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.

中文翻译：

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三层厌氧食物网在化粪池中具有多个底物流入的丰富动力学

对描述厌氧氯酚矿化的三层食物网的数学分析表明，通过其特定的生态相互作用（包括竞争，同养和产品抑制），出现了有趣的动力学行为。先前的数值分析揭示了内部平衡处霍普夫分叉的可能性，以及外部底物输入在缓解周期性溶液的出现和扩大所需的工作区域（正稳态稳定且发生完全矿化）中的作用。 。在这里，我们表明，对于通用模型，多个基质的流入导致更大的动力学复杂性，并显示出由于这些操作参数的变化而导致的超临界霍普夫分叉的发生。进一步，使用数值估计，我们还表明稀释率的变化会导致Bogdanov-Takens和Bautin分叉。最后，我们能够对一系列参数集应用持久性理论，以证明在向系统中多余添加氯酚和氢的情况下的持久性，反映了最近的应用研究，突出了氢在维持稳定的厌氧微生物群落中的作用。