Mathematical Investigation for Two-Bacteria Competition in Presence of a Pathogen With Leachate Recirculation

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Miled El Hajji, Adnan Y. Al-Subhi, Mohammed H. Alharbi


This paper provides a thorough exploration of two-species competition in a continuous bioreactor when adding a pathogen that affects only one species and with leachate recirculation inside the reactor. The dynamics is modelled by a well-constructed system of nonlinear differential equations extending the classical model of the chemostat by adding more realism, enhancing its applicability. The nonnegativity and boundedness of trajectories, the determination of steady states and their local stability strengthens the credibility of the proposed system. The global stability analysis was conducted using uniform persistence theory. The coexistence of both species under somewhat natural assumptions is a key finding, contradicting the well-known competitive exclusion principle. Several numerical examples offer a practical demonstration of the theoretical concepts.

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  1. D.R. Reinhart, T.G. Townsend, Landfill Bioreactor Design and Operation, 1st ed., Routledge, 1998.
  2. F.G. Pohland, B. Al-Yousfi, Design and Operation of Landfills for Optimum Stabilization and Biogas Production, Water Sci. Technol. 30 (1994), 117–124.
  3. D.R. Reinhart, A. Basel Al-Yousfi, The Impact of Leachate Recirculation On Municipal Solid Waste Landfill Operating Characteristics, Waste Manage. Res.: J. Sustain. Circ. Econ. 14 (1996), 337–346.
  4. M. El Hajji, Mathematical Modeling for Anaerobic Digestion Under the Influence of Leachate Recirculation, AIMS Math. 8 (2023), 30287–30312.
  5. P.J. He, X. Qu, L.M. Shao, G.J. Li, D.J. Lee, Leachate Pretreatment for Enhancing Organic Matter Conversion in Landfill Bioreactor, J. Hazard. Mater. 142 (2007), 288–296.
  6. L. Liu, H. Xiong, J. Ma, S. Ge, X. Yu, G. Zeng, Leachate Recirculation for Enhancing Methane Generation within Field Site in China, J. Chem. 2018 (2018), 9056561.
  7. A.A. Alsolami, M. El Hajji, Mathematical Analysis of a Bacterial Competition in a Continuous Reactor in the Presence of a Virus, Mathematics. 11 (2023), 883.
  8. A.H. Albargi, M. El Hajji, Bacterial Competition in the Presence of a Virus in a Chemostat, Mathematics. 11 (2023), 3530.
  9. M. Bisi, M. Groppi, G. Martalò, R. Travaglini, Optimal Control of Leachate Recirculation for Anaerobic Processes in Landfills, Discr. Contin. Dyn. Syst. - Ser. B. 26 (2021), 2957–2976.
  10. O. Laraj, N. El Khattabi, A. Rapaport, Mathematical Model of Anaerobic Digestion With Leachate Recirculation, in: CARI 2022, Tunis, Tunisia, 2022.
  11. M. El Hajji, Influence of the Presence of a Pathogen and Leachate Recirculation on a Bacterial Competition, Int. J. Biomath. In Press.
  12. A.H. Albargi, M.E. Hajji, Mathematical analysis of a two-tiered microbial food-web model for the anaerobic digestion process, Math. Biosci. Eng. 20 (2023), 6591–6611.
  13. H.L. Smith, P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition, Vol. 13, Cambridge Studies in Mathematical Biology, Cambridge University Press, 1995.
  14. M. El Hajji, How Can Inter-Specific Interferences Explain Coexistence or Confirm the Competitive Exclusion Principle in a Chemostat? Int. J. Biomath. 11 (2018), 1850111.
  15. M. El Hajji, N. Chorfi, M. Jleli, Mathematical Model for a Membrane Bioreactor Process, Elec. J. Diff. Equ. 2015(315), 315.
  16. M. El Hajji, N. Chorfi, M. Jleli, Mathematical Modelling and Analysis for a Three-Tiered Microbial Food Web in a Chemostat, Elec. J. Diff. Equ. 2017 (2017), 255.
  17. M. El Hajji, R.M. Alnjrani, Periodic Trajectories for HIV Dynamics in a Seasonal Environment With a General Incidence Rate, Int. J. Anal. Appl. 21 (2023), 96.
  18. M. El Hajji, N.S. Alharbi, M.H. Alharbi, Mathematical Modeling for a CHIKV Transmission Under the Influence of Periodic Environment, Int. J. Anal. Appl. 22 (2024), 6.
  19. M. El Hajji, R.M. Alnjrani, Periodic Behaviour of HIV Dynamics with Three Infection Routes, Mathematics. 12 (2024), 123.
  20. M. El Hajji, Periodic Solutions for Chikungunya Virus Dynamics in a Seasonal Environment With a General Incidence Rate, AIMS Math. 8 (2023), 24888–24913.
  21. G.S.K. Wolkowicz, Successful Invasion of a Food Web in a Chemostat, Math. Biosci. 93 (1989), 249–268.
  22. H.R. Thieme, Convergence Results and a Poincaré-Bendixson Trichotomy for Asymptotically Autonomous Differential Equations, J. Math. Biol. 30 (1992), 755–763.
  23. M. El Hajji, J. Harmand, H. Chaker, C. Lobry, Association Between Competition and Obligate Mutualism in a Chemostat, J. Biol. Dyn. 3 (2009), 635–647.
  24. S. Sobieszek, M. J. Wade, G. S. K. Wolkowicz, Rich Dynamics of a Three-Tiered Anaerobic Food-Web in a Chemostat With Multiple Substrate Inflow, Math. Biosci. Eng. 17 (2020), 7045–7073.
  25. G.J. Butler, H.I. Freedman, P. Waltman, Uniformly Persistent Systems, Proc. Amer. Math. Soc. 96 (1986), 425–429.
  26. G.J. Butler, G.S.K. Wolkowicz, Predator-Mediated Coexistence in a Chemostat: Coexistence and Competition Reversal, Math. Model. 8 (1987), 781–785.