# American Institute of Mathematical Sciences

October  2021, 14(10): 3785-3801. doi: 10.3934/dcdss.2020433

## Oscillatory states and patterns formation in a two-cell cubic autocatalytic reaction-diffusion model subjected to the Dirichlet conditions

 1 Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada 2 Department of Medical Research, China Medical University Hospital, Taichung 40204, Taiwan 3 Department of Mathematics and Informatics, Azerbaijan University, 71 Jeyhun Hajibeyli Street, AZ1007 Baku, Azerbaijan 4 Department of Mathematics, Faculty of Science, Cairo University, Al Orman, Giza 12613, Egypt 5 Department of Mathematics, College of Arts and Sciences, Najran University, Najran, Kingdom of Saudi Arabia 6 Department of Mathematics, Faculty of Applied Science, Taiz University, Taiz, Yemen

* Corresponding author: H. M. Srivastava

Received  November 2019 Revised  June 2020 Published  October 2021 Early access  November 2020

The approximate solutions of a two-cell reaction-diffusion model equation subjected to the Dirichlet conditions are obtained. The reaction is assumed to occur in the presence of cubic autocatalyst which decays to an inert compound in the first cell. Coupling with the reactant is assumed to be cubic in the concentrations. A linear exchange in the concentration of the reactant is taken between the two cells. The formal exact solution is found analytically. Here, in this investigation, use is made of the Picard iterative scheme which is constructed and applied after the exact one. The results obtained are compared with those found by means of a numerical method. It is observed that the solution obtained here is symmetric with respect to the mid-point of the container.The travelling wave is expected due to the parity of the space operator and the symmetric boundary conditions. Symmetric patterns, including among them a parabolic one, are observed for a large time.

When the initial conditions are periodic, the most dominant modes travel at a constant speed for a large time. This phenomenon is highly affected by the rate of decay of the autocatalyst to an inert compound. The present work is of remarkably significant interest in chemical engineering as well as in other physical sciences. For example, in chemical industry, the objective is to achieve a great yield of a given product, which is carried by controlling the initial concentration of the reactant. Furthermore, in the last section on conclusions, we have cited many potentially useful recent works related to the subject-matter of this investigation in order to provide incentive and motivation for making further advances by using space-time fractional derivatives along the lines of the problem of finding approximate analytical solutions of the reaction-diffusion model equations which we have discussed in this article.

