# American Institute of Mathematical Sciences

2015, 14(3): 1147-1167. doi: 10.3934/cpaa.2015.14.1147

## Steady-state solutions and stability for a cubic autocatalysis model

 1 College of Mathematics and Information Science, Shaanxi Normal University, Xi'an, 710062, China 2 College of Mathematics and Information Science, Shaanxi Normal University, Xi’an, Shaanxi 710119 3 Faculty of Science, Xi’an Jiaotong University, Xi’an 710049, China

Received  May 2014 Revised  October 2015 Published  March 2015

A reaction-diffusion system, based on the cubic autocatalytic reaction scheme, with the prescribed concentration boundary conditions is considered. The linear stability of the unique spatially homogeneous steady state solution is discussed in detail to reveal a necessary condition for the bifurcation of this solution. The spatially non-uniform stationary structures, especially bifurcating from the double eigenvalue, are studied by the use of Lyapunov-Schmidt technique and singularity theory. Further information about the multiplicity and stability of the bifurcation solutions are obtained. Numerical examples are presented to support our theoretical results.
Citation: Mei-hua Wei, Jianhua Wu, Yinnian He. Steady-state solutions and stability for a cubic autocatalysis model. Communications on Pure & Applied Analysis, 2015, 14 (3) : 1147-1167. doi: 10.3934/cpaa.2015.14.1147
##### References:
 [1] J. C. Tsai, Existence of traveling waves in a simple isothermal chemical system with the same order for autocatalysis and decay,, \emph{Quart. Appl. Math.}, 69 (2011), 123. [2] R. Peng and F. Yi, On spatiotemporal pattern formation in a diffusive bimolecular model,, \emph{Discrete Contin. Dyn. Syst. Ser. B}, 15 (2011), 217. doi: 10.3934/dcdsb.2011.15.217. [3] Y. You, Dynamics of three-component reversible Gray-Scott model,, \emph{Discrete Contin. Dyn. Syst. Ser. B}, 14 (2010), 1671. doi: 10.3934/dcdsb.2010.14.1671. [4] X. F. Chen and Y. W. Qi, Propagation of local disturbances in reaction diffusion systems modeling quadratic autocatalysis,, \emph{SIAM J. Appl. Math.}, 69 (2008), 273. doi: 10.1137/07070276X. [5] A. L. Kay, D. J. Needham and J. A. Leach, Travelling waves for a coupled, singular reaction-diffusion system arising from a model of fractional order autocatalysis with decay. I. Permanent form travelling waves,, \emph{Nonlinearity}, 16 (2003), 735. doi: 10.1088/0951-7715/16/2/322. [6] J. A. Leach and J. C. Wei, Pattern formation in a simple chemical system with general orders of autocatalysis and decay. I. Stability analysis,, \emph{Phys. D}, 180 (2003), 185. doi: 10.1016/S0167-2789(03)00065-4. [7] P. Gray and S. K. Scott, Autocatalytic reactions in the isothermal continuous stirred tank reactor: isolas and other forms of multistability,, \emph{Chem. Eng. Sci.}, 39 (1983), 29. [8] P. Gray and S. K. Scott, Autocatalytic reactions in the CSTR: oscillations and instabilities in the system $A + 2B\rightarrow 3B$; $B\rightarrow C$,, \emph{Chem. Eng. Sci.}, 39 (1984), 1087. [9] A. D'Anna, P. G. Lignola and S. K. Scott, The application of singularity theory to isothermal autocatalytic open systems,, \emph{Proc. Roy. Soc. A}, 403 (1986), 341. [10] B. Peng, S. K. Scott and K. Showalter, Period doubling and chaos in a three variable autocatalator,, \emph{J. Phys. Chem.}, 94 (1990), 5243. [11] D. T. Lynch, Chaotic behavior of reactions systems: mixed cubic and quadratic autocatalysis,, \emph{Chem. Eng. Sci.