A competition model in the chemostat with allelopathy and substrate inhibition

Mohamed Dellal; Bachir Bar; Mustapha Lakrib

doi:10.3934/dcdsb.2021120

Article Contents

2022, Volume 27, Issue 4: 2025-2050. Doi: 10.3934/dcdsb.2021120

This issue Previous Article Bloch wave approach to almost periodic homogenization and approximations of effective coefficients Next Article On the family of cubic parabolic polynomials

A competition model in the chemostat with allelopathy and substrate inhibition

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

^* Corresponding author: Mohamed Dellal
^* Corresponding author: Mohamed Dellal

Received: July 31, 2020

Revised: January 31, 2021

Early access: April 2021

Published: April 2022

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Abstract

A model of two microbial species in a chemostat competing for a single resource is considered, where one of the competitors that produces a toxin, which is lethal to the other competitor (allelopathic inhibition), is itself inhibited by the substrate. Using general growth rate functions of the species, necessary and sufficient conditions of existence and local stability of all equilibria of the four-dimensional system are determined according to the operating parameters represented by the dilution rate and the input concentration of the substrate. With Michaelis-Menten or Monod growth functions, it is well known that the model can have a unique positive equilibrium which is unstable as long as it exists. If a non monotonic growth rate is considered (which is the case when there is substrate inhibition), it is shown that a new positive equilibrium point exists which can be stable according to the operating parameters of the system. We describe its operating diagram, which is the bifurcation diagram giving the behavior of the system with respect to the operating parameters. By means of this bifurcation diagram, we show that the general model presents a set of fifteen possible behaviors: washout, competitive exclusion of one species, coexistence, multi-stability, occurrence of stable limit cycles through a super-critical Hopf bifurcations, homoclinic bifurcations and flip bifurcation. 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 1. Growth function and definitions of break-even concentrations (a): $ f_1 $ of Monod type; (b): $ f_2 $ of Haldane type

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Figure 2. Graphs of $ f_1 $ (in red) and $ (1\!-\!k)f_2 $ (in blue) when equation $ f_1(S)\! = \!(1\!-\!k)f_2(S) $ has a positive solution $ S\! = \!\overline{S} $ and graphical depiction of $ I_{c_1} $ and $ I_{c_2} $. $ \rm(a) $: $ I_{c_1} = \left(\overline{D},(1\!-\!k)f_2(S_2^m)\right] $ and $ I_{c_2} = \left(0,(1\!-\!k)f_2(S_2^m)\right] $. $ \rm(b) $: $ I_{c_1} = \emptyset $ and $ I_{c_2} = \left(0,\overline{D}\right) $ where $ \overline{D} = f_1\left(\overline{S}\right) = f_2\left(\overline{S}\right) $. Intervals $ I_{c_1} $ and $ I_{c_2} $ are defined by (14)

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Figure 3. Graphs of $ F_1 $ (in green) and $ F_2 $ (in magenta) when equation $ f_1(S)\! = \!(1\!-\!k)f_2(S) $ has a positive solution $ S\! = \!\overline{S} $

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Figure 4. The graphs of $ F_3(x_{c_2}) $ and $ A(x_{c_2}) $, showing the relative positions of the roots $ x_i = x_i(D) $, $ i = 0,2 $, of $ F_3(x_{c_2}) $ with respect to the root $ x_0 = x_0(D) $ of $ A(x_{c_2}) $, when $ D\in I_3 $

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Figure 5. Illustrative operating diagrams corresponding to cases (a) and (b) in Figure 2. The curves $ \Gamma_i $, $ i = 0\cdots 9 $, defined in Table 3, separate the operating plane $ (D,S^0) $ into fifteen regions labeled $ \mathcal J_k $, $ k = 0...14 $. The existence and stability of equilibria $ E_0 $, $ E_1 $, $ E_2^j $ and $ E_c^j $ in the regions $ \mathcal J_0 $, $ \mathcal J_1 $, ...., $ \mathcal J_{14} $ of these diagrams are shown by Table 5

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Figure 6. Operating diagram with biological parmeters given in Table 6, Case 1

