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September  2017, 37(9): 4785-4813. doi: 10.3934/dcds.2017206

## Qualitative analysis on positive steady-states for an autocatalytic reaction model in thermodynamics

 1 School of Mathematics and Information Science, Shaanxi Normal University, Xi'an, Shaanxi 710062, China 2 Department of Mathematics, California State University, Northridge, CA 91325, USA

* Corresponding author (J.Wu): jianhuaw@snnu.edu.cn

Received  July 2016 Revised  April 2017 Published  June 2017

Fund Project: The work is supported by the Natural Science Foundations of China(11271236,11671243,61672021), the Shaanxi New-star Plan of Science and Technology of China(2015KJXX-21) and the Fundamental Research Funds for the Central University(GK201701001).

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.

Citation: Yunfeng Jia, Yi Li, Jianhua Wu. Qualitative analysis on positive steady-states for an autocatalytic reaction model in thermodynamics. Discrete and Continuous Dynamical Systems, 2017, 37 (9) : 4785-4813. doi: 10.3934/dcds.2017206
##### References:
 [1] H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev., 18 (1976), 620-709.  doi: 10.1137/1018114. [2] J. Billingham and D. J. Needham, A note on the properties of a family of travelling wave solutions arising in cubic autocatalysis, Dyn. Stab. Syst., 6 (1991), 33-49.  doi: 10.1080/02681119108806105. [3] T. K. Callahan and E. Knobloch, Pattern formation in three-dimensional reaction-diffusion systems, Phys. D, 132 (1999), 339-362.  doi: 10.1016/S0167-2789(99)00041-X. [4] J. B. Conway, A Course in Functional Analysis, Springer-Verlag, New York, 1985. doi: 10.1007/978-1-4757-3828-5. [5] J. M. Corbel, J. N. van Lingen, J. F. Zevenbergen, O. L. Gijzema and A. Meijerink, Strobes: pyrotechnic compositions that show a curious oscillatory combustion, Angew. Chem. Int. Ed. Engl., 52 (2013), 290-303.  doi: 10.1002/anie.201207398. [6] M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rat. Mech. Anal., 52 (1973), 161-180.  doi: 10.1007/BF00282325. [7] F. A. Davidson and B. P. Rynne, A priori bounds and global existence of solutions of the steady-state Sel'kov model, Proc. Roy. Soc. Edinburgh A, 130 (2000), 507-516.  doi: 10.1017/S0308210500000275. [8] V. Gaspar and M. T. Beck, 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 (1986), 6303-6305.  doi: 10.1021/j100281a048. [9] D. Gilgarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Spring-Verlag, New York, 1977. [10] P. Gray and S. K. Scott, Autocatalytic reactions in the isothermal, continuous stirred tank reactor: Oscillations and instabilities in the system A + 2B → 3B; B → C, Chem. Eng. Sci., 39 (1984), 1087-1097. [11] J. K. Hale, L. A. Peletier and W. C. Troy, Exact homoclinic and heteroclinic solutions of the Gray-Scott model for autocatalysis, SIAM J. Appl. Math., 61 (2000), 102-130.  doi: 10.1137/S0036139998334913. [12] B. D. Hassard, N. D. Kazarinoff and Y. -H. Wan, Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge, 1981. [13] W. Hordijk, P. R. Wills and M. Steel, Autocatalytic sets and biological specificity, Bull. Math. Biol., 76 (2014), 201-224.  doi: 10.1007/s11538-013-9916-4. [14] D. Horváth, V. Petrov, S. K. Scott and K. Showalter, Instabilities in propagating reaction-diffusion fronts, J. Chem. Phys., 98 (1993), 6332-6343. [15] Y. Li and Y. Wu, Stability of traveling front solutions with algebraic spatial decay for some autocatalytic chemical reaction systems, SIAM J. Math. Anal., 44 (2012), 1474-1521.  doi: 10.1137/100814974. [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 (2005), 1400-1406.  doi: 10.1137/S003614100343651X. [17] Y. Lou and W.-M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differential Equations, 131 (1996), 79-131.  doi: 10.1006/jdeq.1996.0157. [18] A. Malevanets, A. Careta and R. Kapral, Biscale chaos in propagating fronts, Phys. Rev. E, 52 (1995), 4724-4735.  doi: 10.1103/PhysRevE.52.4724. [19] J. E. Marsden and M. McCracken, The Hopf Bifurcation and Its Applications, Springer-Verlag, New York, 1976. [20] J. H. Merkin and H. Sevcikova, Travelling waves in the iodate-arsenous acid system, Phys. Chem. Chem. Phys., 1 (1999), 91-97.  doi: 10.1039/a807837h. [21] M. J. Metcalf, J. H. Merkin and S. K. Scott, Oscillating wave fronts in isothermal chemical systems with arbitrary powers of autocatalysis, Proc. Roy. Soc. London A, 447 (1994), 155-174.  doi: 10.1098/rspa.1994.0133. [22] A. H. Msmali, M. I. Nelson and M. P. Edwards, Quadratic autocatalysis with non-linear decay, J. Math. Chem., 52 (2014), 2234-2258.  doi: 10.1007/s10910-014-0382-5. [23] W.-M. Ni and M. Tang, Turing patterns in the Lengyel-Epstein system for the CIMA reactions, Trans. Amer. Math. Soc., 357 (2005), 3953-3969.  doi: 10.1090/S0002-9947-05-04010-9. [24] G. Nicolis, Patterns of spatio-temporal organization in chemical and biochemical kinetics, SIAM-AMS Proc., 8 (1974), 33-58. [25] P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Functional Analysis, 7 (1971), 487-513.  doi: 10.1016/0022-1236(71)90030-9. [26] A. M. Turing, The chemical basis of morphogenesis, Phil. Trans. Roy. Soc. London Ser. B, 237 (1952), 37-72.  doi: 10.1098/rstb.1952.0012. [27] M. Wang, Non-constant positive steady states of the Sel'kov model, J. Differential Equations, 190 (2003), 600-620.  doi: 10.1016/S0022-0396(02)00100-6. [28] J. H. Wu, Global bifurcation of coexistence state for the competition model in the chemostat, Nonlinear Anal., 39 (2000), 817-835.  doi: 10.1016/S0362-546X(98)00250-8. [29] Y. Zhao, Y. Wang and J. Shi, Steady states and dynamics of an autocatalytic chemical reaction model with decay, J. Differential Equations, 253 (2012), 533-552.  doi: 10.1016/j.jde.2012.03.018. [30] J. Zhou and J. Shi, Qualitative analysis of an autocatalytic chemical reaction model with decay, Proc. Roy. Soc. Edinburgh A, 144 (2014), 427-446.  doi: 10.1017/S0308210512001667.

