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

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)

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  • 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.

    Mathematics Subject Classification: Primary: 35J55, 35K57; Secondary: 35Q80, 92C45.


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  • 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 $(u_0, v_0)=(1.2, 1.1)$

  •   H. Amann , Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev., 18 (1976) , 620-709.  doi: 10.1137/1018114.
      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.
      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.
      J. B. Conway, A Course in Functional Analysis, Springer-Verlag, New York, 1985. doi: 10.1007/978-1-4757-3828-5.
      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.
      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.
      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.
      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.
      D. Gilgarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Spring-Verlag, New York, 1977.
      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. 
      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.
      B. D. Hassard, N. D. Kazarinoff and Y. -H. Wan, Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge, 1981.
      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.
      D. Horváth , V. Petrov , S. K. Scott  and  K. Showalter , Instabilities in propagating reaction-diffusion fronts, J. Chem. Phys., 98 (1993) , 6332-6343. 
      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.
      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.
      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.
      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.
      J. E. Marsden and M. McCracken, The Hopf Bifurcation and Its Applications, Springer-Verlag, New York, 1976.
      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.
      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.
      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.
      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.
      G. Nicolis , Patterns of spatio-temporal organization in chemical and biochemical kinetics, SIAM-AMS Proc., 8 (1974) , 33-58. 
      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.
      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.
      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.
      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.
      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.
      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.
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