Citation: H. M. Srivastava, H. I. Abdel-Gawad, Khaled Mohammed Saad. Oscillatory states and patterns formation in a two-cell cubic autocatalytic reaction-diffusion model subjected to the Dirichlet conditions. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3785-3801. doi: 10.3934/dcdss.2020433
##### References:
 [1] H. I. Abdel-Gawad and K. M. Saad, On the behaviour of soultions of the two-cell cubic autocatalator, ANZIAM J., 44 (2002), E1–E32. doi: 10.1017/S1446181100007859. [2] H. I. Abdel-Gawad and H. A. Abdusalam, Approximate solutions of the Kuramoto-Sivashinsky equation for periodic boundary value problems and chaos, Chaos Solitons Fract., 12 (2001), 2039-2050.  doi: 10.1016/S0960-0779(00)00142-9. [3] P. Arcuri and J. D. Murray, Pattern sensitivity to boundary conditions in reaction-diffusion models, J. Math. Biol., 24 (1986), 141-165.  doi: 10.1007/BF00275996. [4] N. F. Britton, Reaction-Diffusion Equations and Their Applications to Biology, Academic Press, New York, 1986. [5] L. Debnath, Nonlinear Partial Differential Equations for Scientists and Engineers, Birkhäuser, Basel and Boston, 1997. doi: 10.1007/978-1-4899-2846-7. [6] I. R. Epstein and J. A. Pojman, An Introduction to Nonlinear Chemical Dynamics: Oscillations, Waves, Patterns and Chaos, Clarendon (Oxford University) Press, Oxford, London and New York, 1998. [7] R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophys. J., 1 (1961), 445-466.  doi: 10.1016/S0006-3495(61)86902-6. [8] L. K. Forbes, On stability and uniqueness of stationary one-dimensional patterns in the Belousov-Zhabotinskii reaction, Physica D: Nonlinear Phenom., 50 (1991), 42-58.  doi: 10.1016/0167-2789(91)90077-M. [9] W. Jager, J. Moser and R. Renmert, Modelling of Patterns in Space and Time, Springer Lectures in Biomathematics, Springer-Verlag, Berlin, Heidelberg and New York, 1984. [10] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematical Studies, 204, Elsevier (North-Holland) Science Publishers, Amsterdam, London and New York, 2006. [11] A. Kolmogoroff, I. Petrovsky and N. Piscounoff, Etude de l'equation de la diffusion avec croissance de la quantité de matie'er et son application a un problém biologique, Moscow Univ. Bull. Math., 1 (1937), 1-25. [12] J. A. Leach, J. H. Merkin and S. K. Scott, Oscillations and waves in the Belousov-Zhabotinskii reaction in a finite medium, J. Math. Chem., 16 (1994), 115-124.  doi: 10.1007/BF01169200. [13] D. Luss, M. Golubitsky and S. Strogatz, Pattern Formation in Continuous and Coupled Systems, IMA Volumes in Mathematics and Its Applications, Springer-Verlag, Berlin, Heidelberg and New York, 1999. doi: 10.1007/978-1-4612-1558-5. [14] P. K. Maini, Spatial and spatiotemporal pattern formation in generalised Turing systems, Comput. Math. Appl., 32 (1996), 71-77.  doi: 10.1016/S0898-1221(96)00198-8. [15] T. R. Marchant, Cubic autocatalytic reaction-diffusion equations$:$ Semi-analytical solutions, Proc. Roy. Soc. London Ser. A Math. Phys. Engrg. Sci., 458 (2002), 1-16.  doi: 10.1098/rspa.2001.0899. [16] J. H. Merkin, D. J. Needham and S. K. Scott, Coupled reaction-diffusion waves in an isothermal autocatalytic chemical system, IMA J. Appl. Math., 50 (1993), 43-76.  doi: 10.1093/imamat/50.1.43. [17] J. D. Murray, Mathematical Biology, Springer-Verlag, Berlin, Heidelberg and New York, 1989. doi: 10.1007/978-3-662-08539-4. [18] I. Prigogine and R. Lefever, Symmetry breaking instabilities in dissipative system. Ⅱ, AIP J. Chem. Phys., 48 (1968), 1-7.  doi: 10.1063/1.1668896. [19] K. M. Saad and A. M. El-Shrae, Travelling waves in a cubic autocatalytic reaction, Adv. Appl. Math. Sci., 8 (2011), 87-104. [20] K. M. Saad and E. H. F. Al-Sharif, Comparative study of a cubic autocatalytic reaction via different analysis methods, Discrete Continuous Dyn. Syst. Ser. S, 12 (2019), 665-684.  doi: 10.3934/dcdss.2019042. [21] K. M. Saad, H. M. Srivastava and J. F. Gómez-Aguilar, A fractional quadratic autocatalysis associated with chemical clock reactions involving linear inhibition, Chaos Solitons Fractals, 132 (2020), 1-9.  doi: 10.1016/j.chaos.2019.109557. [22] R. A. Satnoianu, M. Menzinger and P. K. Maini, Turing instabilities in general systems, J. Math. Biol., 41 (2000), 493-512.  doi: 10.1007/s002850000056. [23] E. E. Sel'kov, Self-oscillations in glycolysis. $1:$ A simple kinetic model, European J. Biochem., 4 (1968), 79-86.  doi: 10.1111/j.1432-1033.1968.tb00175.x. [24] H. M. Srivastava and K. M. Saad, Some new models of the time-fractional gas dynamics equation, Adv. Math. Models Appl., 3 (2018), 5-17. [25] H. M. Srivastava and K. M. Saad, New approximate solution of the time-fractional Nagumo equation involving fractional integrals without singular kernel, Appl. Math. Inform. Sci., 14 (2020), 1-8.  doi: 10.18576/amis/140101. [26] H. M. Srivastava, K. M. Saad and E. H. F. Al-Sharif, New analysis of the time-fractional and space-time fractional-order Nagumo equation, J. Inform. Math. Sci., 10 (2018), 545-561. [27] H. M. Srivastava, H. I. Abdel-Gawad and K. M. Saad, Stability of traveling waves based upon the Evans function and Legendre polynomials, Appl. Sci., 10 (2020), 1-16.  doi: 10.3390/app10030846. [28] H. M. Srivastava, Fractional-order derivatives and integrals: Introductory overview and recent developments, Kyungpook Math. J., 60 (2020), 73-116. [29] A. M. Turing, The chemical basis of morphogenesis, Philos. Trans. Roy. Soc. London Ser. B Biol. Sci., 237 (1952), 37-72.  doi: 10.1098/rstb.1952.0012. [30] J. J. Tyson, Classification of instabilities in chemical reaction systems, AIP J. Chem. Phys., 62 (1975), 1-7.  doi: 10.1063/1.430567.

