[Paper Review] A qualitative study of an anaerobic food-web reveals the importance of hydrogen for microbial stability

Szymon Sobieszek, Gail S. K. Wolkowicz+1 more|arXiv (Cornell University)|Feb 14, 2020

Evolutionary Game Theory and Cooperation22 references16 citations

TL;DR

This study extends a mathematical model of anaerobic chlorophenol mineralisation in a chemostat to include multiple substrate inflows, proving that hydrogen input stabilizes microbial communities by enabling a supercritical Hopf bifurcation and uniform persistence. The analysis reveals that external hydrogen and chlorophenol inflows suppress periodic oscillations and enhance system stability, with numerical bifurcation diagrams identifying Bogdanov-Takens and Bautin bifurcations under dual-parameter variation.

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 for a Hopf bifurcation occurring through the interior equilibrium and the role of extraneous substrate inputs in both mitigating the emergence of periodic solutions and expanding the desired stable positive steady-state, where full mineralisation occurs. Here we show that, for a generalised model, the inflow of multiple substrates results in greater dynamical complexity and prove 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 show apply persistence theory for a range of parameter sets to demonstrate unique persistence in the 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.

Motivation & Objective

• To investigate how multiple substrate inflows—particularly hydrogen—impact the dynamical stability of an anaerobic microbial food-web.
• To extend prior numerical findings on Hopf bifurcations by providing analytical proof of their occurrence under varying inflow parameters.
• To establish conditions for uniform persistence in the system, ensuring long-term survival of all microbial populations.
• To explore the role of dilution rate and substrate inflows in inducing complex bifurcations such as Bogdanov-Takens and Bautin.
• To bridge theoretical modeling with practical bioreactor design by identifying stable operating regimes for full mineralisation.

Proposed method

• Formal mathematical modeling of a three-tiered anaerobic food-web using a dimensionless chemostat system with Monod and double Monod growth kinetics.
• Incorporated external inflows of chlorophenol, phenol, and hydrogen, with generalized growth functions satisfying biological plausibility conditions.
• Applied bifurcation theory to prove the existence and stability of six equilibria and to establish conditions for supercritical Hopf bifurcations.
• Used numerical continuation and two-parameter bifurcation diagrams to identify complex dynamics, including Hopf, saddle-node of limit cycles, and Bogdanov-Takens bifurcations.
• Employed persistence theory to prove uniform persistence under specific inflow configurations, particularly when all three substrates are externally supplied.
• Conducted numerical simulations with parameter values from prior studies to validate theoretical predictions and explore basin of attraction behavior.

Experimental results

Research questions

• RQ1 Under what conditions does a supercritical Hopf bifurcation emerge due to variations in chlorophenol, phenol, or hydrogen inflow rates?
• RQ2 How does the inclusion of multiple substrate inflows affect the number and stability of equilibria in the anaerobic food-web?
• RQ3 What is the role of hydrogen in maintaining microbial community stability and preventing periodic oscillations?
• RQ4 Can uniform persistence be proven for systems with external inflows of chlorophenol and hydrogen, even without phenol input?
• RQ5 What types of complex bifurcations (e.g., Bogdanov-Takens, Bautin) arise when dilution rate and chlorophenol inflow are varied simultaneously?

Key findings

• A supercritical Hopf bifurcation is analytically proven to occur when any of the three substrate inflows (chlorophenol, phenol, hydrogen) are varied as bifurcation parameters.
• Numerical analysis confirms that varying the dilution rate can lead to Bogdanov-Takens and Bautin bifurcations, indicating high dynamical complexity.
• The system exhibits bistability when both a boundary equilibrium (e.g., only chlorophenol degrader) and an interior equilibrium are asymptotically stable.
• Uniform persistence is established for two configurations: when all three substrates are externally supplied, and when phenol is not added (ug = 0).
• External hydrogen and chlorophenol inflows stabilize the system by expanding the region of stable positive steady states and suppressing periodic solutions.
• Low dilution rates combined with high substrate inflows maintain the system in a region of uniform persistence, favoring full mineralisation and community stability.