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Self-sustained dynamics by modeling competing PHA-producers and non-PHA-producers bacteria population for a limited resource: local and homoclinic bifurcation analysis

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Abstract

We propose a mathematical model for two species competing for a limited resource associated to polyhydroxyalkanoate (PHA) production, which possesses regime of stable states: stable equilibrium, and periodic and aperiodic oscillations. Such regimes of stable oscillations are absent in the model without taking into account PHA production but is known to exist in experimental model associated to the production of PHA. It explains the capacity of the system to sustain itself at the lowest value of resource. Thus, the proposed system provides a simpler four-dimensional model containing monod functions with such behaviours. Using analytical tools and numerical bifurcation analysis, we describe parameter regions and bifurcation structures leading to the existence and the coexistence between stable, unstable equilibrium and limit cycle. These explain critical parameter sensitivity impact on the process. Considering the effects on the proposed system under a frequent alternation of the input resource, we investigate how the increase of the length of the feast period in the Feast-Famine conditions, increases the PHA-production or decreases the lowest value of the resource at the equilibrium.

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Data availability

The data from simulations that support the findings of this study are available on request from the corresponding author, B. I. Camara.

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Acknowledgements

The financial supports of this work have been EMS-Simons for Africa and Lorraine University, we are greatly thankful for their financial supports.

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Authors and Affiliations

  1. Fundamental Physics Laboratory, Physics of Complex System Group, Department of Physics, Faculty of Science, University of Douala, Box 24 157, Douala, Cameroon

    I. B. Tagne Nkounga & R. Yamapi

  2. Laboratoire Interdisciplinaire des Environnements Continentaux, Université de Lorraine - CNRS UMR 7360, Campus Bridoux - 8 Rue du Général Delestraint, 57070, Metz, France

    I. B. Tagne Nkounga & P. Bauda

  3. Laboratoire de Conception, Optimisation et Modélisation des Systèmes (LCOMS), Université de Lorraine, Campus Bridoux - 8 Rue du Général Delestraint, 57070, Metz, France

    B. I. Camara

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  1. I. B. Tagne Nkounga
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Principal minors computation of the Jacobian matrix at model equilibria

Principal minors computation of the Jacobian matrix at model equilibria

For each equilibrium, the expressions of the elements of the Jacobian matrix and the principal minors are below:

  1. (a)

    \((N_P, N_O, R, PHA)^*\)=\((0, 0, R, 0)^*\)

    $$\begin{aligned} A= & {} g_P\dfrac{R^*}{R^*+K_P}-m_P;\quad B=-Q_Pg_P\dfrac{R^*}{R^*+K_P};\\ C= & {} 0;\quad D=g_O\dfrac{R^*}{R^*+K_O}-m_O;\\ E= & {} -Q_Og_O\dfrac{R^*}{R^*+K_O};\quad F=0;\\ G= & {} 0;\quad H=-\alpha ; \\ I= & {} 0;\quad J=-m_{PHA}.\\ \end{aligned}$$
  1. (b)

    \((N_P, N_O, R, PHA)^*\)=\((0, N_O, R, 0)^*\)

    $$\begin{aligned} A= & {} g_P\dfrac{R^*}{R^*+K_P}-m_P;\quad B=-Q_Pg_P\dfrac{R^*}{R^*+K_P};\\ C= & {} 0;\quad D=g_O\dfrac{R^*}{R^*+K_O}-m_O;\\ E= & {} -Q_Og_O\dfrac{R^*}{R^*+K_O};\quad F=0;\\ G= & {} N_O^*g_O\dfrac{K_O}{(R^*+K_O)^2};\quad H\\= & {} -\alpha -Q_Og_ON_O^*\dfrac{K_O}{(R^*+K_O)^2};\\ I= & {} 0;\quad J=-m_{PHA}.\\ \end{aligned}$$
  1. (c)

    \((N_P, N_O, R, PHA)^*\)=\((N_P, 0, R, 0)^*\)

    $$\begin{aligned} A= & {} g_P\dfrac{R^*}{R^*+K_P}-m_P;\quad B=-Q_Pg_P\dfrac{R^*}{R^*+K_P};\\ C= & {} 0;\quad D=g_O\dfrac{R^*}{R^*+K_O}-m_O;\\ E= & {} -Q_Og_O\dfrac{R^*}{R^*+K_O}; \quad F=N_P^*g_P\dfrac{K_P}{(R^*+K_P)^2};\\ G= & {} 0;\quad H=-\alpha -Q_Pg_PN_P^*\dfrac{K_P}{(R^*+K_P)^2};\\ I= & {} \dfrac{bN_P^*}{K_{PHA}};\quad J=\dfrac{hN_P^*}{K_{PHA}}-m_{PHA}.\\ \end{aligned}$$
  1. (d)

