# American Institute of Mathematical Sciences

## Mathematical analysis of a three-tiered food-web in the chemostat

 a. University of Tunis El Manar, National Engineering School of Tunis, LAMSIN, 1002, Tunis, Tunisia b. ITAP, Univ Montpellier, INRAE, Institut Agro, Montpellier, France c. University of Monastir, Higher Institute of Computer Science of Mahdia, 5111, Mahdia, Tunisia d. University of Manouba, National School of Computer Science, 2010, Manouba, Tunisia

* Corresponding author: Nahla Abdellatif

Received  June 2020 Published  December 2020

Fund Project: This work was supported by the Euro-Mediterranean research network TREASURE (http://www.inra.fr/treasure).

A mechanistic model describing the anaerobic mineralization of chlorophenol in a three-step food-web is investigated. The model is a six-dimensional system of ordinary differential equations. In our study, the phenol and the hydrogen inflowing concentrations are taken into account as well as the maintenance terms. The case of a large class of growth kinetics is considered, instead of specific kinetics. We show that the system can have up to eight types of steady states and we analytically determine the necessary and sufficient conditions for their existence according to the operating parameters. In the particular case without maintenance, the local stability conditions of all steady states are determined. The bifurcation diagram shows the behavior of the process by varying the concentration of influent chlorophenol as the bifurcating parameter. It shows that the system exhibits a bi-stability where the positive steady state can lose stability undergoing a supercritical Hopf bifurcation with the emergence of a stable limit cycle.

Citation: Sarra Nouaoura, Radhouane Fekih-Salem, Nahla Abdellatif, Tewfik Sari. Mathematical analysis of a three-tiered food-web in the chemostat. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020369
##### References:

show all references

##### References:
Curve of the function $\Psi(.,D)$, where $s_2^{*1}$ and $s_2^{*2}$ are the solutions of equation $\Psi(s_2,D) = (1-\omega)s_0^{\rm in}+s_1^{\rm in}+s_2^{\rm in}$
(a) Magnification of saddle-node bifurcation at $S_{\rm ch}^{\rm in} = \sigma_2$ and the transcritical bifurcation at $S_{\rm ch}^{\rm in} = \sigma_5$ when $S_{\rm ch}^{\rm in}\in\left[0.006,0.02 \right]$. (b) Magnification of the appearance and disappearance of stable limit cycles when $S_{\rm ch}^{\rm in}\in[0.0294,0.0302]$
(a) Projection of the $\omega$-limit set in variable $X_{\rm ch}$ as a function of $S_{\rm ch}^{\rm in}\in[0,0.05]$ (b) A magnification of the transcritical bifurcations occurring at $\sigma_1$, $\sigma_3$ and $\sigma_4$ when $S_{\rm ch}^{\rm in}\in[0,0.015]$
Curve of the function $y = F\left(S_{\rm ch}^{\rm in}\right)$ showing that $F\left(S_{\rm ch}^{\rm in}\right)<0.0013$, for all $S_{\rm ch}^{\rm in}>\sigma_1$
Case $S_{\rm ch}^{\rm in} = 0.029639\in(\sigma^*,\sigma_6)$: bi-stability of the limit cycle (in red) and SS3
showing the sustained oscillations in yellow (a) and blue (b) or the convergence to SS3 in green (c)">Figure 5.  Case $S_{\rm ch}^{\rm in} = 0.029639\in(\sigma^*,\sigma_6)$: trajectories of $X_{\rm ch}$ corresponding to those in Fig. 4 showing the sustained oscillations in yellow (a) and blue (b) or the convergence to SS3 in green (c)
The green line of equation $y = YS_{\rm ch}^{\rm in}$ is above the red and blue curves of the functions $M_0\left(D,s_2^{\ast i}\right)+M_1\left(D,s_2^{\ast i}\right)$, $i = 1,2$
(a) Curve of the function $\phi_4\left(S_{\rm ch}^{\rm in}\right)$ for $S_{\rm ch}^{\rm in}>\sigma_5$ and the solution $\sigma_6$ of equation $\phi_4\left(S_{\rm ch}^{\rm in}\right) = 0$. (b) A magnification for $S_{\rm ch}^{\rm in}\in (\sigma_5,0.034)$
Three eigenvalues of the matrix $\mathbf{J_1}$ evaluated at SS6 as a function of $S_{\rm ch}^{\rm in}$. Real part of the pair of eigenvalues $\lambda_{2,3}$ for $S_{\rm ch}^{\rm in} \in (\sigma^\star,0.05]$ where $\sigma^\star = 0.018$.
