arXiv subjects

Radhouane Fekih-Salem

Publications and source records attributed to Radhouane Fekih-Salem.

6 recordsLinked to original sources

arXiv2026

Qualitative Analysis of a Three-Stage Anaerobic Digestion Model with Microbial Decay

We extend a three-stage anaerobic digestion model by incorporating microbial mortality into the hydrolysis of particulate organic matter. The model describes hydrolysis, acidogenesis, and methanogenesis, each with distinct dilution and decay rates, and accounts for non-monotonic growth in order to capture substrate inhibition. Two hydrolysis mechanisms are considered: a first-order formulation and a biomass-dependent one. The latter distinguishes hydraulic washout from intrinsic mortality and leads to different persistence conditions. For both models, we establish well-posedness and prove the positivity and boundedness of solutions. We then characterize all equilibria and analyze their local stability with respect to key operating parameters. The inclusion of microbial mortality provides a more general and biologically relevant framework, thereby enriching the qualitative dynamics compared to the classical AM2 model.

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arXiv2026

AM2 model with a series configuration of interconnected chemostats and distinct removal rates

We investigate the dynamics of the AM2 model in a serial configuration of two interconnected chemostats with distinct dilution rates. The system is described by nonlinear differential equations, for which the usual reduction from an eight-dimensional system to a four-dimensional one is no longer possible due to the distinct dilution rates. We provide a complete qualitative study by characterizing all steady states and establishing necessary and sufficient conditions for the existence, multiplicity, and local stability of nine types of equilibria, expressed in terms of key operating parameters. We show that coexistence equilibria involving two species in the second bioreactor are not always stable, even when they exist, and require an additional condition related to the trace of the Jacobian matrix. Identifying parameter sets leading to Hopf bifurcations and limit cycles remains an open problem. These results provide new insights into the dynamics of interconnected bioreactors and motivate further investigations on oscillatory behaviors.

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arXiv2026

Global dynamics and bifurcation analysis of a chemostat model with obligate mutualism and mortality

We propose a system of differential equations modeling the competition between two obligate mutualistic species for a single nutrient in a chemostat. Each species promotes the growth of the other, and growth occurs only in the presence of its partner. The three-dimensional model incorporates interspecific density-dependent growth functions and distinct removal rates. We perform a mathematical analysis by characterizing the multiplicity of equilibria and deriving conditions for their existence and stability. Using MatCont, we construct numerical operating diagrams in the parameter space of dilution rate and input substrate concentration, providing a global view of the qualitative dynamics of the system. One-parameter bifurcation diagrams with respect to the input substrate then reveal a variety of dynamical transitions, including saddle-node, Hopf, limit point of cycles LPC, period-doubling PD, and homoclinic bifurcations. When mortality is included, the system exhibits a richer dynamical repertoire than in the mortality-free case, with stable and unstable periodic orbits, tri-stability between equilibria and limit cycles, and several codimension-two bifurcations, including Bogdanov-Takens (BT), cusp of cycles (CPC), resonance points (R1 and R2), and generalized Hopf GH points. These features allow coexistence not only around positive equilibria but also along stable limit cycles, reflecting more realistic ecological dynamics. In contrast, neglecting mortality restricts coexistence to equilibria only. Overall, this study highlights the critical role of mortality in shaping complex dynamics in obligate mutualism, producing multistability and oscillatory coexistence patterns that may better represent natural microbial or ecological systems.

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arXiv2024

Analysis of anaerobic digestion model with two serial interconnected chemostats

In this paper, we study a well known two-step anaerobic digestion model in a configuration of two chemostats in series. This model is an eight-dimensional system of ordinary differential equations. Since the reaction system has a cascade structure, we show that the eight-order model can be reduced to a four-dimensional one. Using general growth rates, we provide an in-depth mathematical analysis of the asymptotic behavior of the system. First, we determine all the steady states of the model where there can be more than fifteen equilibria with a non-monotonic growth rate. Then, the necessary and sufficient conditions of existence and local stability of all steady states are established according to the operating parameters: the dilution rate, the input concentrations of the two nutrients, and the distribution of the total process volume considered. The operating diagrams are then analyzed theoretically to describe the asymptotic behavior of the process according to the four control parameters. There can be seventy regions with rich behavior where the system may exhibit bistability or tristability with the coexistence of both microbial species in the two bioreactors.

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arXiv2024

Analysis and operating diagram of an interspecific density-dependent model

This paper studies a two microbial species model in competition for a single resource in the chemostat including general interspecific density-dependent growth rates with distinct removal rates for each species. We give the necessary and sufficient conditions of existence, uniqueness, and local stability of all steady states. We show that a positive steady state, if it exists, then it is unique and unstable. In this case, the system exhibits a bi-stability where the behavior of the process depends on the initial condition. Our mathematical analysis proves that at most one species can survive which confirms the competitive exclusion principle. We conclude that adding only interspecific competition in the classical chemostat model is not sufficient to show the coexistence of two species even considering mortality in the dynamics of two species. Otherwise, we focus on the study, theoretically and numerically, of the operating diagram which depicts the existence and the stability of each steady state according to the two operating parameters of the process which are the dilution rate and the input concentration of the substrate. Using our mathematical analysis, we construct analytically the operating diagram by plotting the curves that separate their various regions. Our numerical method using MATCONT software validates these theoretical results but it reveals new bifurcations that occur by varying two parameters as Bogdanov-Takens and Zero-Hopf bifurcations. The bifurcation analysis shows that all steady states can appear or disappear only through transcritical bifurcations.

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arXiv2023

The operating diagram of a flocculation model in the chemostat and its dependence on the biological parameters

In this paper, we consider a flocculation model in a chemostat where one species is present in two forms: planktonic and aggregated bacteria with the presence of a single resource. The removal rates of isolated and attached bacteria are distinct and include the specific death rates. Considering distinct yield coefficients with a large class of growth rates, we present a mathematical analysis of the model by establishing the necessary and sufficient conditions of the existence and local asymptotic stability of all steady states according to the two operating parameters which are the dilution rate and the inflowing concentration of the substrate. Using these conditions, we determine first theoretically the operating diagram of the flocculation process describing the asymptotic behavior of the system with respect to two control parameters. The bifurcations analysis shows a rich set of possible types of bifurcations: transcritical bifurcation or branch points of steady states, saddle-node bifurcation or limit points of steady states, Hopf, and homoclinic bifurcations. Using the numerical method with MATCONT software based on a continuation and correction algorithm, we find the same operating diagram obtained theoretically. However, MATCONT detects other types of two-parameter bifurcations such as Bogdanov-Takens and Cusp bifurcations.

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