    \((N_P, N_O, R, PHA)^*\)=\((N_P, 0, R, PHA)^*\)

    $$\begin{aligned} A= & {} g_P\dfrac{R^*}{R^*+K_P}+b\dfrac{PHA^*}{PHA^*+K_{PHA}}-m_P;\quad B\\= & {} -Q_Pg_P\dfrac{R^*}{R^*+K_P};\\ C= & {} h\dfrac{PHA^*}{PHA^*+K_{PHA}};\quad D=g_O\dfrac{R^*}{R^*+K_O}-m_O;\\ E= & {} -Q_Og_O\dfrac{R^*}{R^*+K_O};\quad F=N_P^*g_P\dfrac{K_P}{(R^*+K_P)^2};\\ G= & {} 0;\quad H=-\alpha -Q_Pg_PN_P^*\dfrac{K_P}{(R^*+K_P)^2};\\ I= & {} bN_P^*\dfrac{K_{PHA}}{(PHA^*+K_{PHA})^2};\quad J\\= & {} hN_P^*\dfrac{K_{PHA}}{(PHA^*+K_{PHA})^2}-m_{PHA}.\\ \end{aligned}$$
  1. (e)

    \((N_P, N_O, R, PHA)^*\)=\((N_P, N_O, R, 0)^*\)

    $$\begin{aligned} A= & {} g_P\dfrac{R^*}{R^*+K_P}-m_P;\quad B=-Q_Pg_P\dfrac{R^*}{R^*+K_P};\\ C= & {} 0;\quad D=g_O\dfrac{R^*}{R^*+K_O}-m_O;\\ E= & {} -Q_Og_O\dfrac{R^*}{R^*+K_O};\quad F=N_P^*g_P\dfrac{K_P}{(R^*+K_P)^2};\\ G= & {} N_O^*g_O\dfrac{K_O}{(R^*+K_O)^2};\quad H\\= & {} -\alpha -Q_Pg_PN_P^*\dfrac{K_P}{(R^*+K_P)^2}\\{} & {} -Q_Og_ON_O^*\dfrac{K_O}{(R^*+K_O)^2};\\ I= & {} \dfrac{bN_P^*}{K_{PHA}};\quad J=\dfrac{hN_P^*}{K_{PHA}}-m_{PHA}.\\ \end{aligned}$$
  1. (f)

    \((N_P, N_O, R, PHA)^*\)=\((N_P, N_O, R, PHA)^*\)

    $$\begin{aligned} A= & {} g_P\dfrac{R^*}{R^*+K_P}+b\dfrac{PHA^*}{PHA^*+K_{PHA}}-m_P;\quad B\\= & {} -Q_Pg_P\dfrac{R^*}{R^*+K_P};\\ C= & {} h\dfrac{PHA^*}{PHA^*+K_{PHA}};\quad D=g_O\dfrac{R^*}{R^*+K_O}-m_O;\\ E= & {} -Q_Og_O\dfrac{R^*}{R^*+K_O};\quad F=N_P^*g_P\dfrac{K_P}{(R^*+K_P)^2};\\ G= & {} N_O^*g_O\dfrac{K_O}{(R^*+K_O)^2};\quad H\\= & {} -\alpha -Q_Pg_PN_P^*\dfrac{K_P}{(R^*+K_P)^2}\\{} & {} -Q_Og_ON_O^*\dfrac{K_O}{(R^*+K_O)^2};\\ I= & {} bN_P^*\dfrac{K_{PHA}}{(PHA^*+K_{PHA})^2};\quad J\\= & {} hN_P^*\dfrac{K_{PHA}}{(PHA^*+K_{PHA})^2}-m_{PHA}.\\ \end{aligned}$$

The characteristic polynomial versus the eigen value \(\lambda \) is given by:

$$\begin{aligned}{} & {} P(\lambda )=(\lambda -\lambda _1)(\lambda -\lambda _2)(\lambda -\lambda _3)(\lambda -\lambda _4)\nonumber \\ {}{} & {} \quad =\lambda ^4-T_r(J)\lambda ^3+S_2\lambda ^2-S_3\lambda + Det(J), \end{aligned}$$
(A1)

where,

  1. (i)

