In everlasting memory of our beloved Maestro, Professor Salvatore Rionero, with never-ending admiration and affection. This paper develops a research topic suggested by Him and makes use of methodologies He taught us. “Imagination is more important than knowledge. Knowledge is limited.”
(A. Einstein).
Abstract
A reaction–diffusion model, known as the Sel’kov–Schnakenberg model, is considered. The nonlinear stability of the constant steady state is studied by using a special Liapunov functional and a maximum principle for regular solutions.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
The reaction–diffusion systems of P.D.Es are often used for modelling the chemical reactions (see for example [1,2,3,4,5,6,7,8,9,10,11], and the references quoted therein). In this paper we consider the system
known as the Sel’kov–Schnakenberg model, which is used to model some autocatalytic biochemical and chemical reactions. In (1) the functions U and V represent the reactant concentrations, \(a,\lambda \ge 0\) and \( b, \gamma >0\) are assigned constants and the positive constant d represents the scaled diffusion coefficient. The reactor \(\Omega \subset I\!\!R^3\) is a fixed open bounded domain, with boundary at least \(C^2,\) under the initial-boundary conditions
where \(U^*,\,V^*,\,U_0,\,V_0\) regular functions. System (1) contains as particular case the system introduced by Schnackenberg for trimolecular autocatalytic reactions and is contained in Segel–Jackson [8].
When \(\lambda =a=0,\) system (1) returns the well-known Sel’kov model, introduced in 1968 by Sel’kov [12], while if \(\lambda =0\) and \(a>0\), system (1) returns the well-known Schnackenberg model.
In 1979, Schnakenberg [2] introduced a simple chemical reaction model for glycolysis that showed limit cycle behaviour. The reaction scheme, known as Sel’kov–Schnakenberg model, occurs in the following three steps
where A and B are two chemical sources, C and D are autocatalyst and reactant, respectively. The most common examples of autocatalytic reactions are the chloride-iodide-malonic acid reaction and the reaction of phosphofructokinase glycolysis that includes adenosine triphosphate (ATP), adenosine diphosphate (ADP), and adenosine monophosphate (AMP). A wide literature has been dedicated to the study of (4), including the Sel’kov model and the Schnakenberg model (see for example [13,14,15,16,17,18]). Many papers show that the modeling problem (1) enjoys very rich and complex spatiotemporal patterns and dynamical structures. A generalized Sel’kov–Schnakenberg reaction–diffusion system is analyzed in [19], where the authors investigate the stability of the equilibrium, the effect of the diffusion on the stability and establish various conditions on the existence and nonexistence of nonconstant steady state solutions. In [20, 21] the authors study the steady state problem, show the formation of different interesting spatial patterns and the existence of non-constant steady state solutions. In [22], the authors study the case \(a=0\) and \(\lambda >0\) and perform interesting results such as, the existence of time-periodic orbits and non-constant steady-state solutions, which reveal the effect of various parameters on the existence and non-existence of spatiotemporal patterns. In a more recent work [23] the Selkov–Schnakenberg reaction–diffusion system has been approximated by the Galerkin method in order to perform semi-analytical solutions. In this paper we study the nonlinear stability of the steady state solution of (1); precisely, we investigate under what conditions on the model parameters, the conditions ensuring linear stability also ensure nonlinear stability. We claim that such a study provides challenges and ideas in many other fields of applied mathematics such as ecology, economics in which nonlinear mathematical models having a similar structure are considered (see for instance [15, 24,25,26,27] and references therein). The plan of the paper is as follows. In Sect. 2 we perform a maximum principle for regular solutions of system (1), while in Sect. 3 we introduce the perturbation problem and a peculiar Liapunov functional. A nonlinear stability result, based on the boundedness of the perturbation fields, is obtained in Sect. 4. The paper ends with some Conclusions.
2 General properties of regular solutions
Let \(T>0\) be an arbitrary but fixed time and \(\Omega _T=\Omega \times (0,T]\) the parabolic cylinder, \(\Omega _T\) being the parabolic interior of \(\bar{\Omega }\times [0,T]\) (i.e. \(\Omega _T\) includes the top \(\Omega \times \{t=T\})\). Moreover, we set \(\Gamma _T=\partial \Omega \times [0,T)\). Then, the parabolic boundary of \(\Omega _T\), \(\tilde{\Gamma }_T=\Gamma _T\cup (\Omega \times \{t=0\})\), includes the bottom and the vertical sides of \(\Omega \times [0,T]\), but not the top. The following theorem holds
Theorem 1
Let \(\{U,V\in C^2_1(\Omega _T)\cap C(\bar{\Omega }_T)\}\) be a positive solution of (1), (2), (3) where \(U^*,\,V^*,\,U_0,\,V_0\) positive continuous functions. Then
Proof
Let \(\displaystyle U(\textbf{x}_0,t_0)=\min _{\bar{\Omega }_T}U\). Two cases are possible (see [28])
-
i.
