×

zbMATH — the first resource for mathematics

Qualitative analysis on positive steady-states for an autocatalytic reaction model in thermodynamics. (English) Zbl 1371.35084
Summary: In this paper, a reaction-diffusion system known as an autocatalytic reaction model is considered. The model is characterized by a system of two differential equations which describe a type of complex biochemical reaction. Firstly, some basic characterizations of steady-state solutions of the model are presented. And then, the stability of positive constant steady-state solution and the non-existence, existence of non-constant positive steady-state solutions are discussed. Meanwhile, the bifurcation solution which emanates from positive constant steady-state is investigated, and the global analysis to the system is given in one dimensional case. Finally, a few numerical examples are provided to illustrate some corresponding analytic results.

MSC:
35J57 Boundary value problems for second-order elliptic systems
35K57 Reaction-diffusion equations
92C45 Kinetics in biochemical problems (pharmacokinetics, enzyme kinetics, etc.)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces,, SIAM Rev., 18, 620, (1976) · Zbl 0345.47044
[2] J. Billingham, A note on the properties of a family of travelling wave solutions arising in cubic autocatalysis,, Dyn. Stab. Syst., 6, 33, (1991) · Zbl 0737.35031
[3] T. K. Callahan, Pattern formation in three-dimensional reaction-diffusion systems,, Phys. D, 132, 339, (1999) · Zbl 0935.35065
[4] J. B. Conway, <em>A Course in Functional Analysis</em>,, Springer-Verlag, (1985) · Zbl 0558.46001
[5] J. M. Corbel, Strobes: pyrotechnic compositions that show a curious oscillatory combustion,, Angew. Chem. Int. Ed. Engl., 52, 290, (2013)
[6] M. G. Crandall, Bifurcation, perturbation of simple eigenvalues and linearized stability,, Arch. Rat. Mech. Anal., 52, 161, (1973) · Zbl 0275.47044
[7] F. A. Davidson, A priori bounds and global existence of solutions of the steady-state Sel’kov model,, Proc. Roy. Soc. Edinburgh A, 130, 507, (2000) · Zbl 0960.35026
[8] V. Gaspar, Depressing the bistable behavior of the iodate-arsenous acid reaction in a continuous flow stirred tank reactor by the effect of chloride or bromide ions: A method for determination of rate constants,, J. Phys. Chem., 90, 6303, (1986)
[9] D. Gilgarg, <em>Elliptic Partial Differential Equations of Second Order</em>,, Spring-Verlag, (1977)
[10] P. Gray, Autocatalytic reactions in the isothermal, continuous stirred tank reactor: Oscillations and instabilities in the system \(A+2B → 3B; B→ C\),, Chem. Eng. Sci., 39, 1087, (1984)
[11] J. K. Hale, Exact homoclinic and heteroclinic solutions of the Gray-Scott model for autocatalysis,, SIAM J. Appl. Math., 61, 102, (2000) · Zbl 0965.34037
[12] B. D. Hassard, <em>Theory and Applications of Hopf Bifurcation</em>,, Cambridge University Press, (1981) · Zbl 0474.34002
[13] W. Hordijk, Autocatalytic sets and biological specificity,, Bull. Math. Biol., 76, 201, (2014) · Zbl 1311.92070
[14] D. Horváth, Instabilities in propagating reaction-diffusion fronts,, J. Chem. Phys., 98, 6332, (1993)
[15] Y. Li, Stability of traveling front solutions with algebraic spatial decay for some autocatalytic chemical reaction systems,, SIAM J. Math. Anal., 44, 1474, (2012) · Zbl 1259.35030
[16] G. M. Lieberman, Bounds for the steady-state Sel’kov model for arbitrary \(p\) in any number of dimensions,, SIAM J. Math. Anal., 36, 1400, (2005) · Zbl 1112.35062
[17] Y. Lou, Diffusion, self-diffusion and cross-diffusion,, J. Differential Equations, 131, 79, (1996) · Zbl 0867.35032
[18] A. Malevanets, Biscale chaos in propagating fronts,, Phys. Rev. E, 52, 4724, (1995)
[19] J. E. Marsden, <em>The Hopf Bifurcation and Its Applications</em>,, Springer-Verlag, (1976)
[20] J. H. Merkin, Travelling waves in the iodate-arsenous acid system,, Phys. Chem. Chem. Phys., 1, 91, (1999)
[21] M. J. Metcalf, Oscillating wave fronts in isothermal chemical systems with arbitrary powers of autocatalysis,, Proc. Roy. Soc. London A, 447, 155, (1994) · Zbl 0809.92027
[22] A. H. Msmali, Quadratic autocatalysis with non-linear decay,, J. Math. Chem., 52, 2234, (2014) · Zbl 1307.80007
[23] W.-M. Ni, Turing patterns in the Lengyel-Epstein system for the CIMA reactions,, Trans. Amer. Math. Soc., 357, 3953, (2005) · Zbl 1074.35051
[24] G. Nicolis, Patterns of spatio-temporal organization in chemical and biochemical kinetics,, SIAM-AMS Proc., 8, 33, (1974)
[25] P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems,, J. Functional Analysis, 7, 487, (1971) · Zbl 0212.16504
[26] A. M. Turing, The chemical basis of morphogenesis,, Phil. Trans. Roy. Soc. London Ser. B, 237, 37, (1952) · Zbl 1403.92034
[27] M. Wang, Non-constant positive steady states of the Sel’kov model,, J. Differential Equations, 190, 600, (2003) · Zbl 1163.35362
[28] J. H. Wu, Global bifurcation of coexistence state for the competition model in the chemostat,, Nonlinear Anal., 39, 817, (2000) · Zbl 0940.35114
[29] Y. Zhao, Steady states and dynamics of an autocatalytic chemical reaction model with decay,, J. Differential Equations, 253, 533, (2012) · Zbl 1258.35115
[30] J. Zhou, Qualitative analysis of an autocatalytic chemical reaction model with decay,, Proc. Roy. Soc. Edinburgh A, 144, 427, (2014) · Zbl 1292.35150
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.