    \((N_P, N_O, R, PHA)^*\)=\((0, 0, R, 0)^*\)

    $$\begin{aligned} S_1= & {} A+D+H+J =T_r(J);\\ S_2= & {} \begin{vmatrix}H&0\\0&J\end{vmatrix} +\begin{vmatrix}D&0\\0&J\end{vmatrix}+\begin{vmatrix}D&0\\E&H\end{vmatrix}+\begin{vmatrix}A&0\\0&J\end{vmatrix}+\begin{vmatrix}A&0\\B&H\end{vmatrix}+\begin{vmatrix}A&0\\0&D\end{vmatrix};\\ {}= & {} HJ+DJ+DH+AJ+AH+AD;\\ S_3= & {} \begin{vmatrix}D&0&0\\E&H&0\\0&0&J\end{vmatrix}+\begin{vmatrix}A&0&0\\B&H&0\\0&0&J\end{vmatrix}+\begin{vmatrix}A&0&0\\0&D&0\\0&0&J\end{vmatrix}\\{} & {} +\begin{vmatrix}A&0&0\\0&D&0\\B&E&H\end{vmatrix}; \\ {}= & {} DHJ+AHJ+ADJ+ADH;\\ S_4= & {} \begin{vmatrix}A&0&0&0\\0&D&0&0\\B&E&H&0\\0&0&0&J\end{vmatrix}=AJDH=Det(J); \end{aligned}$$

    A, B, D, E, H, and J have the same expression as in 2.(a).

  1. (ii)

    \((N_P, N_O, R, PHA)^*\)=\((0, N_O, R, 0)^*\)

    $$\begin{aligned} S_1= & {} A+D+H+J =T_r(J);\\ S_2= & {} \begin{vmatrix}H&0\\0&J\end{vmatrix} +\begin{vmatrix}D&0\\0&J\end{vmatrix}+\begin{vmatrix}D&G\\E&H\end{vmatrix}+\begin{vmatrix}A&0\\0&J\end{vmatrix}+\begin{vmatrix}A&0\\B&H\end{vmatrix}+\begin{vmatrix}A&0\\0&D\end{vmatrix};\\ {}= & {} HJ+DJ+DH-EG+AJ+AH+AD;\\ S_3= & {} \begin{vmatrix}D&G&0\\E&H&0\\0&0&J\end{vmatrix}+\begin{vmatrix}A&0&0\\B&H&0\\0&0&J\end{vmatrix}+\begin{vmatrix}A&0&0\\0&D&0\\0&0&J\end{vmatrix}\\{} & {} +\begin{vmatrix}A&0&0\\0&D&G\\B&E&H\end{vmatrix}; \\ {}= & {} DHJ-GEJ+AHJ+ADJ+ADH-AEG;\\ S_4= & {} \begin{vmatrix}A&0&0&0\\0&D&G&0\\B&E&H&0\\0&0&0&J\end{vmatrix}=AJ(DH-GE)=Det(J); \end{aligned}$$

    A, B, D, E, G, H, and J have the same expression as in 2.(b).

  1. (iii)

    \((N_P, N_O, R, PHA)^*\)=\((N_P, 0, R, 0)^*\)

    $$\begin{aligned} S_1= & {} A+D+H+J =T_r(J);\\ S_2= & {} \begin{vmatrix}H&0\\0&J\end{vmatrix} +\begin{vmatrix}D&0\\0&J\end{vmatrix}+\begin{vmatrix}D&0\\E&H\end{vmatrix}+\begin{vmatrix}A&I\\0&J\end{vmatrix}+\begin{vmatrix}A&F\\B&H\end{vmatrix}+\begin{vmatrix}A&0\\0&D\end{vmatrix};\\ {}= & {} HJ+DJ+DH+AJ+AH-BF+AD;\\ S_3= & {} \begin{vmatrix}D&0&0\\E&H&0\\0&0&J\end{vmatrix}+\begin{vmatrix}A&F&I\\B&H&0\\0&0&J\end{vmatrix}+\begin{vmatrix}A&0&I\\0&D&0\\0&0&J\end{vmatrix}+\begin{vmatrix}A&0&F\\0&D&0\\B&E&H\end{vmatrix}; \\ {}= & {} DHJ+AHJ-FBJ+ADJ+ADH-FBD;\\ S_4= & {} \begin{vmatrix}A&0&F&I\\0&D&0&0\\B&E&H&0\\0&0&0&J\end{vmatrix}=AJDH-FDBJ=Det(J); \end{aligned}$$

    A, B, D, E, F, H, I, and J have the same expression as in 2.(c).