show all references

##### References:
 [1] H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev., 18 (1976), 620-709.  doi: 10.1137/1018114. [2] J. Billingham and D. J. Needham, A note on the properties of a family of travelling wave solutions arising in cubic autocatalysis, Dyn. Stab. Syst., 6 (1991), 33-49.  doi: 10.1080/02681119108806105. [3] T. K. Callahan and E. Knobloch, Pattern formation in three-dimensional reaction-diffusion systems, Phys. D, 132 (1999), 339-362.  doi: 10.1016/S0167-2789(99)00041-X. [4] J. B. Conway, A Course in Functional Analysis, Springer-Verlag, New York, 1985. doi: 10.1007/978-1-4757-3828-5. [5] J. M. Corbel, J. N. van Lingen, J. F. Zevenbergen, O. L. Gijzema and A. Meijerink, Strobes: pyrotechnic compositions that show a curious oscillatory combustion, Angew. Chem. Int. Ed. Engl., 52 (2013), 290-303.  doi: 10.1002/anie.201207398. [6] M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rat. Mech. Anal., 52 (1973), 161-180.  doi: 10.1007/BF00282325. [7] F. A. Davidson and B. P. Rynne, A priori bounds and global existence of solutions of the steady-state Sel'kov model, Proc. Roy. Soc. Edinburgh A, 130 (2000), 507-516.  doi: 10.1017/S0308210500000275. [8] V. Gaspar and M. T. Beck, 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 (1986), 6303-6305.  doi: 10.1021/j100281a048. [9] D. Gilgarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Spring-Verlag, New York, 1977. [10] P. Gray and S. K. Scott, Autocatalytic reactions in the isothermal, continuous stirred tank reactor: Oscillations and instabilities in the system A + 2B → 3B; B → C, Chem. Eng. Sci., 39 (1984), 1087-1097. [11] J. K. Hale, L. A. Peletier and W. C. Troy, Exact homoclinic and heteroclinic solutions of the Gray-Scott model for autocatalysis, SIAM J. Appl. Math., 61 (2000), 102-130.  doi: 10.1137/S0036139998334913. [12] B. D. Hassard, N. D. Kazarinoff and Y. -H. Wan, Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge, 1981. [13] W. Hordijk, P. R. Wills and M. Steel, Autocatalytic sets and biological specificity, Bull. Math. Biol., 76 (2014), 201-224.  doi: 10.1007/s11538-013-9916-4. [14] D. Horváth, V. Petrov, S. K. Scott and K. Showalter, Instabilities in propagating reaction-diffusion fronts, J. Chem. Phys., 98 (1993), 6332-6343. [15] Y. Li and Y. Wu, Stability of traveling front solutions with algebraic spatial decay for some autocatalytic chemical reaction systems, SIAM J. Math. Anal., 44 (2012), 1474-1521.  doi: 10.1137/100814974. [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 (2005), 1400-1406.  doi: 10.1137/S003614100343651X. [17] Y. Lou and W.-M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differential Equations, 131 (1996), 79-131.  doi: 10.1006/jdeq.1996.0157. [18] A. Malevanets, A. Careta and R. Kapral, Biscale chaos in propagating fronts, Phys. Rev. E, 52 (1995), 4724-4735.  