show all references

##### References:
 [1] H. I. Abdel-Gawad and K. M. Saad, On the behaviour of soultions of the two-cell cubic autocatalator, ANZIAM J., 44 (2002), E1–E32. doi: 10.1017/S1446181100007859. [2] H. I. Abdel-Gawad and H. A. Abdusalam, Approximate solutions of the Kuramoto-Sivashinsky equation for periodic boundary value problems and chaos, Chaos Solitons Fract., 12 (2001), 2039-2050.  doi: 10.1016/S0960-0779(00)00142-9. [3] P. Arcuri and J. D. Murray, Pattern sensitivity to boundary conditions in reaction-diffusion models, J. Math. Biol., 24 (1986), 141-165.  doi: 10.1007/BF00275996. [4] N. F. Britton, Reaction-Diffusion Equations and Their Applications to Biology, Academic Press, New York, 1986. [5] L. Debnath, Nonlinear Partial Differential Equations for Scientists and Engineers, Birkhäuser, Basel and Boston, 1997. doi: 10.1007/978-1-4899-2846-7. [6] I. R. Epstein and J. A. Pojman, An Introduction to Nonlinear Chemical Dynamics: Oscillations, Waves, Patterns and Chaos, Clarendon (Oxford University) Press, Oxford, London and New York, 1998. [7] R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophys. J., 1 (1961), 445-466.  doi: 10.1016/S0006-3495(61)86902-6. [8] L. K. Forbes, On stability and uniqueness of stationary one-dimensional patterns in the Belousov-Zhabotinskii reaction, Physica D: Nonlinear Phenom., 50 (1991), 42-58.  doi: 10.1016/0167-2789(91)90077-M. [9] W. Jager, J. Moser and R. Renmert, Modelling of Patterns in Space and Time, Springer Lectures in Biomathematics, Springer-Verlag, Berlin, Heidelberg and New York, 1984. [10] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematical Studies, 204, Elsevier (North-Holland) Science Publishers, Amsterdam, London and New York, 2006. [11] A. Kolmogoroff, I. Petrovsky and N. Piscounoff, Etude de l'equation de la diffusion avec croissance de la quantité de matie'er et son application a un problém biologique, Moscow Univ. Bull. Math., 1 (1937), 1-25. [12] J. A. Leach, J. H. Merkin and S. K. Scott, Oscillations and waves in the Belousov-Zhabotinskii reaction in a finite medium, J. Math. Chem., 16 (1994), 115-124.  doi: 10.1007/BF01169200. [13] D. Luss, M. Golubitsky and S. Strogatz, Pattern Formation in Continuous and Coupled Systems, IMA Volumes in Mathematics and Its Applications, Springer-Verlag, Berlin, Heidelberg and New York, 1999. doi: 10.1007/978-1-4612-1558-5. [14] P. K. Maini, Spatial and spatiotemporal pattern formation in generalised Turing systems, Comput. Math. Appl., 32 (1996), 71-77.  doi: 10.1016/S0898-1221(96)00198-8. [15] T. R. Marchant, Cubic autocatalytic reaction-diffusion equations$:$ Semi-analytical solutions, Proc. Roy. Soc. London Ser. A Math. Phys. Engrg. Sci., 458 (2002), 1-16.  doi: 10.1098/rspa.2001.0899. [16] J. H. Merkin, D. J. Needham and S. K. Scott, Coupled reaction-diffusion waves in an isothermal autocatalytic chemical system, IMA J. Appl. Math., 50 (1993), 43-76.  doi: 10.1093/imamat/50.1.43. [17] J. D. Murray, Mathematical Biology, Springer-Verlag, Berlin, Heidelberg and New York, 1989. doi: 10.1007/978-3-662-08539-4. [18] I. Prigogine and R. Lefever, Symmetry breaking instabilities in dissipative system. Ⅱ, AIP J. Chem. Phys., 48 (1968), 1-7.  doi: 10.1063/1.1668896. [19] K. M. Saad and A. M. El-Shrae, Travelling waves in a cubic autocatalytic reaction, Adv. Appl. Math. Sci., 8 (2011), 87-104. [20] K. M. Saad and E. H. F. Al-Sharif, Comparative study of a cubic autocatalytic reaction via different analysis methods, Discrete Continuous Dyn. Syst. Ser. S, 12 (2019), 665-684.  