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Figure 8. (a): $ (S^0,D) = (15,0.53)\in \mathcal J_{12} $. In this case we have bi-stability of $ E_c^2 $ and $ E_2^1 $. (b): $ (S^0,D) = (15,0.51) \in\mathcal J_{11} $. In this case $ E_c^2 $ loses its stability through a super-critical Hopf bifurcation (see Figure 7) creating a stable limit cycle. We use the color codes; Green: initial conditions, Red: local attractors and Blue: unstable equilibria

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Figure 7. Hopf bifurcation. Biological values are in Table 6, Case 1, and $ S^0 = 15 $. (a): Variation of a pair of complex-conjugate eigenvalues. (b): The real part of the eigenvalues showing that its change of stability at $ D = D_{crit}\approx 0.521403 $ indicating a Hopf bifurcation

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Figure 9. Homoclinic bifurcation (a): $ (S^0,D) = (15,0.508105) $. After the Hopf bifurcation, the limit cycle gets larger. (b): $ (S^0,D) = (15,0.50) $. The limit cycle loses its stability (through homoclinic bifurcation) and the only attractor remaining is $ E_2^1 $. We use the color codes, Green: initial conditions, Red: local attractors and Blue: unstable equilibria

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Figure 10. One parameter bifurcation diagram for the homoclinic bifurcation. We plot the projections of the $ \omega $-limit set in variables $ \{S,y\} $ for $ D\in[0.5,0.53] $, which reveals the emergence of limit cycle through a Hopf bifurcation and its disappearance through a homoclinic bifurcation. Solid line is for stable fixed point (dashed when unstable). H: Hopf bifurcation

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Figure 11. (a): Operating diagram corresponding to Table 6, Case 2. (b): A zoom of the operating diagram near regions $ \mathcal J_{13} $ and $ \mathcal J_{14} $

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Figure 12. Tri-stability (a): $ (D,S^0) = (0.26,3.5)\in\mathcal J_{14} $ (see Figure 11(b)). In this case there is tri-stability of equilibria $ E_c^2 $, $ E_2^1 $ and $ E_1 $. (b): $ (D,S^0) = (0.25785,3.5)\in\mathcal J_{13} $ (see Figure 11(b)). Tri-stability of equilibria $ E_2^1 $, $ E_1 $ and a stable limit cycle

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Figure 13. One parameter bifurcation diagram. Biological values are in Table. 6 case 2, and $ S^0 = 5 $. (a): Saddle node bifurcation of $ E_2^1 $ and $ E_2^2 $. (b): Saddle node bifurcation of $ E_c^1 $ and $ E_c^2 $. Solid line is for stable fixed point; dashed when unstable. H: Hopf bifurcation. LP: Limit Point (Saddle-node). PD: Period Doubling

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Figure 14. Bifurcation diagrams of the limit cycle. (a) Continuation of the limit cycle (we fix $ S^0 = 5 $ and plot the projection of the limit cycle on the $ (S,x) $ space as a function of $ D $). (b) Two parameter bifurcation of the limit cycle, the curves (in blue) correspond to the period doubling (flip bifurcation). Matcont was used to produce both of the diagrams. PD: Period Doubling, LPC: Limit Point Cycle

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Figure 15. Period doubling (Flip-bifurcation) before and after $ D_2 $. (a): The limit cycle for $ (D,S^0) = (0.24845,5) $. (b): The limit cycle for $ (D,S^0) = (0.24835,5) $

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Table 1. Existence and stability of equilibria of system (2) when $ \lambda_1 <S^0 $ and $ \lambda_2 <S^0 $. The letter S (resp. U) means stable (resp. unstable) and no letter means that the equilibrium does not exist