Notations, intervals and auxiliary functions
 Definition $s_i = M_i(y,s_2)$ $i = 0,1$ Let $s_2\geqslant 0$. $s_i = M_i(y,s_2)$ is the unique solution of $\mu_i(s_i,s_2) = y$, for all $0\leqslant y<\mu_i(+\infty,s_2)$ $s_2 = M_2(y)$ $s_2 = M_2(y)$ is the unique solution of $\mu_2(s_2) = y$, for all $0\leqslant y<\mu_2(+\infty)$ $s_2 = M_3(s_0,z)$ Let $s_0\geqslant 0$. $s_2 = M_3(s_0,z)$ is the unique solution of $\mu_0(s_0,s_2) = z$, for all $0\leqslant z <\mu_0(s_0,+\infty)$ $s_2^i = s_2^i(D)$ $i = 0,1$ $s_2^i = s_2^i(D)$ is the unique solution of $\mu_i\left(+\infty,s_2\right) = D+a_i$, for all $D+a_0\!<\!\mu_0(+\infty,+\infty)$, $\mu_1(+\infty,+\infty)\!<\!D+a_1<\mu_1(+\infty,0)$, resp. $I_1$, $I_2$ $I_1 = \left\{D \geqslant 0: s_2^0  Definition$ s_i = M_i(y,s_2)  i = 0,1 $Let$ s_2\geqslant 0 $.$ s_i = M_i(y,s_2) $is the unique solution of$ \mu_i(s_i,s_2) = y $, for all$ 0\leqslant y<\mu_i(+\infty,s_2)  s_2 = M_2(y)  s_2 = M_2(y) $is the unique solution of$ \mu_2(s_2) = y $, for all$ 0\leqslant y<\mu_2(+\infty)  s_2 = M_3(s_0,z) $Let$ s_0\geqslant 0 $.$ s_2 = M_3(s_0,z) $is the unique solution of$ \mu_0(s_0,s_2) = z $, for all$ 0\leqslant z <\mu_0(s_0,+\infty)  s_2^i = s_2^i(D)  i = 0,1  s_2^i = s_2^i(D) $is the unique solution of$ \mu_i\left(+\infty,s_2\right) = D+a_i $, for all$ D+a_0\!<\!\mu_0(+\infty,+\infty) $,$ \mu_1(+\infty,+\infty)\!<\!D+a_1<\mu_1(+\infty,0) $, resp.$ I_1 $,$ I_2  I_1 = \left\{D \geqslant 0: s_2^0
Steady states of (2). All functions are defined in Table 1
 $s_0$, $s_1$, $s_2$ and $x_0$, $x_1$, $x_2$ components SS1 $s_0 = s_0^{{\rm in}}$, $s_1 = s_1^{\rm in}$, $s_2 = s_2^{\rm in}$ and $x_0 = 0$, $x_1 = 0$, $x_2 = 0$ SS2 $s_0 = s_0^{{\rm in}}$, $s_1 = s_1^{\rm in}$, $s_2 = M_2(D+a_2)$ and $x_0 = 0$, $x_1 = 0$, $x_2 = \frac{D}{D+a_2}\left(s_2^{\rm in}-s_2\right)$ SS3 $s_1 = s^{\rm in}_1+s^{\rm in}_0-s_0$ and $s_2 = s^{\rm in}_2-\omega\left(s^{\rm in}_0-s_0\right)$, where $s_0$ is a solution of $\psi_0(s_0) = D+a_0$ and $x_0 = \frac{D}{D+a_0}\left(s_0^{\rm in}-s_0\right)$, $x_1 = 0$, $x_2 = 0$ SS4 $s_0 = M_0(D+a_0,s_2)$ and $s_1 = M_1(D+a_1,s_2)$, where $s_2$ is a solution of $\Psi(s_2,D) = (1-\omega)s_0^{{\rm in}}+s_1^{{\rm in}}+s_2^{{\rm in}}$ and $x_0 = \frac{D}{D+a_0}\left(s_0^{{\rm in}}-s_0\right)$, $x_1 = \frac{D}{D+a_1}\left(s_0^{{\rm in}}-s_0+s_1^{{\rm in}}-s_1\right)$, $x_2 = 0$ SS5 $s_0 = \varphi_0(D)$, $s_1 = s_1^{\rm in} + s_0^{\rm in} - s_0$, $s_2 = M_2(D+a_2)$ and $x_0 = \frac{D}{D+a_0}\left(s_0^{\rm in}-s_0\right)$, $x_1 = 0$, $x_2 = \frac{D}{D+a_2}\left(s_2^{\rm in}-s_2-\omega \left(s_0^{\rm in}-s_0 \right)\right)$ SS6 $s_0 = \varphi_0(D)$, $s_1 = \varphi_1(D)$, $s_2 = M_2(D+a_2)$ and $x_0 = \frac{D}{D+a_0}\left(s_0^{\rm in}-s_0\right)$, $x_1 = \frac{D}{D+a_1}\left(s_0^{\rm in}-s_0+s_1^{\rm in}-s_1\right)$, $x_2 = \frac{D}{D+a_2}\left((1-\omega)(s_0^{{\rm in}}-s_0)+s_1^{{\rm in}}-s_1+s_2^{{\rm in}}-s_2\right)$ SS7 $s_0 = s_0^{{\rm in}}$ and $s_2 = s_2^{\rm in}+s_1^{\rm in}-s_1$, where $s_1$ is a solution of $\psi_1(s_1) = D+a_1$ and $x_0 = 0$, $x_1 = \frac{D}{D+a_1}\left(s_1^{\rm in}-s_1\right)$, $x_2 = 0$ SS8 $s_0 = s_0^{{\rm in}}$, $s_1 = \varphi_1(D)$, $s_2 = M_2(D+a_2)$ and $x_0 = 0$, $x_1 = \frac{D}{D+a_1}\left(s_1^{\rm in}-s_1\right)$, $x_2 = \frac{D}{D+a_2}\left(s_1^{\rm in}-s_1+s_2^{\rm in}-s_2\right)$
 $s_0$, $s_1$, $s_2$ and $x_0$, $x_1$, $x_2$ components SS1 $s_0 = s_0^{{\rm in}}$, $s_1 = s_1^{\rm in}$, $s_2 = s_2^{\rm in}$ and $x_0 = 0$, $x_1 = 0$, $x_2 = 0$ SS2 $s_0 = s_0^{{\rm in}}$, $s_1 = s_1^{\rm in}$, $s_2 = M_2(D+a_2)$ and $x_0 = 0$, $x_1 = 0$, $x_2 = \frac{D}{D+a_2}\left(s_2^{\rm in}-s_2\right)$ SS3 $s_1 = s^{\rm in}_1+s^{\rm in}_0-s_0$ and $s_2 = s^{\rm in}_2-\omega\left(s^{\rm in}_0-s_0\right)$, where $s_0$ is a solution of $\psi_0(s_0) = D+a_0$ and $x_0 = \frac{D}{D+a_0}\left(s_0^{\rm in}-s_0\right)$, $x_1 = 0$, $x_2 = 0$ SS4 $s_0 = M_0(D+a_0,s_2)$ and $s_1 = M_1(D+a_1,s_2)$, where $s_2$ is a solution of $\Psi(s_2,D) = (1-\omega)s_0^{{\rm in}}+s_1^{{\rm in}}+s_2^{{\rm in}}$ and $x_0 = \frac{D}{D+a_0}\left(s_0^{{\rm in}}-s_0\right)$, $x_1 = \frac{D}{D+a_1}\left(s_0^{{\rm in}}-s_0+s_1^{{\rm in}}-s_1\right)$, $x_2 = 0$ SS5 $s_0 = \varphi_0(D)$, $s_1 = s_1^{\rm in} + s_0^{\rm in} - s_0$, $s_2 = M_2(D+a_2)$ and $x_0 = \frac{D}{D+a_0}\left(s_0^{\rm in}-s_0\right)$, $x_1 = 0$, $x_2 = \frac{D}{D+a_2}\left(s_2^{\rm in}-s_2-\omega \left(s_0^{\rm in}-s_0 \right)\right)$ SS6 $s_0 = \varphi_0(D)$, $s_1 = \varphi_1(D)$, $s_2 = M_2(D+a_2)$ and $x_0 = \frac{D}{D+a_0}\left(s_0^{\rm in}-s_0\right)$, $x_1 = \frac{D}{D+a_1}\left(s_0^{\rm in}-s_0+s_1^{\rm in}-s_1\right)$, $x_2 = \frac{D}{D+a_2}\left((1-\omega)(s_0^{{\rm in}}-s_0)+s_1^{{\rm in}}-s_1+s_2^{{\rm in}}-s_2\right)$ SS7 $s_0 = s_0^{{\rm in}}$ and $s_2 = s_2^{\rm in}+s_1^{\rm in}-s_1$, where $s_1$ is a solution of $\psi_1(s_1) = D+a_1$ and $x_0 = 0$, $x_1 = \frac{D}{D+a_1}\left(s_1^{\rm in}-s_1\right)$, $x_2 = 0$ SS8 $s_0 = s_0^{{\rm in}}$, $s_1 = \varphi_1(D)$, $s_2 = M_2(D+a_2)$ and $x_0 = 0$, $x_1 = \frac{D}{D+a_1}\left(s_1^{\rm in}-s_1\right)$, $x_2 = \frac{D}{D+a_2}\left(s_1^{\rm in}-s_1+s_2^{\rm in}-s_2\right)$
Existence conditions of steady states of (2). All functions are given in Table 1