doi: 10.1103/PhysRevE.52.4724. [19] J. E. Marsden and M. McCracken, The Hopf Bifurcation and Its Applications, Springer-Verlag, New York, 1976. [20] J. H. Merkin and H. Sevcikova, Travelling waves in the iodate-arsenous acid system, Phys. Chem. Chem. Phys., 1 (1999), 91-97.  doi: 10.1039/a807837h. [21] M. J. Metcalf, J. H. Merkin and S. K. Scott, Oscillating wave fronts in isothermal chemical systems with arbitrary powers of autocatalysis, Proc. Roy. Soc. London A, 447 (1994), 155-174.  doi: 10.1098/rspa.1994.0133. [22] A. H. Msmali, M. I. Nelson and M. P. Edwards, Quadratic autocatalysis with non-linear decay, J. Math. Chem., 52 (2014), 2234-2258.  doi: 10.1007/s10910-014-0382-5. [23] W.-M. Ni and M. Tang, Turing patterns in the Lengyel-Epstein system for the CIMA reactions, Trans. Amer. Math. Soc., 357 (2005), 3953-3969.  doi: 10.1090/S0002-9947-05-04010-9. [24] G. Nicolis, Patterns of spatio-temporal organization in chemical and biochemical kinetics, SIAM-AMS Proc., 8 (1974), 33-58. [25] P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Functional Analysis, 7 (1971), 487-513.  doi: 10.1016/0022-1236(71)90030-9. [26] A. M. Turing, The chemical basis of morphogenesis, Phil. Trans. Roy. Soc. London Ser. B, 237 (1952), 37-72.  doi: 10.1098/rstb.1952.0012. [27] M. Wang, Non-constant positive steady states of the Sel'kov model, J. Differential Equations, 190 (2003), 600-620.  doi: 10.1016/S0022-0396(02)00100-6. [28] J. H. Wu, Global bifurcation of coexistence state for the competition model in the chemostat, Nonlinear Anal., 39 (2000), 817-835.  doi: 10.1016/S0362-546X(98)00250-8. [29] Y. Zhao, Y. Wang and J. Shi, Steady states and dynamics of an autocatalytic chemical reaction model with decay, J. Differential Equations, 253 (2012), 533-552.  doi: 10.1016/j.jde.2012.03.018. [30] J. Zhou and J. Shi, Qualitative analysis of an autocatalytic chemical reaction model with decay, Proc. Roy. Soc. Edinburgh A, 144 (2014), 427-446.  doi: 10.1017/S0308210512001667.
A three dimensional view of spatiotemporal pattern of solution of system (1). The equilibrium $U^*\approx(0.77, 1.3)$. The parameter values are endowed with $d_1=1, d_2=0.05, a=1.3$, and the initial conditions are set as $(u_0, v_0)=(1.2, 0.9)$
A three dimensional view of spatiotemporal pattern of solution of system (1). The equilibrium $U^*\approx(0.91, 1.1)$. The parameter values are endowed with $d_1=1, d_2=0.06, a=1.1$ and the initial conditions are set as $(u_0, v_0)=(1.3, 0.8)$
A three dimensional view of spatiotemporal periodic pattern of solution of system (1). The equilibrium $U^*\approx(1.02, 0.98)$. The parameter values are endowed with $d_1=1.2, d_2=0.08, a=0.98$ and the initial conditions are set as $(u_0, v_0)=(2.1, 1.9)$
A three dimensional view of spatiotemporal periodic pattern of solution of system (1). The equilibrium $U^*\approx(0.97, 1.03)$. The parameter values are endowed with $d_1=1.2, d_2=0.09, a=1.03$ and the initial conditions are set as $(u_0, v_0)=(1.2, 1.1)$

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