  1. (iv)

    \((N_P, N_O, R, PHA)^*\)=\((N_P, 0, R, PHA)^*\)

    $$\begin{aligned} S_1= & {} A+D+H+J =T_r(J);\\ S_2= & {} \begin{vmatrix}H&0\\0&J\end{vmatrix} +\begin{vmatrix}D&0\\0&J\end{vmatrix}+\begin{vmatrix}D&0\\E&H\end{vmatrix}+\begin{vmatrix}A&I\\C&J\end{vmatrix}+\begin{vmatrix}A&F\\B&H\end{vmatrix}+\begin{vmatrix}A&0\\0&D\end{vmatrix};\\ {}= & {} HJ+DJ+DH+AJ-CI+AH-BF+AD;\\ S_3= & {} \begin{vmatrix}D&0&0\\E&H&0\\0&0&J\end{vmatrix}+\begin{vmatrix}A&F&I\\B&H&0\\C&0&J\end{vmatrix}+\begin{vmatrix}A&0&I\\0&D&0\\C&0&J\end{vmatrix}+\begin{vmatrix}A&0&F\\0&D&0\\B&E&H\end{vmatrix}; \\ {}= & {} DHJ+AHJ-FBJ-ICH+ADJ-ICD\\ {}{} & {} +ADH-FBD;\\ S_4= & {} \begin{vmatrix}A&0&F&I\\0&D&0&0\\B&E&H&0\\C&0&0&J\end{vmatrix}=AJDH-FDBJ-ICDH=Det(J); \end{aligned}$$

    A, B, C, D, E, F, H, I, and J have the same expression as in 2.(d).

  1. (v)

    \((N_P, N_O, R, PHA)^*\)=\((N_P, N_O, R, 0)^*\)

    $$\begin{aligned} S_1= & {} A+D+H+J =T_r(J);\\ S_2= & {} \begin{vmatrix}H&0\\0&J\end{vmatrix} +\begin{vmatrix}D&0\\0&J\end{vmatrix}+\begin{vmatrix}D&G\\E&H\end{vmatrix}+\begin{vmatrix}A&I\\C&J\end{vmatrix}+\begin{vmatrix}A&F\\B&H\end{vmatrix}+\begin{vmatrix}A&0\\0&D\end{vmatrix};\\ {}= & {} HJ+DJ+DH-EG+AJ-CI+AH-BF+AD;\\ S_3= & {} \begin{vmatrix}D&G&0\\E&H&0\\0&0&J\end{vmatrix}+\begin{vmatrix}A&F&I\\B&H&0\\C&0&J\end{vmatrix}+\begin{vmatrix}A&0&I\\0&D&0\\C&0&J\end{vmatrix}+\begin{vmatrix}A&0&F\\0&D&G\\B&E&H\end{vmatrix};\\= & {} DHJ-GEJ+AHJ-FBJ-ICH+ADJ\\ {}{} & {} -ICD+ADH-AEG-FBD;\\ S_4= & {} \begin{vmatrix}A&0&F&I\\0&D&G&0\\B&E&H&0\\C&0&0&J\end{vmatrix}\\= & {} AJ(DH-GE)-FDBJ-IC(DH-GE)=Det(J); \end{aligned}$$

    A, B, D, E, F, G, H, I, and J have the same expression as in 2.(e).

  1. (vi)

    \((N_P, N_O, R, PHA)^*\)=\((N_P, N_O, R, PHA)^*\)

    $$\begin{aligned} S_1= & {} A+D+H+J =T_r(J);\\ S_2= & {} \begin{vmatrix}H&0\\0&J\end{vmatrix} +\begin{vmatrix}D&0\\0&J\end{vmatrix}+\begin{vmatrix}D&G\\E&H\end{vmatrix}+\begin{vmatrix}A&I\\0&J\end{vmatrix}+\begin{vmatrix}A&F\\B&H\end{vmatrix}+\begin{vmatrix}A&0\\0&D\end{vmatrix};\\ {}= & {} HJ+DJ+DH-EG+AJ+AH-BF+AD;\\ S_3= & {} \begin{vmatrix}D&G&0\\E&H&0\\0&0&J\end{vmatrix}+\begin{vmatrix}A&F&I\\B&H&0\\0&0&J\end{vmatrix}+\begin{vmatrix}A&0&I\\0&D&0\\0&0&J\end{vmatrix}+\begin{vmatrix}A&0&F\\0&D&G\\B&E&H\end{vmatrix}; \\ {}= & {} DHJ-GEJ+AHJ-FBJ+ADJ\\ {}{} & {} +ADH-AEG-FBD;\\ S_4= & {} \begin{vmatrix}A&0&F&I\\0&D&G&0\\B&E&H&0\\0&0&0&J\end{vmatrix}\\= & {} AJ(DH-GE)-FDBJ=Det(J); \end{aligned}$$

    A, B, C, D, E, F, G, H, I, and J have the same expression as in 2.(f).

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Tagne Nkounga, I.B., Bauda, P., Yamapi, R. et al. Self-sustained dynamics by modeling competing PHA-producers and non-PHA-producers bacteria population for a limited resource: local and homoclinic bifurcation analysis. Nonlinear Dyn 112, 9673–9701 (2024). https://doi.org/10.1007/s11071-024-09541-8

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