Case	Condition	Equilibria and nature
		$ E_1 $	$ E_2^1 $	$ E_2^2 $	$ E_c^1 $	$ E_c^2 $
$ \mu_2>S^0 $	$ \lambda_1<\widehat{\lambda}<\lambda_2<\mu_2 $	S	U
	$ \lambda_1 <\lambda_2 <\widehat{\lambda} <\mu_2 $	S	S		U
	$ \lambda_2 <\lambda_1 <\widehat{\lambda} <\mu_2 $	U	S
$ \mu_2<S^0 $	$ \lambda_1 <\widehat{\lambda} <\lambda_2 <\mu_2 $	S	U	U
	$ \lambda_1 <\lambda_2 <\widehat{\lambda} <\mu_2 $	S	S	U	U
	$ \lambda_2<\lambda_1<\widehat{\lambda}<\mu_2 $	U	S	U
	$ \lambda_1<\lambda_2<\mu_2<\widehat{\lambda} $ & $ AB >C \ \hbox{and} \ A >0 $	S	S	U	U	S
	$ \lambda_1<\lambda_2<\mu_2<\widehat{\lambda} $ & $ AB<C \ \hbox{or} \ A<0 $	S	S	U	U	U
	$ \lambda_2<\lambda_1<\mu_2<\widehat{\lambda} $ & $ AB >C \ \hbox{and} \ A >0 $	U	S	U		S
	$ \lambda_2<\lambda_1<\mu_2<\widehat{\lambda} $ & $ AB< C \ \hbox{or} \ A<0 $	U	S	U		U
	$ \lambda_2<\mu_2<\lambda_1<\widehat{\lambda} $	S	S	U

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Table 2. Existence and stability of equilibria of system (2) with respect to the operating parameters

Equilibria	Existence	Local exponential stability
$ E_0 $	Always	$D >\!\max(f_1(S^0),(1-k)f_2(S^0)) $
$ E_1 $	$ S^0>\lambda_1(D) $	$ \lambda_1(D)<\lambda_2(D) $ or $ \lambda_1(D)>\mu_2(D) $
$ E_2^1 $	$ S^0>\lambda_2(D) $	$ S^0>F_1(D) $
$ E_2^2 $	$ S^0>\mu_2(D) $	Unstable if it exists
$ E_c^1 $	$ \lambda_1(D)<\lambda_2(D)<S^0 $ & $ S^0>F_1(D) $	Unstable if it exists
$ E_c^2 $	$ \lambda_1(D)<\mu_2(D)<S^0 $ & $ S^0>F_2(D) $	$ F_3(D,S^0)>0 \ \hbox{and} \ A(D,S^0)>0 $

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Table 3. Boundaries of the regions in the operating diagram

The curve $ \Gamma_i $, $ i=1...9 $	Boundary
$ \Gamma_1=\left\{(D,S^0):S^0=\lambda_1(D)\right\} $	is the border to which $ E_1 $ exists
$ \Gamma_2=\left\{(D,S^0):S^0=\lambda_2(D)\right\} $	is the border to which $ E_2^1 $ exists
$ \Gamma_3=\left\{(D,S^0):S^0=\mu_2(D)\right\} $	is the border to which $ E_2^2 $ exists
$ \Gamma_4=\left\{(D,S^0)\!:\lambda_1(D)\!=\!\lambda_2(D), S^0\!>\!\lambda_1(\!D)\right\} $	is the border to which $ E_1 $ is stable
	and at the same time $ E_c^1 $ exists
$ \Gamma_5=\left\{(D,S^0)\!:\lambda_1(D)\!=\!\mu_2(D), S^0\!>\!\lambda_1(\!D)\right\} $	is the border to which $ E_1 $ is stable
	and at the same time $ E_c^2 $ exists
$ \Gamma_6=\left\{(D,S^0):S^0=F_1(D), S^0>\lambda_2(D)\right\} $	is the border to which $ E_2^1 $ is stable
	and at the same time $ E_c^1 $ exists
$ \Gamma_7=\left\{(D,S^0):S^0=F_2(D), S^0>\mu_2(D)\right\} $	is the border to which $ E_c^2 $ exists
$ \Gamma_8=\left\{(D,S^0):S^0=F_5(D) \right\} $	is the border to which $ E_c^2 $ is stable
$ \Gamma_9=\left\{(D,S^0)\!: \lambda_2(D)\!=\!\mu_2(D), S^0\!>\!\lambda_2(\!D) \right\} $	Horizontal line $ D=(\!1\!-\!k\!)f_2(S_2^m) $

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Table 4. Definitions of the regions $ \mathcal J_k $, $ k = 0...14 $, in the operating diagrams in Figures 5, Figure 6 and 11