}, 47 (1992), 4435. [12] K. Alhumaizi and R. Aris, Chaos in a simple two-phase reactor,, \emph{Chaos Solitons Fractals}, 4 (1994), 1985. [13] H. I. Abdel-Gawad and A. M. El-Shrae, Approximate solutions to the two-cell cubic autocatalytic reaction model,, \emph{Kyungpook Math. J.}, 44 (2004), 187. [14] E. A. Elrifai, On cubic autocatalytic chemical reaction model, CSTR and invariants of knots,, \emph{Far East J. Appl. Math.}, 32 (2008), 435. [15] J. H. Merkin, D. J. Needham and S. K. Scott, Oscillatory chemical reactions in closed vessels,, \emph{Proc. Roy. Soc. London Ser. A}, 406 (1986), 299. [16] A. B. Finlayson and J. H. Merkin, Creation of spatial structure by an electric field applied to an ionic cubic autocatalator system,, \emph{J. Engrg. Math.}, 38 (2000), 279. doi: 10.1023/A:1004799200173. [17] L. S. Chen and D. D. Wang, A biochemical oscillation,, \emph{Acta Math. Sci. Ser. B Engl. Ed.}, 5 (1985), 261. [18] J. H. Merkin, D. J. Needham and S. K. Scott, On the creation, growth and extinction of oscillatory solutions for a simple pooled chemical reaction scheme,, \emph{SIAM J. Appl. Math.}, 47 (1987), 1040. doi: 10.1137/0147068. [19] J. H. Merkin and D. J. Needham, Reaction-diffusion in a simple pooled chemical system,, \emph{Dyn. Stab. Syst.}, 4 (1989), 141. doi: 10.1080/02681118908806069. [20] D. J. Needham and J. H. Merkin, Pattern formation through reaction and diffusion in a simple pooled-chemical system,, \emph{Dyn. Stab. Syst.}, 4 (1989), 259. doi: 10.1080/02681118908806076. [21] R. Hill, J. H. Merkin and D. J. Needham, Stable pattern and standing wave formation in a simple isothermal cubic autocatalytic reaction scheme,, \emph{J. Engrg. Math.}, 29 (1995), 413. doi: 10.1007/BF00043976. [22] J. Jang, W. M. Ni and M. Tang, Global bifurcation and structure of Turing patterns in the 1-D Lengyel-Epstein model,, \emph{J. Dynam. Differential Equations}, 16 (2005), 297. doi: 10.1007/s10884-004-2782-x. [23] M. H. Wei, J. H. Wu and G. H. Guo, Turing structures and stability for the 1-D Lengyel-Epstein system,, \emph{J. Math. Chem.}, 50 (2012), 2374. doi: 10.1007/s10910-012-0037-3. [24] M. G. Crandall and P. Rabinowitz, Bifurcation from simple eigenvalue,, \emph{J. Funct. Anal.}, 8 (1971), 321. [25] K. J. Brown, Local and global bifurcation results for a semilinear boundary value problem,, \emph{J. Differential Equations}, 239 (2007), 296. doi: 10.1016/j.jde.2007.05.013. [26] D. Schaeffer and M. Golubitsky, Bifurcation analysis near a double eigenvalue of a model chemical reaction,, \emph{Arch. Rational Mech. Anal.}, 75 (1981), 315. doi: 10.1007/BF00256382. [27] M. Golubitsky and D. Schaeffer, Singularities and Groups in Bifurcation Theory, Vol. I,, Springer, (1985). doi: 10.1007/978-1-4612-5034-0. [28] M. Golubitsky and D. Schaeffer, Imperfect bifurcation in the presence of symmetry,, \emph{Comm. Math. Phys.}, 67 (1979), 205. [29] P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems,, \emph{J. Funct. Anal.}, 7 (1971), 487. [30] J. H. Wu, Global bifurcation of coexistence state for the competition model in the chemostat,, \emph{Nonlinear Anal.}, 39 (2000), 817. doi: 10.1016/S0362-546X(98)00250-8. [31] M. Golubitsky and D. Schaeffer, A theory for imperfect bifurcation via singularity theory,, \emph{Comm. Pure Appl. Math.}, 32 (1979), 21. doi: 10.1002/cpa.3160320103.