 Existence conditions SS1 Always exists SS2 $\mu_2\left(s_2^{\rm in}\right)>D+a_2$ SS3 $\mu_0\left(s^{\rm in}_0,s^{\rm in}_2\right)>D+a_0$ SS4 $(1-\omega)s_0^{\rm in}+s_1^{\rm in}+s_2^{\rm in}\geqslant \phi_1(D)$, $s_0^{\rm in} > M_0(D+a_0,s_2)$, $s_0^{\rm in} + s_1^{\rm in} >M_0(D+a_0,s_2) + M_1(D+a_1,s_2)$ with $s_2$ solution of equation $\Psi(s_2,D) = (1-\omega)s_0^{\rm in}+s_1^{\rm in}+s_2^{\rm in}$ SS5 $s_0^{\rm in}>\varphi_0(D)$, $s_2^{\rm in}-\omega s_0^{\rm in}>M_2(D+a_2)-\omega \varphi_0(D)$ SS6 $(1-\omega)s^{\rm in}_0+s_1^{\rm in}+s_2^{\rm in}>\phi_2(D)$, $s_0^{\rm in}>\varphi_0(D)$, $s_0^{\rm in}+s_1^{\rm in} >\varphi_0(D)+\varphi_1(D)$ SS7 $\mu_1\left(s^{\rm in}_1,s^{\rm in}_2\right)>D+a_1$ SS8 $s^{\rm in}_1>\varphi_1(D)$, $s_1^{\rm in}+s_2^{\rm in}>\varphi_1(D)+M_2(D+a_2)$
 Existence conditions SS1 Always exists SS2 $\mu_2\left(s_2^{\rm in}\right)>D+a_2$ SS3 $\mu_0\left(s^{\rm in}_0,s^{\rm in}_2\right)>D+a_0$ SS4 $(1-\omega)s_0^{\rm in}+s_1^{\rm in}+s_2^{\rm in}\geqslant \phi_1(D)$, $s_0^{\rm in} > M_0(D+a_0,s_2)$, $s_0^{\rm in} + s_1^{\rm in} >M_0(D+a_0,s_2) + M_1(D+a_1,s_2)$ with $s_2$ solution of equation $\Psi(s_2,D) = (1-\omega)s_0^{\rm in}+s_1^{\rm in}+s_2^{\rm in}$ SS5 $s_0^{\rm in}>\varphi_0(D)$, $s_2^{\rm in}-\omega s_0^{\rm in}>M_2(D+a_2)-\omega \varphi_0(D)$ SS6 $(1-\omega)s^{\rm in}_0+s_1^{\rm in}+s_2^{\rm in}>\phi_2(D)$, $s_0^{\rm in}>\varphi_0(D)$, $s_0^{\rm in}+s_1^{\rm in} >\varphi_0(D)+\varphi_1(D)$ SS7 $\mu_1\left(s^{\rm in}_1,s^{\rm in}_2\right)>D+a_1$ SS8 $s^{\rm in}_1>\varphi_1(D)$, $s_1^{\rm in}+s_2^{\rm in}>\varphi_1(D)+M_2(D+a_2)$
Maintenance free case: the stability conditions of steady states of (2). All functions are given in Table 1 with $a_0 = a_1 = a_2 = 0$, while $\phi_4$ is defined by (4)
 Stability conditions SS1 $\mu_0\left(s_0^{\rm in},s_2^{\rm in}\right)0$, $\phi_3(D)>0$, with $s_2$ solution of equation $\Psi(s_2,D) = (1-\omega)s_0^{\rm in}+s_1^{\rm in}+s_2^{\rm in}$ SS5 $s_0^{\rm in}+s_1^{\rm in} < \varphi_0(D) + \varphi_1(D)$ SS6 $\phi_3(D)\geqslant 0$, or $\phi_3(D)<0$ and $\phi_4\left(D,s^{\rm in}_0,s_1^{\rm in},s_2^{\rm in}\right)>0$ SS7 $s_1^{\rm in}+s_2^{\rm in}< M_3 \left( s_0^{\rm in},D\right) + M_1\left(D,M_3\left( s_0^{\rm in},D\right)\right)$, $s_1^{\rm in}+s_2^{\rm in}  Stability conditions SS1$ \mu_0\left(s_0^{\rm in},s_2^{\rm in}\right)0 $,$ \phi_3(D)>0 $, with$ s_2 $solution of equation$ \Psi(s_2,D) = (1-\omega)s_0^{\rm in}+s_1^{\rm in}+s_2^{\rm in} $SS5$ s_0^{\rm in}+s_1^{\rm in} < \varphi_0(D) + \varphi_1(D) $SS6$ \phi_3(D)\geqslant 0 $, or$ \phi_3(D)<0 $and$ \phi_4\left(D,s^{\rm in}_0,s_1^{\rm in},s_2^{\rm in}\right)>0 $SS7$ s_1^{\rm in}+s_2^{\rm in}< M_3 \left( s_0^{\rm in},D\right) + M_1\left(D,M_3\left( s_0^{\rm in},D\right)\right) $,$ s_1^{\rm in}+s_2^{\rm in}
Nominal parameter values. Units are expressed in Chemical Oxygen Demand (COD)
