Steady state analysis of a syntrophic model: the effect of a new input substrate concentration

Abstract

In this work, we are interested in a reduced and simplified model of the anaerobic digestion process. We focus on the acetogenesis and hydrogenetrophic methanogenesis phases. The model describes a syntrophic relashionship between two microbial species (the acetogenic bacteria and the hydrogenetrophic methanogenic bacteria) with two input substrates (the fatty acids and the hydrogen) including both decay terms and inhibition of the acetogenic bacteria growth by an excess of hydrogen in the system. The existence and stability analysis of the steady states of the model points out the existence of a new equilibrium point which can be stable according to the operating parameters of the system. By means of operating diagrams, we show that, whatever the region of space considered, there exists only one locally exponentially stable steady state.

Mathematical analysis of a three-tiered food-web in the chemostat

“…This has led to a great deal of mathematical research aimed at extending the chemostat model to better match theory and observations. Among the mechanisms that promote the coexistence of species, we can cite : the crowding effects (see [1,8] and the references therein), the role of density-dependent growth functions (see [15] and the references therein), more complex food webs (see [2,19,33] and the reference therein), the presence of inhibitors that affects the strongest competitor (see [3,9,10,32], and the references therein), the commensalistic relationship where a second species (the commensal) needs the first one (the host) to grow while the host species is not affected by the growth of the commensal one (see [5,6,11,26] and the references therein), the syntrophic relationship where two microbial species depend on each other for survival (see [7,12,14,23,24,34] and the reference therein).…”

“…Recently, a rigorous mathematical analysis of this model (1) was done in [25] with general growth rates but only the chlorophenol inflowing concentration has been taken into account. Using the linear change of variables given by (7) and (8), model (1) can be written as follows:…”

A Mathematical Model of Anaerobic Digestion with Syntrophic Relationship, Substrate Inhibition, and Distinct Removal Rates

“…Decreasing in s 0 , increasing in s 1 Harvey et al [24] 0 D Increasing in s 0 , decreasing in s 1 Nonmonotonic Sari and Harmand [40] 0 D + a i Increasing in s 0 , decreasing in s 1 Increasing Fekih et al [18] 0 D + a i Increasing in s 0 , decreasing in s 1 Nonmonotonic Daoud et al [11] ≥ 0 D + a i Increasing in s 0 , decreasing in s 1 Increasing Harvey et al [24] have studied model (1.2) in the particular case where s in 1 = 0, D i = D, and the growth rate µ 0 (s 0 , s 1 ) = f (s 0 ).g(s 1 ) with f is increasing in s 0 and g is decreasing in s 1 . Our study provides an extension of the results in [24] to the case where D 1 and D 2 are distinct from D. Notice that most of the studies in the existing literature (see Table 2) consider the case of equal removal rates (D 1 = D 2 = D), where the model can be reduced to a two-dimensional system.…”

Mathematical Analysis of a Three-Tiered Model of Anaerobic Digestion

“…Two-tiered models with commensalistic relationship including or not substrate inhibition of the second population are widely considered [2,3,17,21] where the second population (the commensal population) benefits for its growth from the first population (the host population) while the host population is not affected by the growth of the commensal population. On the contrary, when the growth of the first population is affected by the growth of the second population, the system describes a syntrophic relationship [5,7,9,18,19,24,29].…”