Region	Definition
$ \mathcal J_0 $	$ S^0<\lambda_1(D) $ and $ S^0<\lambda_2(D) $
$ \mathcal J_1 $	$ S^0<\lambda_1(D) $ and $ \lambda_2(D)<S^0<\mu_2(D) $
$ \mathcal J_2 $	$ S^0<\lambda_1(D) $ and $ S^0>\mu_2(D) $
$ \mathcal J_3 $	$ S^0>\lambda_1(D) $ and $ S^0<\lambda_2(D) $
$ \mathcal J_4 $	$ S^0>\lambda_1(D) $, $ \lambda_2(D)<S^0<\mu_2(D) $ and $ S^0<F_1(D) $
$ \mathcal J_5 $	$ S^0>\lambda_1(D) $, $ \lambda_2(D)<S^0<\mu_2(D) $, $ S^0>F_1(D) $ and $ \lambda_1(D)<\lambda_2(D) $
$ \mathcal J_6 $	$ S^0>\lambda_1(D) $, $ \lambda_2(D)<S^0<\mu_2(D) $, $ S^0>F_1(D) $ and $ \lambda_2(D)<\lambda_1(D) $
$ \mathcal J_7 $	$ S^0>\lambda_1(D) $, $ S^0>\mu_2(D) $ and $ S^0<F_1(D) $
$ \mathcal J_8 $	$ S^0>\lambda_1(D) $, $ S^0>\mu_2(D) $, $ F_1(D)<S^0<F_2(D) $ and $ \lambda_1(D)<\lambda_2(D) $
$ \mathcal J_9 $	$ S^0>\lambda_1(D) $, $ S^0>\mu_2(D) $, $ F_1(D)<S^0<F_2(D) $ and $ \lambda_2(D)<\lambda_1(D) $
$ \mathcal J_{10} $	$ S^0>\lambda_1(D) $, $ S^0>\mu_2(D) $ and $ \lambda_1(D)>\mu_2(D) $
$ \mathcal J_{11} $	$ S^0>\lambda_1(D) $, $ S^0>\mu_2(D) $, $ S^0\!>\!F_2(D) $, $ S^0\!<\!F_5(D) $ and $ \lambda_2(D)<\lambda_1(D) $
$ \mathcal J_{12} $	$ S^0>\lambda_1(D) $, $ S^0>\mu_2(D) $, $ S^0\!>\!F_2(D) $, $ S^0\!>\!F_5(D) $ and $ \lambda_2(D)<\lambda_1(D) $
$ \mathcal{J}_{13} $	$ S^0>\lambda_1(D) $, $ S^0>\mu_2(D) $, $ S^0\!>\!F_2(D) $, $ S^0\!<\!F_5(D) $ and $ \lambda_1(D)<\lambda_2(D) $
$ \mathcal J_{14} $	$ S^0>\lambda_1(D) $, $ S^0>\mu_2(D) $, $ S^0\!>\!F_2(D) $, $ S^0\!>\!F_5(D) $ and $ \lambda_1(D)<\lambda_2(D) $

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Table 5. Existence and stability of equilibria in the regions of the operating diagrams in Figsures 5, 6 and 11

Region	$ \mathcal{J}_0 $	$ \mathcal{J}_1 $	$ \mathcal{J}_2 $	$ \mathcal{J}_3 $	$ \mathcal{J}_4 $	$ \mathcal{J}_5 $	$ \mathcal{J}_6 $	$ \mathcal{J}_7 $	$ \mathcal{J}_8 $	$ \mathcal{J}_9 $	$ \mathcal{J}_{10} $	$ \mathcal{J}_{11} $	$ \mathcal{J}_{12} $	$ \mathcal J_{13} $	$ \mathcal{J}_{14} $
$ E_0 $	S	U	S	U	U	U	U	U	U	U	U	U	U	U	U
$ E_1 $				S	S	S	U	S	S	U	S	U	U	S	S
$ E_2^1 $		S	S		U	S	S	U	S	S	S	S	S	S	S
$ E_2^2 $			U					U	U	U	U	U	U	U	U
$ E_c^1 $						U			U					U	U
$ E_c^2 $												U	S	U	S

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Table 6. $[![]!]

Case	$ m_1 $	$ m_2 $	$ K_1 $	$ K_2 $	$ K_3 $	$ k $	$ \gamma $	Figs
1	1.0	4.0	1.0	1.0	0.5	0.2	0.3	6, 7, 8, 9, 10
2	1.5	2.7	1.0	1.0	0.08	0.2	0.3	11, 12, 14, 15.

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