If \((\textbf{x}_0,t_0)\in \Omega _T\), then
$$\begin{aligned} \left( \frac{\partial U}{\partial t}\right) _{(\textbf{x}_0,t_0)}=0,\quad \left( \Delta U\right) _{(\textbf{x}_0,t_0)}\ge 0, \end{aligned}$$(6)from (1)\(_1\) it follows that
$$\begin{aligned} a-U(\textbf{x}_0,t_0)\le 0,\end{aligned}$$(7)and hence
$$\begin{aligned} U(\textbf{x}_0,t_0)\ge a. \end{aligned}$$(8) -
ii.
On the other hand, if \((\textbf{x}_0,t_0)\in \tilde{\Gamma }_T\) then
$$\begin{aligned} U(\textbf{x}_0,t_0)=\inf \left\{ \min _{\bar{\Omega }}U_0,\min _{\partial \Omega \times [0,T]}U^*\right\} . \end{aligned}$$(9)
From (1) and (2) we get (5)\(_1\).
Passing to V, let \(\displaystyle V(\bar{\textbf{x}},{\bar{t}})=\max _{\bar{\Omega }_T}V\).
-
a.
If \((\bar{\textbf{x}},{\bar{t}})\in \Omega _T,\) we find
$$\begin{aligned} \left( \frac{\partial V}{\partial t}\right) _{(\bar{\textbf{x}},{\bar{t}})}=0,\quad \left( \Delta V\right) _{(\bar{\textbf{x}},{\bar{t}})}\le 0,\end{aligned}$$(10)from (1)\(_2\) it turns out
$$\begin{aligned} b-\lambda V-U^2V\ge 0 \end{aligned}$$(11)and finally
$$\begin{aligned} V(\bar{\textbf{x}},{\bar{t}})\le \frac{b}{\lambda +m^2}. \end{aligned}$$(12) -
b.
On the other hand, if \((\bar{\textbf{x}},{\bar{t}})\in \tilde{\Gamma }_T,\)
$$\begin{aligned} V(\bar{\textbf{x}},{\bar{t}})=\sup \left\{ \max _{\bar{\Omega }}V_0,\max _{\partial \Omega \times [0,T]}V^*\right\} .\end{aligned}$$(13)
As in the previous case, we find (5)\(_2\). \(\square \)
3 Preliminaries to longtime behaviour of solutions
Constant steady states are the non-negative solutions of the system
System (14) admits the following equilibrium
Studying the longtime behaviour of solutions of system (1), is then reduced to study the stability of (15). It can be shown easily that the equations governing the perturbations \((C_1,C_2)\) to the basic steady state \((U^*,V^*)\) are
under the homogeneous Dirichlet boundary conditions
with
Following the Rionero’s method [29, 30] we introduce the scaling
with \(\mu \) a constant to be chosen suitably later. By using the above scaling in (16), denoting by \(\bar{\alpha }\) the lowest eigenvalue \(\beta \) of
in \(H_0^1(\Omega ),\) we get
where
Setting
the characteristic equation is given by
whose solutions are the eigenvalues of the following matrix
It is worth recalling that, the conditions implying the linear stability of (\(U^*,V^*\)) are the conditions guaranteeing \(\{I<0 \,\,\text {and}\,\, A>0\},\) i.e. the validity of the Routh–Hurwitz conditions, necessary and sufficient to guarantee that all the roots of (24) have negative real part [31].
Our goal is to study the nonlinear stability of the steady state solution of (1), and to this end, we introduce the following Liapunov functional
Along the solutions of (21) it turns out
where
Remark 1
Observe that W is equivalent to the usual \(L^2\)-norm [30]. In other words, there exist positive constants \(k_1\) and \(k_2\) such that
4 Nonlinear stability analysis via the boundedness of V
Theorem 2
Let
then \((U^*,V^*)\) is nonlinearly asymptotically stable with respect the \(L^2(\Omega )\)-norm.
