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

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.