 Parameter Wade et al. [31] Unit $k_{m,\rm{ch}}$ $k_{m,\rm{ph}} $k_{m,\rm{H_{2}}} 292635 \rm{kgCOD_{S}/kgCOD_{X}/d} K_{S,\rm{ch}} K_{S,\rm{H_{2},c}} K_{S,\rm{ph}} K_{I,\rm{H_{2}}} K_{S,\rm{H_{2}}} 0.053 10^{-6} 0.302 3.5 \!\times\!10^{-6} 2.5 \!\times\!10^{-5} \rm{kgCOD/m^{3}} Y_{\rm{ch}} Y_{\rm{ph}} Y_{\rm{H_{2}}} 0.019 0.04 0.06 \rm{kgCOD_X/kgCOD_S}  Parameter Wade et al. [31] Unit k_{m,\rm{ch}} k_{m,\rm{ph}}$ k_{m,\rm{H_{2}}}$ 292635 $\rm{kgCOD_{S}/kgCOD_{X}/d}$ $K_{S,\rm{ch}}$ $K_{S,\rm{H_{2},c}}$ $K_{S,\rm{ph}}$ $K_{I,\rm{H_{2}}}$ $K_{S,\rm{H_{2}}}$ 0.053 $10^{-6}$ 0.302 3.5$\!\times\!10^{-6}$ 2.5$\!\times\!10^{-5}$ $\rm{kgCOD/m^{3}}$ $Y_{\rm{ch}}$ $Y_{\rm{ph}}$ $Y_{\rm{H_{2}}}$ 0.019 0.04 0.06 $\rm{kgCOD_X/kgCOD_S}$
Auxiliary functions in the case of growth functions given by (9)
 Auxiliary function Definition domain $M_0(y,s_2)=\frac{yK_0(L_0+s_2)}{m_0s_2-y(L_0+s_2)}$ $0\leqslant y<\frac{m_0s_2}{L_0+s_2}$ $M_1(y,s_2)=\frac{yK_1(K_I+s_2)}{m_1K_I-y(K_I+s_2)}$ $0\leqslant y<\frac{m_1K_I}{K_I+s_2}$ $M_2(y)=\frac{yK_2}{m_2-y}$ $0\leqslant y  Auxiliary function Definition domain$ M_0(y,s_2)=\frac{yK_0(L_0+s_2)}{m_0s_2-y(L_0+s_2)}  0\leqslant y<\frac{m_0s_2}{L_0+s_2}  M_1(y,s_2)=\frac{yK_1(K_I+s_2)}{m_1K_I-y(K_I+s_2)}  0\leqslant y<\frac{m_1K_I}{K_I+s_2}  M_2(y)=\frac{yK_2}{m_2-y}  0\leqslant y
Definitions of the critical values of $\sigma_i$, $i = 1,\ldots,6$
 Definition Value $\sigma_1=M_0\left(D,S_{\rm H_2}^{\rm in} \right)/Y$ $0.001017$ $\sigma_2=(\phi_1(D)-S_{\rm H_2}^{\rm in})/((1-\omega)Y)$ $0.009159$ $\sigma_3=\varphi_0(D)/Y$ $0.010846$ $\sigma_4=(S_{\rm H_2}^{\rm in}-M_2(D)+\omega\varphi_0(D))/(\omega Y)$ $0.011191$ $\sigma_5=(\phi_2(D)-S_{\rm H_2}^{\rm in})/((1-\omega)Y)$ $0.016575$ $\sigma_6$ is the solution of equation $\phi_4(S_{\rm ch}^{\rm in})=0$ $0.029877$
 Definition Value $\sigma_1=M_0\left(D,S_{\rm H_2}^{\rm in} \right)/Y$ $0.001017$ $\sigma_2=(\phi_1(D)-S_{\rm H_2}^{\rm in})/((1-\omega)Y)$ $0.009159$ $\sigma_3=\varphi_0(D)/Y$ $0.010846$ $\sigma_4=(S_{\rm H_2}^{\rm in}-M_2(D)+\omega\varphi_0(D))/(\omega Y)$ $0.011191$ $\sigma_5=(\phi_2(D)-S_{\rm H_2}^{\rm in})/((1-\omega)Y)$ $0.016575$ $\sigma_6$ is the solution of equation $\phi_4(S_{\rm ch}^{\rm in})=0$ $0.029877$
Existence and stability of steady states, with respect to $S_{\rm ch}^{\rm in}$. The letter S (resp. U) means that the corresponding steady state is LES (resp. unstable). No letter means that the steady state does not exist
 Interval of $S_{\rm ch}^{\rm in}$ SS1 SS2 SS3 $\rm SS4^1$ $\rm SS4^2$ SS5 SS6 $(0,\sigma_1)$ U S $(\sigma_1,\sigma_2)$ U S U $(\sigma_2,\sigma_3)$ U S U U U $(\sigma_3,\sigma_4)$ U U U U U S $(\sigma_4,\sigma_5)$ U U S U U $(\sigma_5,\sigma_6)$ U U S U U U $(\sigma_6,+\infty)$ U U S U U S