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##### References:
 [1] J. C. Tsai, Existence of traveling waves in a simple isothermal chemical system with the same order for autocatalysis and decay,, \emph{Quart. Appl. Math.}, 69 (2011), 123. [2] R. Peng and F. Yi, On spatiotemporal pattern formation in a diffusive bimolecular model,, \emph{Discrete Contin. Dyn. Syst. Ser. B}, 15 (2011), 217. doi: 10.3934/dcdsb.2011.15.217. [3] Y. You, Dynamics of three-component reversible Gray-Scott model,, \emph{Discrete Contin. Dyn. Syst. Ser. B}, 14 (2010), 1671. doi: 10.3934/dcdsb.2010.14.1671. [4] X. F. Chen and Y. W. Qi, Propagation of local disturbances in reaction diffusion systems modeling quadratic autocatalysis,, \emph{SIAM J. Appl. Math.}, 69 (2008), 273. doi: 10.1137/07070276X. [5] A. L. Kay, D. J. Needham and J. A. Leach, Travelling waves for a coupled, singular reaction-diffusion system arising from a model of fractional order autocatalysis with decay. I. Permanent form travelling waves,, \emph{Nonlinearity}, 16 (2003), 735. doi: 10.1088/0951-7715/16/2/322. [6] J. A. Leach and J. C. Wei, Pattern formation in a simple chemical system with general orders of autocatalysis and decay. I. Stability analysis,, \emph{Phys. D}, 180 (2003), 185. doi: 10.1016/S0167-2789(03)00065-4. [7] P. Gray and S. K. Scott, Autocatalytic reactions in the isothermal continuous stirred tank reactor: isolas and other forms of multistability,, \emph{Chem. Eng. Sci.}, 39 (1983), 29. [8] P. Gray and S. K. Scott, Autocatalytic reactions in the CSTR: oscillations and instabilities in the system $A + 2B\rightarrow 3B$; $B\rightarrow C$,, \emph{Chem. Eng. Sci.}, 39 (1984), 1087. [9] A. D'Anna, P. G. Lignola and S. K. Scott, The application of singularity theory to isothermal autocatalytic open systems,, \emph{Proc. Roy. Soc. A}, 403 (1986), 341. [10] B. Peng, S. K. Scott and K. Showalter, Period doubling and chaos in a three variable autocatalator,, \emph{J. Phys. Chem.}, 94 (1990), 5243. [11] D. T. Lynch, Chaotic behavior of reactions systems: mixed cubic and quadratic autocatalysis,, \emph{Chem. Eng. Sci.}, 47 (1992), 4435. [12] K. Alhumaizi and R. Aris, Chaos in a simple two-phase reactor,, \emph{Chaos Solitons Fractals}, 4 (1994), 1985. [13] H. I. Abdel-Gawad and A. M. El-Shrae, Approximate solutions to the two-cell cubic autocatalytic reaction model,, \emph{Kyungpook Math. J.}, 44 (2004), 187. [14] E. A. Elrifai, On cubic autocatalytic chemical reaction model, CSTR and invariants of knots,, \emph{Far East J. Appl. Math.}, 32 (2008), 435. [15] J. H. Merkin, D. J. Needham and S. K. Scott, Oscillatory chemical reactions in closed vessels,, \emph{Proc. Roy. Soc. London Ser. A}, 406 (1986), 299. [16] A. B. Finlayson and J. H. Merkin, Creation of spatial structure by an electric field applied to an ionic cubic autocatalator system,, \emph{J. Engrg. Math.}, 38 (2000), 279. doi: 10.1023/A:1004799200173. [17] L. S. Chen and D. D. Wang, A biochemical oscillation,, \emph{Acta Math. Sci. Ser. B Engl. Ed.}, 5 (1985), 261. [18] J. H. Merkin, D. J. Needham and S. K. Scott, On the creation, growth and extinction of oscillatory solutions for a simple pooled chemical reaction scheme,, \emph{SIAM J. Appl. Math.}, 47 (1987), 1040. doi: 10.1137/0147068. [19] J. H. Merkin and D. J. Needham, Reaction-diffusion in a simple pooled chemical system,, \emph{Dyn. Stab. Syst.}, 4 (1989), 141. doi: 10.1080/02681118908806069. [20] D. J. Needham and J. H. Merkin, Pattern formation through reaction and diffusion in a simple pooled-chemical system,, \emph{Dyn. Stab. Syst.}, 4 (1989), 259. doi: 10.1080/02681118908806076. [21] R. Hill, J. H. Merkin and D. J. Needham, Stable pattern and standing wave formation in a simple isothermal cubic autocatalytic reaction scheme,, \emph{J. Engrg. Math.}, 29 (1995), 413. doi: 10.1007/BF00043976. [22] J. Jang, W. M. Ni and M. Tang, Global bifurcation and structure of Turing patterns in the 1-D Lengyel-Epstein model,, \emph{J. Dynam. Differential Equations}, 16 (2005), 297. doi: 10.1007/s10884-004-2782-x. [23] M. H. Wei, J. H. Wu and G. H. Guo, Turing structures and stability for the 1-D Lengyel-Epstein system,, \emph{J. Math. Chem.}, 50 (2012), 2374. doi: 10.1007/s10910-012-0037-3. [24] M. G. Crandall and P. Rabinowitz, Bifurcation from simple eigenvalue,, \emph{J. Funct. Anal.}, 8 (1971), 321. [25] K. J. Brown, Local and global bifurcation results for a semilinear boundary value problem,, \emph{J. Differential Equations}, 239 (2007), 296. doi: 10.1016/j.jde.2007.05.013. [26] D. Schaeffer and M. Golubitsky, Bifurcation analysis near a double eigenvalue of a model chemical reaction,, \emph{Arch. Rational Mech. Anal.}, 75 (1981), 315. doi: 10.1007/BF00256382. [27] M. Golubitsky and D. Schaeffer, Singularities and Groups in Bifurcation Theory, Vol. I,, Springer, (1985). doi: 10.1007/978-1-4612-5034-0. [28] M. Golubitsky and D. Schaeffer, Imperfect bifurcation in the presence of symmetry,, \emph{Comm. Math. Phys.}, 67 (1979), 205. [29] P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems,, \emph{J. Funct. Anal.}, 7 (1971), 487. [30] J. H. Wu, Global bifurcation of coexistence state for the competition model in the chemostat,, \emph{Nonlinear Anal.}, 39 (2000), 817. doi: 10.1016/S0362-546X(98)00250-8. [31] M. Golubitsky and D. Schaeffer, A theory for imperfect bifurcation via singularity theory,, \emph{Comm. Pure Appl. Math.}, 32 (1979), 21. doi: 10.1002/cpa.3160320103.
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