Global dynamics in a tumor-immune model with an immune checkpoint inhibitor
Mohamed Dellal a,c,, and Bachir Bar b,c,
TAG_SUP Université Ibn Khaldoun, Tiaret 14000, Algérie
TAG_SUP Ecole Normale Supérieure, Mostaganem 27000, Algérie
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)
Figure(7) / Table(5)
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.
Keywords: Chemostat, competition, inhibitor, stability, operating diagram.
Mathematics Subject Classification: Primary: 34C23, 34D20; Secondary: 92B05, 92D25.
Citation: Mohamed Dellal, Bachir Bar. Global analysis of a model of competition in the chemostat with internal inhibitor. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1129-1148. doi: 10.3934/dcdsb.2020156
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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 $
Figs. 5(a, b) are exactly the same except that the values of $ \alpha_i $ have been inverted">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
Figs. 6(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 $
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 $">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_1<S^0 $ $ F_1(S_1)>F_2(S_1) $
$ E_{c} $ (23) has a solution $ (\alpha_1\gamma_1-\alpha_2\gamma_2)[F'_2(S_c)-F'_1(S_c)]>0 $
Table Options
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<S_c^2<S_i, i=1,2 $ $ ^{**} $ U U S
$ (S^0,D)\in\mathcal J_4 $ $ S_c^1<S_i<S_c^2, i=1,2 $ U S U
$ (S^0,D)\in\mathcal J_6 $ $ S_c^1<S_2<S_c^2<S_1 $ U U U S
$ (S^0,D)\in\mathcal J_7 $ $ S_c^1<S_1<S_c^2<S_2 $ U S S U
**When Sc1 and Sc2 do not exist, the condition reduces to Si, i = 1, 2 exist.
Table 4. Existence and stability of equilibria in the regions of the operating diagrams of Fig. 6, when the curves $ \Gamma_i^c $ intersect
$ (S^0,D)\in\mathcal J_3 $ $ S_i<S_c^1<S_c^2, i=1,2 $ $ ^{**} $ U U S
$ (S^0,D)\in\mathcal J_6^a $ $ S_1<S_c^1<S_2<S_c^2 $ U U U S
$ (S^0,D)\in\mathcal J_7^a $ $ S_c^1<S_1<S_c^2<S_2 $ U S S U
$ (S^0,D)\in\mathcal J_8^a $ $ S_1<S_c^1<S_c^2<S_2 $ U U S S U
$ (S^0,D)\in\mathcal J_6^b $ $ S_c^1<S_2<S_c^2<S_1 $ U U U S
$ (S^0,D)\in\mathcal J_7^b $ $ S_2<S_c^1<S_1<S_c^2 $ U S S U
$ (S^0,D)\in\mathcal J_8^b $ $ S_2<S_c^1<S_c^2<S_1 $ U U S U S
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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Mohamed Dellal Bachir Bar