 Interval of $S_{\rm ch}^{\rm in}$ SS1 SS2 SS3 $\rm SS4^1$ $\rm SS4^2$ SS5 SS6 $(0,\sigma_1)$ U S $(\sigma_1,\sigma_2)$ U S U $(\sigma_2,\sigma_3)$ U S U U U $(\sigma_3,\sigma_4)$ U U U U U S $(\sigma_4,\sigma_5)$ U U S U U $(\sigma_5,\sigma_6)$ U U S U U U $(\sigma_6,+\infty)$ U U S U U S
Nature of the bifurcations corresponding to the critical values of $\sigma_i$, $i = 1,\ldots,6$, defined in Table 5. There exists also a critical value $\sigma^*\simeq 0.029638$ corresponding to the value of $S_{\rm ch}^{\rm in}$ where the stable limit cycle disappears when $S_{\rm ch}^{\rm in}$ is decreasing
 Type of the bifurcation $\sigma_1$ Transcritical bifurcation of SS1 and SS3 $\sigma_2$ Saddle-node bifurcation of $\rm SS4^1$ and $\rm SS4^2$ $\sigma_3$ Transcritical bifurcation of SS2 and SS5 $\sigma_4$ Transcritical bifurcation of SS3 and SS5 $\sigma_5$ Transcritical bifurcation of $\rm SS4^1$ and SS6 $\sigma^*$ Disappearance of the stable limit cycle $\sigma_6$ Supercritical Hopf bifurcation
 Type of the bifurcation $\sigma_1$ Transcritical bifurcation of SS1 and SS3 $\sigma_2$ Saddle-node bifurcation of $\rm SS4^1$ and $\rm SS4^2$ $\sigma_3$ Transcritical bifurcation of SS2 and SS5 $\sigma_4$ Transcritical bifurcation of SS3 and SS5 $\sigma_5$ Transcritical bifurcation of $\rm SS4^1$ and SS6 $\sigma^*$ Disappearance of the stable limit cycle $\sigma_6$ Supercritical Hopf bifurcation
Colors used in Figs. 2 and 3. The solid (resp. dashed) lines are used for LES (resp. unstable) steady states
 SS1 SS2 SS3 $\rm SS4^1$ $\rm SS4^2$ SS5 SS6 Red Blue Purple Dark Green Magenta Green Cyan
 SS1 SS2 SS3 $\rm SS4^1$ $\rm SS4^2$ SS5 SS6 Red Blue Purple Dark Green Magenta Green Cyan
Existence and local stability conditions of steady states of (1), when $S_{\rm ph}^{\rm in} = 0$ and $k_{\rm dec, ch} = k_{\rm dec, ph} = k_{\rm dec, H_2} = 0$. All functions are given in Tables 1 and 11, while $\phi_4$ and $\mu_i$ are given by (4) and (9)
 Existence conditions Stability conditions SS1 Always exists $\mu_0\left(YS_{\rm ch}^{\rm in},S_{\rm H_2}^{\rm in}\right)D$ $YS_{\rm ch}^{\rm in} < \varphi_0(D)$ SS3 $\mu_0\left(YS_{\rm ch}^{\rm in},S_{\rm H_2}^{\rm in}\right)>D$ $\mu_1\left(YS_{\rm ch}^{\rm in}-s_0,S_{\rm H_2}^{\rm in}-\omega \left(YS_{\rm ch}^{\rm in}-s_0\right)\right) M_0(D,s_2)+ M_1(D,s_2)$ with $s_2$ solution of $\Psi(s_2,D) = (1-\omega)YS_{\rm ch}^{\rm in}+S_{\rm H_2}^{\rm in}$ $(1-\omega)YS_{\rm ch}^{\rm in}+S_{\rm H_2}^{\rm in}< \phi_2(D)$, $\frac{\partial\Psi}{\partial s_2} \left(s_2,D \right) >0$, $\phi_3(D) >0$ SS5 $YS_{\rm ch}^{\rm in}>\varphi_0(D)$, $S_{\rm H_2}^{\rm in}-\omega YS_{\rm ch}^{\rm in} \! > \! M_2(D)-\omega \varphi_0(D)$ $YS_{\rm ch}^{\rm in}<\varphi_0(D)+\varphi_1(D)$ SS6 $(1-\omega)YS_{\rm ch}^{\rm in}+S_{\rm H_2}^{\rm in}>\phi_2(D)$, $YS_{\rm ch}^{\rm in}>\varphi_0(D)+\varphi_1(D)$ $\phi_3(D)\!\geqslant\! 0$ or $\phi_3(D)\!<\!0$ and $\phi_4(D,S_{\rm ch}^{\rm in},S_{\rm H_2}^{\rm in})\!>\!0$
