perspectives in mathematical modelling for microbial ecology, Ecological Modelling, 321 (2016), 64-74.  doi: 10.1016/j.ecolmodel.2015.11.002. [29] M. Weedermann, Analysis of a model for the effects of an external toxin on anaerobic digestion, Mathematical Biosciences and Engineering, 9 (2012), 445-459.  doi: 10.3934/mbe.2012.9.445. [30] G. S. K. Wolkowicz and Z. Lu, Global dynamics of a mathematical model of competition in the chemostat: general response functions and differential death rates, SIAM Journal on Applied Mathematics, 52 (1992), 222-233.  doi: 10.1137/0152012.
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
The existence of multiple positive equilibria when the condition (20) $\rm(a)$: is not satisfied; $\rm(b)$: is satisfied
Case $n = 3$: Local stability of $E_1$, $E_2$, $E_c^2$ and instability of $E_c^1$, $E_c^2$, $E_0$
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
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
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 : $S_c^1<S_i<S_c^2$
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 $(S^0,D) = (0.02,1.15)\in\mathcal{J}_6^a$
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 $(\alpha_1\gamma_1-\alpha_2\gamma_2)[F'_2(S_c)-F'_1(S_c)]>0$
 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 $(\alpha_1\gamma_1-\alpha_2\gamma_2)[F'_2(S_c)-F'_1(S_c)]>0$
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)
 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)
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 $(S^0,D)\in\mathcal J_3$ $S_c^1  Region The relative positions of$ 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
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 $(S^0,D)\in\mathcal J_3$ $S_i  Region The relative positions of$ 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
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)
 $(S^0,D)$ 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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