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Super-critical and sub-critical bifurcations in a reaction-diffusion Schnakenberg model with linear cross-diffusion

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Abstract

In this paper the Turing pattern formation mechanism of a two components reaction-diffusion system modeling the Schnakenberg chemical reaction is considered. In Ref. (Madzavamuse et al., J Math Biol 70(4):709–743, 2015) it was shown how the presence of linear cross-diffusion terms favors the destabilization of the constant steady state. We perform the weakly nonlinear multiple scales analysis to derive the equations for the amplitude of the Turing patterns and to show how the cross-diffusion coefficients influence the occurrence of super-critical or sub-critical bifurcations. We present a numerical exploration of far from equilibrium regimes and prove the existence of multistable stationary solutions.

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Acknowledgments

The work of GG and SL was partially supported by GNFM-INdAM through a Progetto Giovani Grant. The work of MCL and MS was partially supported by GNFM-INdAM. The authors thank the anonymous reviewer for the comments and the suggestions that helped improve the paper.

Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, University of Palermo, Via Archirafi, 34, 90123, Palermo, Italy

    G. Gambino, M. C. Lombardo, S. Lupo & M. Sammartino

Authors
  1. G. Gambino
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  2. M. C. Lombardo
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  3. S. Lupo
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  4. M. Sammartino
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Corresponding author

Correspondence to M. Sammartino.

Appendix: The quintic Stuart–Landau equation

Appendix: The quintic Stuart–Landau equation

Taking into account that (24) still holds for the amplitude A (although now the derivative with respect to T is a partial derivative), the solvability condition \(\left\langle \mathbf{G}, \mathbf{\psi } \right\rangle =0\) for (19) is satisfied and the solution is:

$$\begin{aligned} \mathbf w _3=\left( A\mathbf{w}_{31}+A^3 \mathbf{w}_{32}\right) \cos {(k_c x)} +A^3 \mathbf{w}_{33}\cos {(3 k_c x)}\, , \end{aligned}$$
(28)

where the expression for the vectors \(\mathbf{w}_{3i}, i=1, 2, 3\) can be computed solving the following linear systems:

$$\begin{aligned} L_1\mathbf{w}_{31}=\sigma {\mathbf {r}}+\mathbf {G}_1^{(1)}, \quad L_2\mathbf{w}_{32}\!=\!-L{\mathbf {r}}\!+\!\mathbf {G}_1^{(3)}, \quad L_3\mathbf{w}_{33}\!=\!\mathbf {G}_3, \end{aligned}$$

where we have defined \(L_{i}=\varGamma J-i^2k_c^2D^{d_c}\).

At \(O(\varepsilon ^4)\) the resulting equation is \( \mathcal {L}^{d_c} \mathbf{w}_3=\mathbf {H}\), where:

$$\begin{aligned} \mathbf{H}&=2A\frac{\partial A}{\partial T_2}{\mathbf {w}_{20}}+A^2\mathbf{H}_0^{(2)}+A^4\mathbf{H}_0^{(4)}+\left( 2A\frac{\partial A}{\partial T}{\mathbf {w}_{22}}+A^2\mathbf{H}_2^{(2)}+A^4\mathbf{H}_2^{(4)}\right) \cos (2\bar{k}_cx)\\&\quad +A^4\mathbf{H}_4^{(4)}\cos (4\bar{k}_cx), \end{aligned}$$

and:

