, S * 2 (r, z) (b) B * 2 (r, z)
, Representation of the steady-state solution
2) (µ(1)) := 1/µ(1) ,
URL : https://hal.archives-ouvertes.fr/in2p3-00148681
,
According to Remark 2.2 and Definition 4.1, the stability analysis of system (1.2) (shown in Section 1) can be rewritten as ? If Da < Da W (1.2) (µ(1)), then the equilibrium solution (1, 0) of system (1.2) is asymptotically stable. ? If µ satisfies (A1) and Da > Da NW (A1),(1.2) (µ(1)), then the equilibrium solution ,
, ? If µ satisfies (A2) and Da > Da NW (A2)
, (1.2) = 1 (and the difference, when µ(1) = 0.5, between the variable Da W (2.4) (Th B , µ(1)) and the constant Da W (1.2) (µ(1)) = 2). In both cases Th B ?, Th B ) and the constant Da NW (A2)
, Similarly, for the particular case when µ(1) = 0.5, we observe that log(Da W (2.4) (Th B , 0.5)) ? log(2) also for values smaller than log(Th B ) ? ?2 (Th B ? 0.1). This comparison, performed with other reaction values µ(1) ? {i/20} 20 i=1
A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates, Biotechnology and Bioengineering, vol.10, issue.6, pp.707-723, 1968. ,
Phenomena of multiplicity, stability and symmetry, Annals of the New York Academy of Sciences, vol.231, issue.1, pp.86-98, 1974. ,
, The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts, vol.1, 1975.
, Partial Differential Equations with Fourier Series and Boundary Value Problems, 2005.
, Biochemical Engineering Fundamentals. Chemical engineering. McGraw-Hill, 1986.
The effect of time delay and growth rate inhibition in the bacterial treatment of wastewater, Journal of Theoretical Biology, vol.63, issue.2, pp.385-395, 1976. ,
Existence and uniqueness of solution of a continuous flow bioreactor model with two species. Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A: Matemáticas, vol.110, issue.2, pp.357-377, 2016. ,
A. Rapaport; Modeling and optimization of activated sludge bioreactors for wastewater treatment taking into account spatial inhomogeneities, Journal of Process Control, vol.54, pp.118-128, 2017. ,
URL : https://hal.archives-ouvertes.fr/in2p3-00010761
Continuous flow systems, Chemical Engineering Science, vol.2, issue.1, pp.1-13, 1953. ,
Remarks on a precedent paper by Crespo, Ivorra and Ramos on the stability of bioreactor processes ,
On the principle of pseudo-linearized stability: Applications to some delayed nonlinear parabolic equations, Nonlinear Analysis, vol.63, issue.5, pp.997-1007, 2005. ,
Automatic Control of Bioprocesses, 2010. ,
Dynamical Modelling and Estimation in Wastewater Treatment Processes, 2001. ,
A semilinear parabolic boundary-value problem in bioreactors theory, Electronic Journal of Differential Equations, 2004. ,
, Multiple stable equilibrium profiles in tubular bioreactors. Mathematical and Computer Modelling, vol.48, pp.1840-1853, 1112.
Linearization stability of nonlinear partial differential equations, Proceedings of Symposia in Pure Mathematics, vol.27, pp.219-263, 1975. ,
, Minimal time bioremediation of natural water resources, vol.47, pp.1764-1769, 2011.
URL : https://hal.archives-ouvertes.fr/hal-00521118
Nonlinear population dynamics in the chemostat, Computing in Science Engineering, vol.3, 2001. ,
Asymptotic Behavior of Dissipative Systems. Mathematical Surveys and Monographs, 2010. ,
Telling; The continuous culture of bacteria; a theoretical and experimental study, Journal of general microbiology, 1956. ,
,
, Métodos Numéricos. Teoría, problemas y prácticas con MATLAB. Ediciones Pirámide. Grupo Anaya, 2015.
