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# Pattern formation of Brusselator in the reaction-diffusion system

• * Corresponding author: Jianwei Shen
• Time delay profoundly impacts reaction-diffusion systems, which has been considered in many areas, especially infectious diseases, neurodynamics, and chemistry. This paper aims to investigate the pattern dynamics of the reaction-diffusion model with time delay. We obtain the condition in which the system induced the Hopf bifurcation and Turing instability as the parameter of the diffusion term and time delay changed. Meanwhile, the amplitude equation of the reaction-diffusion system with time delay is also derived based on the Friedholm solvability condition and the multi-scale analysis method near the critical point of phase transition. We discussed the stability of the amplitude equation. Theoretical results demonstrate that the delay can induce rich pattern dynamics in the Brusselator reaction-diffusion system, such as strip and hexagonal patterns. It is evident that time delay causes steady-state changes in the spatial pattern under certain conditions but does not cause changes in pattern selection under certain conditions. However, diffusion and delayed feedback affect pattern formation and pattern selection. This paper provides a feasible method to study reaction-diffusion systems with time delay and the development of the amplitude equation. The numerical simulation well verifies and supports the theoretical results.

Mathematics Subject Classification: Primary: 35K57, 35B36, 35Q92; Secondary: 34H20.

 Citation:

• Figure 1.  (a) Hopf bifurcation of $u$ about $b$ when $\theta = 0.05$ and $\tau = 0$. The time evolution corresponding to system (2) when $b = 1.5$(b), $b = 2.5$(c) and $b = 7.5$(d)

Figure 2.  The curve of the bifurcation critical point $b_{H}$ of the system (2) with the feedback strength $\theta$ when without diffusion and $\tau = 0$

Figure 3.  Concentration-time profiles of variables $u$ and $v$ in the system (2) for different $\tau$ when $d_1 = d_2 = 0$, $a = 1$, $\theta = 0.05$ and $b = 1.5$. The time delays are (a) $\tau = 1$, (b) $\tau = 1.5$ and (c) $\tau = 2$ respectively. (d) Bifurcation of the concentration $u$ versus $\tau$

Figure 4.  The pattern formation of $u$ and $v$ in the system(2) when $a = 1$, $b = 1.5$, $d_1 = 0.004$, $\tau = 0$, $\theta = 0$, (a)(b) $d_2 = 0.3$; (c)(d) $d_2 = 0.4$; (e)(f) $d_2 = 0.5$

Figure 5.  The pattern formation of $u$ and $v$ in the system(2) when $a = 1$, $b = 1.5$, $d_1 = 0.15$, $d_2 = 2.1$, (a)(b) $\tau = 0$, $\theta = 0$; (c)(d) $\tau = 0$, $\theta = 0.45$; (e)(f) $\tau = 0.1$, $\theta = 0.45$

Figure 6.  The pattern formation of $u$ and $v$ in the system(2) when $a = 1$, $b = 2.5$, $d_1 = 0.04$, $d_2 = 0.2$, $\tau = 0$, $\theta = 0.35$

Figure 7.  The pattern formation of $u$ and $v$ in the system(2) when $a = 1$, $b = 2.5$, $d_1 = 0.01$, $d_2 = 0.2$, $\tau = 0$, $\theta = 0.35$

Figure 8.  The pattern formation of $u$ and $v$ in the system(2) when $a = 1$, $b = 2.5$, $d_1 = 0.01$, $d_2 = 0.8$, $\tau = 0, \theta = 0.35$

Figure 9.  The pattern formation of $u$ and $v$ in the system(2) when $a = 1$, $b = 2.5$, $d_1 = 0.01$, $d_2 = 0.2$, $\tau = 0.05, \theta = 0.5$

Figure 10.  The pattern formation of $u$ and $v$ in the system(2) when $a = 1$, $b = 2.5$, $d_1 = 0.04$, $d_2 = 0.2$, $\tau = 0.05, \theta = 0.5$

Figure 11.  The pattern formation of $u$ and $v$ in the system(2) when $a = 1$, $b = 2.5$, $d_1 = 0.01$, $d_2 = 0.2$, $\tau = 5, \theta = 0.25$

Figure 12.  The pattern formation of $u$ and $v$ in the system(2) when $a = 1$, $b = 2.5$, $d_1 = 0.04$, $d_2 = 0.2$, $\tau = 0.5, \theta = 0.25$

Figure 13.  The pattern formation of $u$ and $v$ in the system(2) when $a = 1$, $b = 2.5$, $d_1 = 0.5$, $d_2 = 2$, $\tau = 0$ (a)(b) $\theta = 0.15$, (c)(d) $\theta = 0.25$, (e)(f) $\theta = 0.45$

Figure 14.  The pattern formation of $u$ and $v$ in the system(2) when $a = 1$, $b = 2.5$, $d_1 = 0.5$, $d_2 = 2$ and $\theta = 0.35$, (a)(b) $\tau = 0$; (c)(d) $\tau = 0.1$; (e)(f) $\tau = 1$

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