$$\begin{aligned} \mathbf{H}_0^{(2)}&= \displaystyle \frac{1}{2}\gamma \begin{pmatrix} \frac{2b}{(a+b)^2}+2(a+b)M&{}\quad 2(a+b) \\ -\frac{2b}{(a+b)^2}-2(a+b)M &{}\quad -2(a+b) \end{pmatrix}{\mathbf {w}_{31}},\\ \mathbf{H}_0^{(4)}&= \displaystyle \frac{1}{2}\gamma \begin{pmatrix} \frac{2b}{(a+b)^2}+2(a+b)M &{}\quad 2(a+b) \\ -\frac{2b}{(a+b)^2}-2(a+b)M &{}\quad -2(a+b) \end{pmatrix} {\mathbf {w}_{32}}\\&\quad \ +\gamma \begin{pmatrix} \frac{b}{(a+b)^2}{\mathbf {w}_{20}}(1)+2(a+b){\mathbf {w}_{20}}(2)+M &{}\quad \frac{1}{2}\\ -\frac{b}{(a+b)^2}{\mathbf {w}_{20}}(1)-2(a+b){\mathbf {w}_{20}}(2)-M &{}\quad -\frac{1}{2} \end{pmatrix} {\mathbf {w}_{20}}\\&\quad \ +\displaystyle \frac{1}{2}\gamma \begin{pmatrix} \frac{b}{(a+b)^2}{\mathbf {w}_{22}}(1)+2(a+b){\mathbf {w}_{22}}(2)+M &{}\quad \frac{1}{2}\\ -\frac{b}{(a+b)^2}{\mathbf {w}_{22}}(1)-2(a+b){\mathbf {w}_{22}}(2)-M &{}\quad -\frac{1}{2} \end{pmatrix} {\mathbf {w}_{22}}\\ \mathbf{H}_2^{(2)}&= \begin{pmatrix} 0&{}\quad 0\\ 0&{}\quad 4d^{(2)}k_c^2 \end{pmatrix}{\mathbf {w}_{22}}+\displaystyle \frac{1}{2}\gamma \begin{pmatrix} \frac{2b}{(a+b)^2}+2(a+b)M &{}\quad 2(a+b) \\ -\frac{2b}{(a+b)^2}-2(a+b)M &{}\quad -2(a+b) \end{pmatrix} {\mathbf {w}_{31}},\\ \\ \end{aligned}$$
$$\begin{aligned} \mathbf{H}_2^{(4)}&= \displaystyle \frac{1}{2}\gamma \begin{pmatrix} \frac{2b}{(a+b)^2}+2(a+b)M &{}\quad 2(a+b) \\ -\frac{2b}{(a+b)^2}-2(a+b)M &{}\quad -2(a+b) \end{pmatrix}({\mathbf {w}_{32}}+{\mathbf {w}_{33}})\\&\quad \ +\gamma \begin{pmatrix} \frac{b}{(a+b)^2}{\mathbf {w}_{20}}(1)+2(a+b){\mathbf {w}_{20}}(2)+M &{}\quad \frac{1}{2}\\ -\frac{b}{(a+b)^2}{\mathbf {w}_{20}}(1)-2(a+b){\mathbf {w}_{20}}(2)-M &{}\quad -\frac{1}{2} \end{pmatrix} {\mathbf {w}_{22}}\\&\quad \ +\gamma \begin{pmatrix} \frac{b}{(a+b)^2}{\mathbf {w}_{22}}(1)+2(a+b){\mathbf {w}_{22}}(2)+M &{}\quad \frac{1}{2}\\ -\frac{b}{(a+b)^2}{\mathbf {w}_{22}}(1)-2(a+b){\mathbf {w}_{22}}(2)-M &{}\quad -\frac{1}{2} \end{pmatrix}{\mathbf {w}_{20}},\\ \mathbf{H}_4^{(4)}&= \displaystyle \frac{1}{2}\gamma \begin{pmatrix}\frac{2b}{(a+b)^2}+2(a+b)M &{}\quad 2(a+b) \\ -\frac{2b}{(a+b)^2}-2(a+b)M &{}\quad -2(a+b) \end{pmatrix}{\mathbf {w}_{33}}\\&\quad \ +\displaystyle \frac{1}{2}\gamma \begin{pmatrix} \frac{b}{(a+b)^2}{\mathbf {w}_{22}}(1)+2(a+b){\mathbf {w}_{22}}(2)+M &{}\quad \frac{1}{2}\\ -\frac{b}{(a+b)^2}{\mathbf {w}_{22}}(1)-2(a+b){\mathbf {w}_{22}}(2)-M &{}\quad -\frac{1}{2} \end{pmatrix} {\mathbf {w}_{22}}. \end{aligned}$$

The solvability condition for is automatically satisfied and the solution is:

$$\begin{aligned} \mathbf{w}_4=A^2 \mathbf{w}_{40}+A^4\mathbf{w}_{41}+\left( A^2 \mathbf{w}_{42}+A^4\mathbf{w}_{43}\right) \cos (2k_c x)+A^4 \mathbf{w}_{44}\cos (4 k_c x)\, , \end{aligned}$$
(29)

where the vector \(\mathbf{w}_{4i}\), \(i=1,\ldots ,4\), are the solutions of the following linear systems:

$$\begin{aligned} \varGamma K\mathbf{w}_{40}= & {} 2 \sigma \mathbf{w}_{20}+\mathbf{H}_0^{(2)},\quad \varGamma K\mathbf{w}_{41}= -2 L \mathbf{w}_{20}+ \mathbf{H}_0^{(4)}, \\ L_2 \mathbf{w}_{42}= & {} 2 \sigma \mathbf{w}_{22}+\mathbf{H}_2^{(2)}, \quad L_3 \mathbf{w}_{43}= -2 L \mathbf{w}_{22}+ \mathbf{H}_2^{(4)}, \quad L_4 \mathbf{w}_{44}= \mathbf{H}_4. \end{aligned}$$

