Rich dynamics of a three-tiered anaerobic food-web in a chemostat with multiple substrate inflow

Szymon Sobieszek; Matthew J. Wade; Gail S. K. Wolkowicz; Szymon Sobieszek; Matthew J. Wade; Gail S. K. Wolkowicz

doi:10.3934/mbe.2020363

Mathematical Biosciences and Engineering

2020, Volume 17, Issue 6: 7045-7073. doi: 10.3934/mbe.2020363

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Rich dynamics of a three-tiered anaerobic food-web in a chemostat with multiple substrate inflow

1.
Department of Mathematics and Statistics, McMaster University, Hamilton, ON L8S 4K1, Canada
2.
School of Engineering, Newcastle University, Newcastle-upon-Tyne NE1 7RU, United Kingdom

Received: 15 June 2020 Accepted: 29 September 2020 Published: 19 October 2020

The mathematical analysis of a three-tiered food-web describing anaerobic chlorophenol mineralisation has suggested the emergence of interesting dynamical behaviour through its specific ecological interactions, which include competition, syntrophy, and product inhibition. Previous numerical analyses have revealed the possibility of a Hopf bifurcation at the interior equilibrium as well as the role of extraneous substrate inputs in both mitigating the emergence of periodic solutions and expanding the desired operating region where the positive steady-state is stable and full mineralisation occurs. Here we show that, for a generalised model, the inflow of multiple substrates results in greater dynamical complexity and show the occurrence of a supercritical Hopf bifurcation resulting from variations in these operating parameters. Further, using numerical estimation, we also show that variations in the dilution rate can lead to Bogdanov-Takens and Bautin bifurcations. Finally, we are able to apply persistence theory for a range of parameter sets to demonstrate persistence in cases where chlorophenol and hydrogen are extraneously added to the system, mirroring recent applied studies highlighting the role of hydrogen in maintaining stable anaerobic microbial communities.
- chlorophenol mineralisation,
- mathematical modelling,
- bifurcation analysis,
- Hopf bifurcation,
- hydrogen
Citation: Szymon Sobieszek, Matthew J. Wade, Gail S. K. Wolkowicz. Rich dynamics of a three-tiered anaerobic food-web in a chemostat with multiple substrate inflow[J]. Mathematical Biosciences and Engineering, 2020, 17(6): 7045-7073. doi: 10.3934/mbe.2020363

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Abstract

The mathematical analysis of a three-tiered food-web describing anaerobic chlorophenol mineralisation has suggested the emergence of interesting dynamical behaviour through its specific ecological interactions, which include competition, syntrophy, and product inhibition. Previous numerical analyses have revealed the possibility of a Hopf bifurcation at the interior equilibrium as well as the role of extraneous substrate inputs in both mitigating the emergence of periodic solutions and expanding the desired operating region where the positive steady-state is stable and full mineralisation occurs. Here we show that, for a generalised model, the inflow of multiple substrates results in greater dynamical complexity and show the occurrence of a supercritical Hopf bifurcation resulting from variations in these operating parameters. Further, using numerical estimation, we also show that variations in the dilution rate can lead to Bogdanov-Takens and Bautin bifurcations. Finally, we are able to apply persistence theory for a range of parameter sets to demonstrate persistence in cases where chlorophenol and hydrogen are extraneously added to the system, mirroring recent applied studies highlighting the role of hydrogen in maintaining stable anaerobic microbial communities.

