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

  • * Corresponding author: Nahla Abdellatif

    * Corresponding author: Nahla Abdellatif 
This work was supported by the Euro-Mediterranean research network TREASURE (http://www.inra.fr/treasure).
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  • 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.

    Mathematics Subject Classification: Primary: 34A34, 34D20; Secondary: 92B05, 92D25.

    Citation:

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  • Figure 1.  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} $

    Figure 3.  (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] $

    Figure 2.  (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] $

    Figure 6.  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 $

    Figure 4.  Case $ S_{\rm ch}^{\rm in} = 0.029639\in(\sigma^*,\sigma_6) $: bi-stability of the limit cycle (in red) and SS3

    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)

    Figure 7.  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 $

    Figure 8.  (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) $

    Figure 9.  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 $.

    Table 1.  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<s_2^1\right\} $, $ I_2 = \left\{D \in I_1: s_2^0<M_2(D+a_2)<s_2^1\right\} $
    $ \Psi(s_2,D) $$ \Psi\left(s_2,D\right) = (1-\omega)M_0(D+a_0,s_2)+M_1(D+a_1,s_2)+s_2 $, for all $ D\in I_1 $ and $ s_2^0<s_2<s_2^1 $
    $ \phi_1(D) $$ \phi_1(D) = \inf_{s_2^0<s_2<s_2^1} \Psi(s_2,D) $, for all $ D\in I_1 $
    $ \phi_2(D) $$ \phi_2(D) = \Psi\left(M_2(D+a_2),D\right) $, for all $ D\in I_2 $
    $ \phi_3(D) $$ \phi_3(D) = \frac{\partial\Psi}{\partial s_2}\left(M_2(D+a_2),D\right) $, for all $ D\in I_2 $
    $ J_0 $, $ J_1 $$ J_0 = \left(\max\left(0,s^{\rm in}_0-{s^{\rm in}_2}/{\omega}\right),s^{\rm in}_0\right) $, $ J_1 = \left( 0,s_1^{\rm in}\right) $
    $ \psi_0(s_0) $$ \psi_0(s_0) = \mu _0\left(s_0,s_2^{\rm in}-\omega \left(s_0^{\rm in}-s_0\right) \right) $, for all $ s_0\geqslant \max\left(0,s^{\rm in}_0-{s^{\rm in}_2}/{\omega}\right) $
    $ \psi_1(s_1) $$ \psi_1(s_1) = \mu _1\left(s_1,s_2^{\rm in}+s_1^{\rm in}-s_1\right) $, for all $ s_1\in \left[0,s^{\rm in}_1+{s^{\rm in}_2}\right] $
    $ \varphi_i(D) $
    $ i = 0,1 $
    $ \varphi_i(D)=M_i\left(D+a_i,M_2(D+a_2) \right) $, resp., for all, $ D\in\left\{D\geqslant 0 : s_2^0<M_2(D+a_2)\right\} $, $ D\in\left\{D\geqslant 0 : M_2(D+a_2)<s_2^1\right\} $
     | Show Table
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    Table 2.  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) $
     | Show Table
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    Table 3.  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) $
     | Show Table
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    Table 4.  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)<D $, $ \mu_1\left(s_1^{\rm in},s_2^{\rm in}\right)<D $, $ \mu_2\left(s_2^{\rm in}\right)<D $
    SS2$ s_0^{\rm in} < \varphi_0(D) $, $ s_1^{\rm in} < \varphi_1(D) $
    SS3$ \mu_1\left(s_1^{\rm in}+s_0^{\rm in}-s_0,s_2^{\rm in}-\omega \left(s_0^{\rm in}-s_0\right)\right)<D $, $ s_2^{\rm in}-\omega s_0^{\rm in} < M_2(D) -\omega \varphi_0(D) $, with $ s_0 $ solution of equation $ \psi_0(s_0) = D $
    SS4$ (1-\omega)s_0^{\rm in}+s_1^{\rm in}+s_2^{\rm in} < \phi_2(D) $, $ \frac{\partial\Psi}{\partial s_2} \left( s_2,D \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}<M_2(D)+\varphi_1(D) $
    SS8$ s_0^{\rm in} < \varphi_0(D) $
     | Show Table
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    Table 10.  Nominal parameter values. Units are expressed in Chemical Oxygen Demand (COD)

    ParameterWade et al. [31]Unit
    $ k_{m,\rm{ch}} $
    $ k_{m,\rm{ph}} $
    $ k_{m,\rm{H_{2}}} $
    29
    26
    35
    $ \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} $
     | Show Table
    DownLoad: CSV

    Table 11.  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<m_2 $
    $ M_3(s_0,z)=\frac{zL_0(K_0+s_0)}{m_0s_0-z(K_0+s_0)} $ $ 0\leqslant z<\frac{m_0s_0}{K_0+s_0} $
    $ s_2^0(D)=\frac{L_0(D+a_0)}{m_0-D-a_0} $ $ D+a_0<m_0 $
    $ s_2^1(D)=\frac{K_I(m_1-D-a_1)}{D+a_1} $ $ D+a_1<m_1 $
    $ \Psi(s_2,D) $ =$ (1-\omega)\frac{(D+a_0)K_0(L_0+s_2)}{m_0s_2-(D+a_0)(L_0+s_2)} $ + $ \frac{(D+a_1)K_1(K_I+s_2)}{m_1K_I-(D+a_1)(K_I+s_2)}+s_2 $ $ \left\{D\in I_1 : s_2^0<s_2<s_2^1 \right\} $
    $ \psi_0(s_0)=\frac{m_0s_0\left(s_2^{\rm in}-\omega\left(s_0^{\rm in}-s_0\right)\right)}{(K_0+s_0)\left(L_0+s_2^{\rm in}-\omega\left(s_0^{\rm in}-s_0\right)\right)} $$ s_0\in\left[\max\left(0,s^{\rm in}_0\!-\!{s^{\rm in}_2}/{\omega}\right),+\infty\right) $
    $ \psi_1(s_1)= \frac{m_1s_1K_I}{\left(K_1+s_1\right)\left(K_I+s_2^{\rm in}+s_1^{\rm in}-s_1\right)} $$ s_1\in \left[0,s^{\rm in}_1+{s^{\rm in}_2}\right] $
     | Show Table
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    Table 5.  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 $
     | Show Table
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    Table 6.  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
     | Show Table
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    Table 7.  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
     | Show Table
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    Table 8.  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
     | Show Table
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    Table 9.  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 $, $ \mu_2\left(S_{\rm H_2}^{\rm in}\right)<D $
    SS2$ \mu_2\left(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)<D $
    $ S_{\rm H_2}^{\rm in}-\omega YS_{\rm ch}^{\rm in}<M_2(D)-\omega \varphi_0(D) $
    with $ s_0 $ solution of $ \psi_0(s_0) = D $
    SS4$ (1-\omega)YS_{\rm ch}^{\rm in}+S_{\rm H_2}^{\rm in}\geqslant \phi_1(D) $, $ YS_{\rm ch}^{\rm in} > 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 $
     | Show Table
    DownLoad: CSV
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