At \(O(\varepsilon ^5)\) the resulting equation is \(\mathcal {L}^{d_c} \mathbf{w}_3=\mathbf {P}\), where:

$$\begin{aligned} \mathbf{P}&=\left( \frac{\partial A}{\partial T_4}{\mathbf {r}}+\frac{\partial A}{\partial T_2}{\mathbf {w}_{31}}+3A^2\frac{\partial A}{\partial T_2}{\mathbf {w}_{32}}+A\mathbf{P}_1^{(1)}+A^3\mathbf{P}_1^{(3)}+A^5\mathbf{P}_1^{(5)}\right) \cos (\bar{k}_cx) \end{aligned}$$
(30)
$$\begin{aligned}&\quad \ +\left( 3A^2\frac{\partial A}{\partial T_2}{\mathbf {w}_{33}}+A^3\mathbf{P}_3^{(3)}+A^5\mathbf{P}_3^{(5)}\right) \cos (3\bar{k}_cx)+A^5\mathbf{P}_5^{(5)}\cos (5\bar{k}_cx), \end{aligned}$$
(31)

and:

$$\begin{aligned} \mathbf{P}_1^{(1)}&= \begin{pmatrix} 0 &{}\quad 0 \\ 0 &{}\quad d^{(2)}k_c^2 \end{pmatrix}{\mathbf {w}_{31}}+ \begin{pmatrix} 0\\ d^{(4)}k_c^2 M \end{pmatrix},\\ \mathbf{P}_1^{(3)}&=\begin{pmatrix} 0 &{}\quad 0 \\ 0 &{}\quad d^{(2)}k_c^2 \end{pmatrix} {\mathbf {w}_{32}}- \gamma \begin{pmatrix} \frac{2b}{(a+b)^2}+2(a+b)M &{}\quad 2(a+b) \\ -\frac{2b}{(a+b)^2}-2(a+b)M &{}\quad -2(a+b) \end{pmatrix} \left( {\mathbf {w}_{40}}+\displaystyle \frac{1}{2}{\mathbf {w}_{42}}\right) \\&\quad \ -\gamma \begin{pmatrix} \frac{2b}{(a+b)^2}{\mathbf {w}_{20}}(1)+2(a+b){\mathbf {w}_{20}}(2) &{}\quad 2(a+b){\mathbf {w}_{20}}(1)+\frac{1}{2}\\ -\frac{2b}{(a+b)^2}{\mathbf {w}_{20}}(1)-2(a+b){\mathbf {w}_{20}}(2) &{}\quad -2(a+b){\mathbf {w}_{20}}(1)-\frac{1}{2} \end{pmatrix} {\mathbf {w}_{31}}\\&\quad \ -\displaystyle \frac{1}{2}\gamma \begin{pmatrix} \frac{2b}{(a+b)^2}{\mathbf {w}_{22}}(1)+2(a+b){\mathbf {w}_{22}}(2) &{} 2(a+b){\mathbf {w}_{22}}(1)+\frac{1}{2}\\ -\frac{2b}{(a+b)^2}{\mathbf {w}_{22}}(1)-2(a+b){\mathbf {w}_{22}}(2) &{}\quad -2(a+b){\mathbf {w}_{22}}(1)-\frac{1}{2} \end{pmatrix}{\mathbf {w}_{31}}, \end{aligned}$$
$$\begin{aligned} \mathbf{P}_1^{(5)}&=-\gamma \begin{pmatrix} \frac{2b}{(a+b)^2}+2(a+b)M &{}\quad 2(a+b) \\ -\frac{2b}{(a+b)^2}-2(a+b)M &{}\quad -2(a+b) \end{pmatrix} \left( {\mathbf {w}_{41}}+\displaystyle \frac{1}{2}{\mathbf {w}_{43}}\right) \\&\quad \ -\gamma \begin{pmatrix} \frac{2b}{(a+b)^2}{\mathbf {w}_{20}}(1)+2(a+b){\mathbf {w}_{20}}(2) &{}\quad 2(a+b){\mathbf {w}_{20}}(1)+\frac{1}{2}\\ -\frac{2b}{(a+b)^2}{\mathbf {w}_{20}}(1)-2(a+b){\mathbf {w}_{20}}(2) &{}\quad -2(a+b) {\mathbf {w}_{20}}(1)-\frac{1}{2} \end{pmatrix}{\mathbf {w}_{32}}\\&\quad \ -\displaystyle \gamma \begin{pmatrix} \frac{2b}{(a+b)^2}{\mathbf {w}_{20}}(1)+2(a+b){\mathbf {w}_{20}}(2) &{}\quad 2(a+b){\mathbf {w}_{20}}(1)+\frac{1}{2}\\ -\frac{2b}{(a+b)^2}{\mathbf {w}_{20}}(1)-2(a+b){\mathbf {w}_{20}}(2) &{}\quad -2(a+b){\mathbf {w}_{20}}(1)-\frac{1}{2} \end{pmatrix} ({\mathbf {w}_{32}}+{\mathbf {w}_{33}})\\&\quad \ -\gamma \begin{pmatrix} \left( {\mathbf {w}_{20}}(1)+\frac{1}{2}{\mathbf {w}_{22}}(1)\right) M &{}\quad 2{\mathbf {w}_{20}}(1)+{\mathbf {w}_{22}}(1)\\ -\left( {\mathbf {w}_{20}}(1)+\frac{1}{2}{\mathbf {w}_{22}}(1)\right) M &{}\quad -2{\mathbf {w}_{20}}(1)-{\mathbf {w}_{22}}(1) \end{pmatrix} {\mathbf {w}_{20}}\\&\quad \ -\displaystyle \frac{1}{2}\gamma \begin{pmatrix}\Big ({\mathbf {w}_{20}}(1)+{\mathbf {w}_{22}}(1)\Big )M &{}\quad 2\Big ({\mathbf {w}_{20}}(1)+{\mathbf {w}_{22}}(1)\Big )\\ -\Big ({\mathbf {w}_{20}}(1)+{\mathbf {w}_{22}}(1)\Big )M &{}\quad -2\Big ({\mathbf {w}_{20}}(1)\!