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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 & Continuous Dynamical Systems - A, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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J. B. Conway, A Course in Functional Analysis, Springer-Verlag, New York, 1985. doi: 10.1007/978-1-4757-3828-5.  Google Scholar

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J. M. CorbelJ. N. van LingenJ. F. ZevenbergenO. 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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

D. Gilgarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Spring-Verlag, New York, 1977.  Google Scholar

[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; BC, Chem. Eng. Sci., 39 (1984), 1087-1097.   Google Scholar

[11]

J. K. HaleL. 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.  Google Scholar

[12]

B. D. Hassard, N. D. Kazarinoff and Y. -H. Wan, Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge, 1981.  Google Scholar

[13]

W. HordijkP. R. Wills and M. Steel, Autocatalytic sets and biological specificity, Bull. Math. Biol., 76 (2014), 201-224.  doi: 10.1007/s11538-013-9916-4.  Google Scholar

[14]

D. HorváthV. PetrovS. K. Scott and K. Showalter, Instabilities in propagating reaction-diffusion fronts, J. Chem. Phys., 98 (1993), 6332-6343.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

A. MalevanetsA. Careta and R. Kapral, Biscale chaos in propagating fronts, Phys. Rev. E, 52 (1995), 4724-4735.  doi: 10.1103/PhysRevE.52.4724.  Google Scholar

[19]

J. E. Marsden and M. McCracken, The Hopf Bifurcation and Its Applications, Springer-Verlag, New York, 1976.  Google Scholar

[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.  Google Scholar

[21]

M. J. MetcalfJ. 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.  Google Scholar

[22]

A. H. MsmaliM. 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.  Google Scholar

[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.  Google Scholar

[24]

G. Nicolis, Patterns of spatio-temporal organization in chemical and biochemical kinetics, SIAM-AMS Proc., 8 (1974), 33-58.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

Y. ZhaoY. 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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

J. B. Conway, A Course in Functional Analysis, Springer-Verlag, New York, 1985. doi: 10.1007/978-1-4757-3828-5.  Google Scholar

[5]

J. M. CorbelJ. N. van LingenJ. F. ZevenbergenO. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

D. Gilgarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Spring-Verlag, New York, 1977.  Google Scholar

[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; BC, Chem. Eng. Sci., 39 (1984), 1087-1097.   Google Scholar

[11]

J. K. HaleL. 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.  Google Scholar

[12]

B. D. Hassard, N. D. Kazarinoff and Y. -H. Wan, Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge, 1981.  Google Scholar

[13]

W. HordijkP. R. Wills and M. Steel, Autocatalytic sets and biological specificity, Bull. Math. Biol., 76 (2014), 201-224.  doi: 10.1007/s11538-013-9916-4.  Google Scholar

[14]

D. HorváthV. PetrovS. K. Scott and K. Showalter, Instabilities in propagating reaction-diffusion fronts, J. Chem. Phys., 98 (1993), 6332-6343.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

A. MalevanetsA. Careta and R. Kapral, Biscale chaos in propagating fronts, Phys. Rev. E, 52 (1995), 4724-4735.  doi: 10.1103/PhysRevE.52.4724.  Google Scholar

[19]

J. E. Marsden and M. McCracken, The Hopf Bifurcation and Its Applications, Springer-Verlag, New York, 1976.  Google Scholar

[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.  Google Scholar

[21]

M. J. MetcalfJ. 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.  Google Scholar

[22]

A. H. MsmaliM. 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.  Google Scholar

[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.  Google Scholar

[24]

G. Nicolis, Patterns of spatio-temporal organization in chemical and biochemical kinetics, SIAM-AMS Proc., 8 (1974), 33-58.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

Y. ZhaoY. 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.  Google Scholar

[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.  Google Scholar

Figure 1.  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 $(u_0, v_0)=(1.2, 0.9)$
Figure 2.  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 $(u_0, v_0)=(1.3, 0.8)$
Figure 3.  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 $(u_0, v_0)=(2.1, 1.9)$
Figure 4.  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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