Proof
Let us observe that
and that \(b_1<0\) implies that
Following [30], for any constant \(\bar{\varepsilon }\) such that
setting
we can write (21) as follows
where
and we observe that from conditions (32) the following inequalities hold
Along the solutions of (35), it turns out
where
Now choosing
it follows that \(\bar{\alpha }_3=0\) and hence
But
with \(k^*=\bar{\varepsilon }\,\inf (\bar{\alpha }_1,\bar{\alpha }_2).\)
Moreover
In order to prove the decay of W, and then the stability of (15), we have to control suitably the nonlinear terms in (43). By using the usual embedding theorems, this can be done for all the terms except, as far as we know, the strong nonlinear term \(<C_1^3 C_2>.\)
In the present section, we solve this problem by using the boundedness of the perturbation fields.
Coming back to (43), recall that
-
(a)
from Theorem 1, there exists a positive constant \(\Gamma _2\) such that
$$\begin{aligned} \vert C_2(\textbf{x},t)\vert \le \Gamma _2,\qquad \forall (\textbf{x},t)\in \Omega \times [0,T]; \end{aligned}$$(44) -
(b)
from Sobolev embedding theorem, there exists a positive constant \(k(\Omega )\) such that
$$\begin{aligned} (<\phi ^4>)^{1/2}\le k(\Omega )\parallel \nabla \phi \parallel ^2. \end{aligned}$$(45)
By means of the above inequalities and the Cauchy–Schwarz inequality it turns out that
From (43) and inequalities (46), we find
where
Finally, from (38), (42), (47) it turns out that
where \(\Gamma ^*=\Gamma \left( \sup \{\mu ^2,1\}\right) ^\frac{3}{2}.\) So, provided that
by means of recursive arguments it turns out
where
\(\square \)
Remark 2
We observe that if
then by virtue of Theorem 2, \((U^*,V^*)\) is nonlinearly asymptotically stable with respect the \(L^2(\Omega )\)-norm.
5 Conclusions
The Sel’kov–Schnakenberg model is used to model some autocatalytic biochemical and chemical reactions. In this work our attention is devoted to the study of the above-mentioned model. In particular, we have first showed some a priori estimates on the solutions and then the main theorem, in which sufficient conditions guaranteeing nonlinear asymptotic stability of the constant steady state are found. Such conditions are very sharp since it can be shown that they guarantee also the linear stability of the constant solution (15), and hence its instability. Therefore, Theorem 2 gives necessary and sufficient conditions for nonlinear stability of the solution (15).
References
Flavin, J.N., Rionero, S.: Qualitative Estimates for Partial Differential Equations: An Introduction. CRC Press, Boca Raton (1996)
Schnackenberg, J.: Simple chemical reaction systems with limit cycle behaviour. J. Theor. Biol. 81, 389–400 (1979)
Turing, A.M.: The chemical basis of morphogenesis. Philos. Trans. R. Soc. B 237, 37–72 (1952)
Dillon, R., Maini, P.K., Othmer, H.G.: Pattern formation in generalized Turing systems. J. Math. Biol. 32, 345–393 (1994)
Madzvamuse, A., Thomas, R.D.K., Maini, P.K., Wathen, A.J.: A numerical approach to the study of spatial pattern formation in the ligaments of arcoid bivalves. Bull. Math. Biol. 64, 501–530 (2002)
Murray, J.D.: Mathematical Biology I. An introduction. Third edition, Interdisciplinary Applied Mathematics, 17. Springer, New York (2003)
Murray, J.D.: Mathematical Biology II. Spatial model and Biomedical Applications. Third edition, Interdisciplinary Applied Mathematics 18. Springer, New York (2003)
Segel, L., Jackson, J.: Dissipative structure: an explanation and an ecological example. J. Theor. Biol. 37, 545–559 (1972)
Capone, F., De Luca, R., Torcicollo, I.: Influence of diffusion on the stability of a full Brusselator model. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 29(4), 661–678 (2018)
Rionero, S.: Long Time Behaviour of Three Competing Species and Mutualistic Communities. Asymptotic Methods in Nonlinear Wave Phenomena in honor of the 65th birthday of A. Greco, pp. 171–185. World Scientific, Singapore (2007)