+\!{\mathbf {w}_{22}}(1)\Big ) \end{pmatrix}{\mathbf {w}_{22}},\\ \mathbf{P}_3^{(3)}&= \begin{pmatrix} 0 &{}\quad 0 \\ 0 &{}\quad 9d^{(2)}k_c^2 \end{pmatrix} {\mathbf {w}_{33}}-\displaystyle \frac{1}{2}\gamma \begin{pmatrix} \frac{2b}{(a+b)^2}+2(a+b)M &{}\quad 2(a+b) \\ -\frac{2b}{(a+b)^2}-2(a+b)M &{}\quad -2(a+b) \end{pmatrix} {\mathbf {w}_{42}}\\&\quad \ -\displaystyle \gamma \begin{pmatrix} \frac{2b}{(a+b)^2}{\mathbf {w}_{20}}(1)+2(a+b){\mathbf {w}_{20}}(2) &{}\quad 2(a+b){\mathbf {w}_{20}}(1)+\frac{1}{2}\\ -\frac{2b}{(a+b)^2}{\mathbf {w}_{20}}(1)-2(a+b){\mathbf {w}_{20}}(2) &{}\quad -2(a+b){\mathbf {w}_{20}}(1)-\frac{1}{2} \end{pmatrix}{\mathbf {w}_{31}}, \end{aligned}$$
$$\begin{aligned} \mathbf{P}_3^{(5)}&= -\displaystyle \frac{1}{2}\gamma \begin{pmatrix}\frac{2b}{(a+b)^2}+2(a+b)M &{}\quad 2(a+b) \\ -\frac{2b}{(a+b)^2}-2(a+b)M &{}\quad -2(a+b) \end{pmatrix}({\mathbf {w}_{43}}+{\mathbf {w}_{44}})\\&\quad -\gamma \begin{pmatrix} \frac{2b}{(a+b)^2}{\mathbf {w}_{20}}(1)+2(a+b){\mathbf {w}_{20}}(2) &{}\quad 2(a+b){\mathbf {w}_{20}}(1)+\frac{1}{2}\\ -\frac{2b}{(a+b)^2}{\mathbf {w}_{20}}(1)-2(a+b){\mathbf {w}_{20}}(2) &{}\quad -2(a+b){\mathbf {w}_{20}}(1)-\frac{1}{2}\end{pmatrix}{\mathbf {w}_{33}}\\&\quad -\displaystyle \frac{1}{2}\gamma \begin{pmatrix} \frac{2b}{(a+b)^2}{\mathbf {w}_{20}}(1)+2(a+b){\mathbf {w}_{20}}(2) &{}\quad 2(a+b){\mathbf {w}_{20}}(1)+\frac{1}{2}\\ -\frac{2b}{(a+b)^2}{\mathbf {w}_{20}}(1)-2(a+b){\mathbf {w}_{20}}(2) &{}\quad -2(a+b){\mathbf {w}_{20}}(1)-\frac{1}{2}\end{pmatrix}{\mathbf {w}_{32}}\\&\quad -\displaystyle \frac{1}{2}\gamma \begin{pmatrix}M{\mathbf {w}_{20}}(1) &{}\quad 2{\mathbf {w}_{20}}(1)\\ -M{\mathbf {w}_{20}}(1) &{}\quad -2{\mathbf {w}_{20}}(1) \end{pmatrix}{\mathbf {w}_{22}}\\&\quad -\displaystyle \frac{1}{2}\gamma \begin{pmatrix} M{\mathbf {w}_{22}}(1) &{}\quad 2{\mathbf {w}_{22}}(1) \\ -M{\mathbf {w}_{22}}(1) &{}\quad -2{\mathbf {w}_{22}}(1) \end{pmatrix}\left( {\mathbf {w}_{20}}+\displaystyle \frac{1}{2}{\mathbf {w}_{22}}\right) ,\\ \mathbf{P}_5^{(5)}&= -\displaystyle \frac{1}{2}\gamma \begin{pmatrix}\frac{2b}{(a+b)^2}+2(a+b)M &{}\quad 2(a+b) \\ -\frac{2b}{(a+b)^2}-2(a+b)M &{}\quad -2(a+b) \end{pmatrix}{\mathbf {w}_{44}}\\&\quad -\displaystyle \frac{1}{2}\gamma \begin{pmatrix} \frac{2b}{(a+b)^2}{\mathbf {w}_{20}}(1)+2(a+b){\mathbf {w}_{20}}(2) &{}\quad 2(a+b){\mathbf {w}_{20}}(1)+\frac{1}{2}\\ -\frac{2b}{(a+b)^2}{\mathbf {w}_{20}}(1)-2(a+b){\mathbf {w}_{20}}(2) &{}\quad -2(a+b){\mathbf {w}_{20}}(1)-\frac{1}{2} \end{pmatrix}{\mathbf {w}_{33}}\\&\quad -\displaystyle \frac{1}{4}\gamma \begin{pmatrix} M{\mathbf {w}_{22}}(1) &{}\quad 2{\mathbf {w}_{22}}(1) \\ -M{\mathbf {w}_{22}}(1) &{}\quad -2{\mathbf {w}_{22}}(1) \end{pmatrix}{\mathbf {w}_{22}}. \end{aligned}$$