 Existence conditions Stability conditions SS1 Always exists $\mu_0\left(YS_{\rm ch}^{\rm in},S_{\rm H_2}^{\rm in}\right)D$ $YS_{\rm ch}^{\rm in} < \varphi_0(D)$ SS3 $\mu_0\left(YS_{\rm ch}^{\rm in},S_{\rm H_2}^{\rm in}\right)>D$ $\mu_1\left(YS_{\rm ch}^{\rm in}-s_0,S_{\rm H_2}^{\rm in}-\omega \left(YS_{\rm ch}^{\rm in}-s_0\right)\right) M_0(D,s_2)+ M_1(D,s_2)$ with $s_2$ solution of $\Psi(s_2,D) = (1-\omega)YS_{\rm ch}^{\rm in}+S_{\rm H_2}^{\rm in}$ $(1-\omega)YS_{\rm ch}^{\rm in}+S_{\rm H_2}^{\rm in}< \phi_2(D)$, $\frac{\partial\Psi}{\partial s_2} \left(s_2,D \right) >0$, $\phi_3(D) >0$ SS5 $YS_{\rm ch}^{\rm in}>\varphi_0(D)$, $S_{\rm H_2}^{\rm in}-\omega YS_{\rm ch}^{\rm in} \! > \! M_2(D)-\omega \varphi_0(D)$ $YS_{\rm ch}^{\rm in}<\varphi_0(D)+\varphi_1(D)$ SS6 $(1-\omega)YS_{\rm ch}^{\rm in}+S_{\rm H_2}^{\rm in}>\phi_2(D)$, $YS_{\rm ch}^{\rm in}>\varphi_0(D)+\varphi_1(D)$ $\phi_3(D)\!\geqslant\! 0$ or $\phi_3(D)\!<\!0$ and $\phi_4(D,S_{\rm ch}^{\rm in},S_{\rm H_2}^{\rm in})\!>\!0$
 [1] Rui Hu, Yuan Yuan. Stability, bifurcation analysis in a neural network model with delay and diffusion. Conference Publications, 2009, 2009 (Special) : 367-376. doi: 10.3934/proc.2009.2009.367 [2] Akio Matsumoto, Ferenc Szidarovszky. Stability switching and its directions in cournot duopoly game with three delays. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021069 [3] Marzia Bisi, Maria Groppi, Giorgio Martalò, Romina Travaglini. Optimal control of leachate recirculation for anaerobic processes in landfills. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2957-2976. doi: 10.3934/dcdsb.2020215 [4] Massimo Barnabei, Alessandro Borri, Andrea De Gaetano, Costanzo Manes, Pasquale Palumbo, Jorge Guerra Pires. A short-term food intake model involving glucose, insulin and ghrelin. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021114 [5] Qigang Yuan, Jingli Ren. Periodic forcing on degenerate Hopf bifurcation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2857-2877. doi: 10.3934/dcdsb.2020208 [6] Wen-Bin Yang, Yan-Ling Li, Jianhua Wu, Hai-Xia Li. Dynamics of a food chain model with ratio-dependent and modified Leslie-Gower functional responses. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2269-2290. doi: 10.3934/dcdsb.2015.20.2269 [7] Tian Hou, Yi Wang, Xizhuang Xie. Instability and bifurcation of a cooperative system with periodic coefficients. Electronic Research Archive, , () : -. doi: 10.3934/era.2021026 [8] Hildeberto E. Cabral, Zhihong Xia. Subharmonic solutions in the restricted three-body problem. Discrete & Continuous Dynamical Systems, 1995, 1 (4) : 463-474. doi: 10.3934/dcds.1995.1.463 [9] Qixiang Wen, Shenquan Liu, Bo Lu. Firing patterns and bifurcation analysis of neurons under electromagnetic induction. Electronic Research Archive, , () : -. doi: 10.3934/era.2021034 [10] Skyler Simmons. Stability of Broucke's isosceles orbit. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3759-3779. doi: 10.3934/dcds.2021015 [11] Shangzhi Li, Shangjiang Guo. Permanence and extinction of a stochastic SIS epidemic model with three independent Brownian motions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2693-2719. doi: 10.3934/dcdsb.2020201 [12] Yuyue Zhang, Jicai Huang, Qihua Huang. The impact of toxins on competition dynamics of three species in a polluted aquatic environment. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3043-3068. doi: 10.3934/dcdsb.2020219 [13] Nouressadat Touafek, Durhasan Turgut Tollu, Youssouf Akrour. On a general homogeneous three-dimensional system of difference equations. Electronic Research Archive, , () : -. doi: 10.3934/era.2021017 [14] Chiun-Chuan Chen, Hung-Yu Chien, Chih-Chiang Huang. A variational approach to three-phase traveling waves for a gradient system. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021055 [15] Qian Liu. The lower bounds on the second-order nonlinearity of three classes of Boolean functions. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2020136 [16] Rafael Luís, Sandra Mendonça. A note on global stability in the periodic logistic map. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4211-4220. doi: 10.3934/dcdsb.2020094 [17] Lakmi Niwanthi Wadippuli, Ivan Gudoshnikov, Oleg Makarenkov. Global asymptotic stability of nonconvex sweeping processes. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1129-1139. doi: 10.3934/dcdsb.2019212 [18] Michael Grinfeld, Amy Novick-Cohen. Some remarks on stability for a phase field model with memory. Discrete & Continuous Dynamical Systems, 2006, 15 (4) : 1089-1117. doi: 10.3934/dcds.2006.15.1089 [19] Guangying Lv, Jinlong Wei, Guang-an Zou. Noise and stability in reaction-diffusion equations. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021005 [20] Yuzhou Tian, Yulin Zhao. Global phase portraits and bifurcation diagrams for reversible equivariant Hamiltonian systems of linear plus quartic homogeneous polynomials. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2941-2956. doi: 10.3934/dcdsb.2020214

2019 Impact Factor: 1.27

## Tools

Article outline

Figures and Tables