Abstract
The complexity of the anaerobic digestion process has motivated the development of complex models, such as the widely used Anaerobic Digestion Model No. 1. However, this complexity makes it intractable to identify the stability profile coupled to the asymptotic behaviour of existing steady-states as a function of conventional chemostat operating parameters (substrate inflow concentration and dilution rate). In a previous study this model was simplified and reduced to its very backbone to describe a three-tiered chlorophenol mineralising food-web, with its stability analysed numerically using consensus values for the various biological parameters of the Monod growth functions. Steady-states where all organisms exist were always stable and non-oscillatory. Here we investigate a generalised form of this three-tiered food-web, whose kinetics do not rely on the specific kinetics of Monod form. The results are valid for a large class of growth kinetics as long as they keep the signs of their derivatives. We examine the existence and stability of the identified steady-states and find that, without a maintenance term, the stability of the system may be characterised analytically. These findings permit a better understanding of the operating region of the bifurcation diagram where all organisms exist, and its dependence on the biological parameters of the model. For the previously studied Monod kinetics, we identify four interesting cases that show this dependence of the operating diagram with respect to the biological parameters. When maintenance is included, it is necessary to perform numerical analysis. In both cases we verify the discovery of two important phenomena; i) the washout steady-state is always stable, and ii) a switch in dominance between two organisms competing for hydrogen results in the system becoming unstable and a loss in viability. We show that our approach results in the discovery of an unstable operating region in its positive steady-state, where all three organisms exist, a fact that has not been reported in a previous numerical study. This type of analysis can be used to determine critical behaviour in microbial communities in response to changing operating conditions.
Highlights
The mathematical modelling of engineered biological systems has entered a new era in recent years with the expansion and standardisation of existing models aimed at collating disparate components of these processes and provide scientists, engineers and practitioners with the tools to better predict, control and optimise them
The development of Anaerobic Digestion Model No 1 (ADM1) was enabled largely due to the possibilities for better identification and characterisation of functional groups responsible for the discrete degradation steps operating in series within anaerobic digesters
We show that our approach leads to the discovery of five operating regions, in which one leads to the possibility of instability of the positive steady state, where all three organisms exist, a fact that has not be reported by Wade et al (2015)
Summary
The mathematical modelling of engineered biological systems has entered a new era in recent years with the expansion and standardisation of existing models aimed at collating disparate components of these processes and provide scientists, engineers and practitioners with the tools to better predict, control and optimise them. Whilst simpler models are approximations of real systems, it can be beneficial to consider a reduced model to better understand biological phenomena of sub-processes without the need to consider extraneous system parameters and variables, which tend to make mathematical analysis intractable and cumbersome In their recent paper, Weedermann et al (2015) found an unexpected biological phenomenon in their reduced system describing biogas yield, in which the co-existence steady-state was sub-optimal with regard to maximisation of biogas production. The model described by Xu et al (2011) was extended by the addition of a third organism and substrate to create a three-tiered ‘foodweb’ (Wade et al 2015) In this model, the stability of some steady-states could be determined analytically, but due to the complexity of the Jacobian matrix for certain steady-states, local solutions were necessary using numerical analysis, when considering the full system behaviour.
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