Transient and steady state of mass-conserved reactiondiffusion systems, Phys. Rev. E, vol.75, p.15203, 2007. ,
, Advanced Real Analysis. Cornerstones. Birkhäuser Boston, 2005.
Invariant sets for nonlinear elliptic and parabolic systems, Journal on Mathematical Analysis, vol.11, issue.6, pp.1075-1103, 1980. ,
On the Robin boundary condition for Laplace's equation in Lipschitz domains, Communications in Partial Differential Equations, vol.29, issue.1 & 2, pp.91-109, 2004. ,
Elementary Feedback Stabilization of the Linear Reaction-Convection-Diffusion Equation and the Wave Equation, Mathématiques et Applications. Springer Berlin Heidelberg, 2009. ,
Advances in Chemical Engineering: Multiscale analysis. Advances in Chemical Engineering, 2005. ,
Asymptotic behavior for semilinear differential equations in Banach spaces, SIAM Journal on Mathematical Analysis, vol.9, issue.6, pp.1105-1119, 1978. ,
Convergence of solutions of one-dimensional semilinear parabolic equations, J. Math. Kyoto Univ, vol.18, issue.2, pp.221-227, 1978. ,
Asymptotic behavior and stability of solutions of semilinear diffusion equations, Publ. Res. Inst. Math. Sci, vol.15, pp.401-454, 1979. ,
Tubular reactor steady state and stability characteristics. I. Effect of axial mixing, AIChE Journal, vol.17, issue.4, pp.831-837, 1971. ,
Optimal time control of bioreactors for the wastewater treatment, Optimal Control Applications and Methods, vol.20, issue.3, pp.145-164, 1999. ,
Stability and bifurcation of nonconstant solutions to a reaction diffusion system with conservation of mass, Nonlinearity, vol.23, issue.6, p.1387, 2010. ,
Asymptotic stability of reaction-diffusion systems in chemical reactor and combustion theory, Journal of Mathematical Analysis and Applications, vol.82, issue.2, pp.503-526, 1981. ,
, Nonlinear Parabolic and Elliptic Equations. Fems Symposium, 1992.
, Advances in Numerical Simulation in Physics and Engineering: Lecture Notes of the XV 'Jacques-Louis Lions, 2014.
Phase differences in reaction-diffusionadvection systems and applications to morphogenesis, IMA Journal of Applied Mathematics, vol.55, pp.19-33, 1995. ,
, Introducción al Análisis Matemático del Método de Elementos Finitos. Editorial Complutense, 2012.
Basic chemostat model revisted. Differential Equations and Dynamical Systems, vol.17, pp.3-16, 2009. ,
Global dynamics of the buffered chemostat for a general class of response functions, Journal of Mathematical Biology, vol.71, issue.1, pp.69-98, 2015. ,
URL : https://hal.archives-ouvertes.fr/hal-00923826
Dynamics of reaction-diffusion patterns controlled by asymmetric nonlocal coupling as a limiting case of differential advection, Phys. Rev. E, vol.89, p.52909, 2014. ,
The theory of the Chemostat, Cambridge studies in Mathematical biology, vol.13, 1995. ,
Analysis and simplification of a mathematical model for high-pressure food processes, Applied Mathematics and Computation, vol.226, pp.20-37, 2014. ,
An experimental study of steady state multiplicity and stability in an adiabatic stirred reactor, AIChE Journal, vol.16, issue.3, pp.410-419, 1970. ,
, A First Course in Partial Differential Equations with Complex Variables and Transform Methods, 1995.
Dynamical analysis of distributed parameter tubular reactors, Automatica, vol.36, issue.3, pp.349-361, 2000. ,
,
INRA/SupAgro). 2, Place P.Viala, 34060 Montpellier, France E-mail address: maria.crespo-moya@umontpellier.fr Benjamin Ivorra Departamento de Matemática Aplicada & Instituto de Matemática Interisciplinar, Informatique et Statistique pour lÉnvironnement et lÁgronomie ,