References

[1]	M. J. Wade, R. W. Pattinson, N. G. Parker, J. Dolfing, Emergent behaviour in a chlorophenolmineralising three-tiered microbial 'food web', J. Theor. Biol., 389 (2016), 171-186. doi: 10.1016/j.jtbi.2015.10.032
[2]	C. Mazur, W. Jones, C. Tebes-Stevens, H₂ consumption during the microbial reductive dehalogenation of chlorinated phenols and tetrachloroethene, Biodegradation, 14 (2003), 285-295. doi: 10.1023/A:1024765706617
[3]	L. Levén, K. Nyberg, A. Schnürer, Conversion of phenols during anaerobic digestion of organic solid waste - a review of important microorganisms and impact of temperature, J. Env. Manage., 95 (2012), 99-103.
[4]	T. Großkopf, O. Soyer, Microbial diversity arising from thermodynamic constraints, ISME J., 10 (2016), 2725-2733. doi: 10.1038/ismej.2016.49
[5]	B. Schink, Energetics of syntrophic cooperation in methanogenic degradation, Microbiol. Mol. Biol. Rev., 61 (1997), 262-280. doi: 10.1128/.61.2.262-280.1997
[6]	I. Bassani, P. G. Kougias, L. Treu, I. Angelidaki, Biogas upgrading via hydrogenotrophic methanogenesis in two-stage continuous stirred tank reactors at mesophilic and thermophilic conditions, Environ. Sci. Technol., 49 (2015), 12585-12593. doi: 10.1021/acs.est.5b03451
[7]	J. Chen, M. J. Wade, J. Dolfing, O. S. Soyer, Increasing sulfate levels show a differential impact on synthetic communities comprising different methanogens and a sulfate reducer, J. Royal Soc. Interface, 16 (2019), 20190129.
[8]	N. W. Smith, P. R. Shorten, E. H. Altermann, N. C. Roy, W. C. McNabb, Hydrogen cross-feeders of the human gastrointestinal tract, Gut Microbes, 10 (2019), 270-288. doi: 10.1080/19490976.2018.1546522
[9]	T. Sari, M. J. Wade, Generalised approach to modelling a three-tiered microbial food-web, Math. Biosci., 291 (2017), 21-37. doi: 10.1016/j.mbs.2017.07.005
[10]	M. El Hajji, N. Chorfi, M. Jleli, Mathematical modelling and analysis for a three-tiered microbial food web in a chemostat, Electron. J. Differ. Eq., 255 (2017), 1-13.
[11]	S. Nouaoura, N. Abdellatif, R. Fekih-Salem, T. Sari, Mathematical analysis of a three-tiered model of anaerobic digestion, Preprint, hal-02540350v2.
[12]	S. Nouaoura, R. Fekih-Salem, N. Abdellatif, T. Sari, Mathematical analysis of a three-tiered food-web in the chemostat, Preprint, hal-02878246.
[13]	Y. Kuznestov, Elements of Applied Bifurcation Theory, Springer-Verlag, New York, 2004.
[14]	Maple [Software]. Version 18.02. Waterloo Maple Inc., Waterloo, Ontario, 2018. Available from: https://maplesoft.com.
[15]	MATLAB [Software]. Version 9.5.0.944444 (R2018b). The MathWorks Inc., Natick, Massachusetts, 2018. Available from: https://www.mathworks.com.
[16]	H. L. Smith, P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition (Cambridge Studies in Mathematical Biology), Cambridge University Press, 1995.
[17]	G. J. Butler, H. I. Freedman, P. Waltman, Uniformly persistent systems, P. Am. Math. Soc., 96 (1986), 425-430.
[18]	G. J. Butler, G. S. K. Wolkowicz, Predator-mediated competition in the chemostat, J. Math. Biol., 24 (1986), 167–191.
[19]	H. R. Thieme, Convergence results and a Poincare-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755-763.
[20]	XPPAUT [Software]. Version 8.0. Dr. Bard Ermentrout, Dept of Mathematics, University of Pittsburgh, Pittsburgh PA, 2016. Available from: http://www.math.pitt.edu/bard/xpp/xpp.html.
[21]	M. El Karoui, M. Hoyos-Flight, L. Fletcher, Future trends in synthetic biology—a report, Front. Bioeng. Biotechnol., 7 (2019), 175.
[22]	H. Delattre, J. Chen, M. J. Wade, O. S. Soyer, Thermodynamic modelling of synthetic communities predicts minimum free energy requirements for sulfate reduction and methanogenesis, J. R. Soc. Interface, 17 (2020), 20200053.

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