Rionero, S.: \(L^2\) stability of the solutions to a nonlinear binary reaction–diffusion system of P.D.ES. Rend. Accad. Naz. Lincei Serie IX XVI, 227–238 (2005)
Sel’kov, E.E.: Self-oscillations in glycolysis. Eur. J. Biochem. 4, 79–86 (1968)
Gambino, G., Lombardo, M.C., Lupo, S., Sammartino, M.: Super-critical and sub-critical bifurcations in a reaction–diffusion Schnakenberg model with linear cross-diffusion. Ricerche Mat. 65, 449–467 (2016)
Gentile, M., Tataranni, A.: On the nonlinear stability for a rection–diffusion system concerning chemical reactions. In: Proceedings “Waves and Stability in Continuous Media”, pp. 315–320 (2008)
Gentile, M., Tataranni, A.: Turing instability for the Schnackenberg system. In: Proceedings “Waves and Stability in Continuous Media”, pp. 309–314 (2008)
Zhao, Y.H., Iqbal, M.S., Baber, M.Z., Inc, M., Ahmed, M., Khurshid, H.: On traveling wave solutions of an autocatalytic reaction–diffusion Sel’kov–Schnakenberg system. Results Phys. 44, 106129 (2023)
Khan, F.M., Ali, A., Hamadneh, N., Abdullah Alam, M.N.: Numerical investigation of chemical schnakenberg mathematical model. J. Nanomater. Article ID 9152972 (2021)
Iqbal, M.S., Seadawy, A.R., Baber, M.Z.: Demonstration of unique problems from Soliton solutions to nonlinear Sel’kov–Schnakenberg system. Chaos Solitons Fractals 162, 112485 (2022)
Li, B., Wang, F., Zhang, X.: Analysis on a generalized Sel’kov–Schnakenberg reaction–diffusion system. Nonlinear Anal. Real World Appl. 44, 537–558 (2018)
Li, B., Zhang, X.: Steady states of a Sel’kov–Schnakenberg reaction–diffusion system. Discrete Contin. Dyn. Syst. Ser. S. 10(5), 1009–1023 (2017)
Uecker, H., Wetzel, D.: Numerical results for snaking of patterns over patterns in some 2D Selkov–Schnakenberg reaction–diffusion systems. SIAM J. Appl. Dyn. Syst. 13(1), 94–128 (2014)
Zhou, J., Shi, J.P.: Pattern formation in a general glycolysis reaction–diffusion system. IMA J. Appl. Math. 80, 1703–1738 (2015)
Noufaey, K.: Stability analysis for Selkov–Schnakenberg reaction–diffusion system. Open Math. 19, 46–62 (2021)
Torcicollo, I.: On the non-linear stability of a continuous duopoly model with constant conjectural variation. Int. J. Non-Linear Mech. 81, 268–273 (2016)
Rionero, S., Torcicollo, I.: On the dynamics of a nonlinear reaction–diffusion duopoly model. Int. J. Non-Linear Mech. 99, 105–111 (2018)
Carfora, M.F., Torcicollo, I.: Identification of epidemiological models: the case study of Yemen cholera outbreak. Appl. Anal. 101(10), 3744–3754 (2022)
Carfora, M.F., Torcicollo, I.: Cross-diffusion-driven instability in a predator–prey system with fear and group defense. Mathematics 8(8), 1244 (2020)
Protter, M.H., Weinberger, H.F.: Maximum Principles in Differential Equations. Springer, Berlin (1984)
Rionero, S.: A rigorous reduction of the \(L^2(\Omega )\)-stability of the solutions to a nonlinear binary reaction–diffusion system of P.D.E.s. to the stability of the solutions to a linear binary system of O.D.E.’s. J. Math. Anal. Appl. 319, 377–397 (2006)
Rionero, S.: A nonlinear \(L^2\)-stability analysis for two-species population dynamics with dispersal. Math. Biosci. Eng. 3, 189–204 (2006)
Merkin, D.R.: Introduction to the Theory of Stability. Text in Applied Mathematics, vol. 24. Springer, Berlin (1997)
Acknowledgements
This work has been performed under the auspicies of the GNFM of INdAM.
Funding
Open access funding provided by Universitá degli Studi di Napoli Federico II within the CRUI-CARE Agreement.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Gentile, M., Torcicollo, I. Nonlinear stability analysis of a chemical reaction–diffusion system. Ricerche mat 73 (Suppl 1), 189–200 (2024). https://doi.org/10.1007/s11587-023-00793-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11587-023-00793-x