Putting:

$$\begin{aligned}&\displaystyle \tilde{\sigma }=-\frac{\left\langle \mathbf{P}_1^{(1)}, {\varvec{\psi }}\right\rangle }{\left\langle {\mathbf {r}}, {\varvec{\psi }}\right\rangle },\qquad L=\frac{\left\langle {3\sigma {\mathbf {w}_{32}}-L{\mathbf {w}_{31}}+\mathbf P}_1^{(3)}, {\varvec{\psi }}\right\rangle }{\left\langle {\mathbf {r}},{\varvec{\psi }}\right\rangle },\qquad \nonumber \\&\displaystyle R=\frac{\left\langle {3L{\mathbf {w}_{32}}+\mathbf P}_1^{(5)}, {\varvec{\psi }}\right\rangle }{\left\langle {\mathbf {r}}, {\varvec{\psi }}\right\rangle } \end{aligned}$$
(32)

the Fredholm alternative \(\left\langle \mathbf{P}, {\varvec{\psi }}\right\rangle \) for the Eq. (31) leads to:

$$\begin{aligned} \frac{\partial A}{\partial T_4}=\tilde{\sigma }A -\tilde{L}A^3+\tilde{R}A^5. \end{aligned}$$
(33)

Adding up (33) to (24) one gets (26), with:

$$\begin{aligned} \bar{\sigma }=\sigma +\varepsilon ^2\tilde{\sigma }, \qquad \bar{L}=L+\varepsilon ^2\tilde{L}, \qquad \bar{R}=\varepsilon ^2\tilde{R}. \end{aligned}$$
(34)

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Gambino, G., Lombardo, M.C., Lupo, S. et al. Super-critical and sub-critical bifurcations in a reaction-diffusion Schnakenberg model with linear cross-diffusion. Ricerche mat 65, 449–467 (2016). https://doi.org/10.1007/s11587-016-0267-y

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  • DOI: https://doi.org/10.1007/s11587-016-0267-y

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