doi: 10.3934/dcdss.2019042. [21] K. M. Saad, H. M. Srivastava and J. F. Gómez-Aguilar, A fractional quadratic autocatalysis associated with chemical clock reactions involving linear inhibition, Chaos Solitons Fractals, 132 (2020), 1-9.  doi: 10.1016/j.chaos.2019.109557. [22] R. A. Satnoianu, M. Menzinger and P. K. Maini, Turing instabilities in general systems, J. Math. Biol., 41 (2000), 493-512.  doi: 10.1007/s002850000056. [23] E. E. Sel'kov, Self-oscillations in glycolysis. $1:$ A simple kinetic model, European J. Biochem., 4 (1968), 79-86.  doi: 10.1111/j.1432-1033.1968.tb00175.x. [24] H. M. Srivastava and K. M. Saad, Some new models of the time-fractional gas dynamics equation, Adv. Math. Models Appl., 3 (2018), 5-17. [25] H. M. Srivastava and K. M. Saad, New approximate solution of the time-fractional Nagumo equation involving fractional integrals without singular kernel, Appl. Math. Inform. Sci., 14 (2020), 1-8.  doi: 10.18576/amis/140101. [26] H. M. Srivastava, K. M. Saad and E. H. F. Al-Sharif, New analysis of the time-fractional and space-time fractional-order Nagumo equation, J. Inform. Math. Sci., 10 (2018), 545-561. [27] H. M. Srivastava, H. I. Abdel-Gawad and K. M. Saad, Stability of traveling waves based upon the Evans function and Legendre polynomials, Appl. Sci., 10 (2020), 1-16.  doi: 10.3390/app10030846. [28] H. M. Srivastava, Fractional-order derivatives and integrals: Introductory overview and recent developments, Kyungpook Math. J., 60 (2020), 73-116. [29] A. M. Turing, The chemical basis of morphogenesis, Philos. Trans. Roy. Soc. London Ser. B Biol. Sci., 237 (1952), 37-72.  doi: 10.1098/rstb.1952.0012. [30] J. J. Tyson, Classification of instabilities in chemical reaction systems, AIP J. Chem. Phys., 62 (1975), 1-7.  doi: 10.1063/1.430567.
The concentration of the reactant $\alpha_{1}$ displayed against $x$ for $\gamma = 0.1$, $k = 0.01$ and $L = 100$ in (a), (b) and (c) for $t = 30$, and $t = 100$, respectively
The concentration of the autocatalyst $\beta$ displayed against $x$ for $\gamma = 0.1$, $k = 0.01$ and $L = 100$ in (a), (b) and (c) for $t = 30$, and $t = 100$, respectively
The concentration of the reactant $\alpha_{2}$ displayed against $x$ for $\gamma = 0.1$, $k = 0.01$ and $L = 100$ in (a), (b) and (c) for $t = 30$, and $100$, respectively
The concentration of $\alpha_{1}$, $\beta$ and $\alpha_{2}$ displayed against $x$ for $\gamma = 0.6$, $k = 0.09$ and $L = 100$. $(\cdots)$ for $t = 0$, $(–)$ for $t = 30$ and $(-)$ for $t = 100$, respectively
Contour plots of $\alpha_{1}$, $\beta$ and $\alpha_{2}$, respectively, for $\gamma = 0.09$, $k = 0.01$ and $L = 100$
Approximate analytical and numerical solutions of $\alpha_{i}(x,t)$ and $\beta(x,t)$ displayed against $x$ for $t = 50$, $k = 0.09$, $\gamma = 0.03$ and $L = 10$. $(-)$ for approximate analytical solutions; $(–)$ for numerical solutions. The error estimate is of order $($$10^{-8})$
Approximate analytical and numerical solutions of $\alpha_{i}(x,t)$ and $\beta(x,t)$ displayed against $x$ for $t = 50$, $k = 0.01$, $\gamma = 0.001$ and $L = 100$. $(-)$ for approximate analytical solutions; $(–)$ for numerical solutions.Th error estimate is of order $(10^{-4})$
The absolute error between the approximate analytical and numerical solutions of $\alpha_{i}(x,t)$ and $\beta(x,t)$ displayed against $x$, $\alpha_{1}(x,t)$ (solid line), $\beta(x,t)$ (doted-line) and $\alpha_{2}(x,t)$ (dashed line) for $k = 0.09$, $\gamma = 0.1$ and $L = 100$. (a) $t = 30$, (b) $t = 50$ and (c) $t = 100$. The error estimate is of order $(10^{-3}$)
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