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摘要: Schnakenberg系统是一类典型的化学反应扩散控制系统. 目前国内外研究仅局限于Schnakenberg系统的Turing不稳定与分岔, 而关于其化学斑图演化机理的报道较少. 斑图机理分析可以准确揭示化学反应中自组织现象的产生和空间模式的演化规律. 本文研究交叉扩散驱动下Schnakenberg系统斑图的结构蜕变、演化速度及时间依赖性, 重点探讨交叉扩散对其动力学与斑图演化的响应机制. 研究发现, 当自扩散诱导的系统稳定时, 交叉扩散可以激发斑图的产生; 当自扩散诱导的系统不稳定时, 交叉扩散可以实现斑图结构的蜕变; 对于环波结构, 不同组分的交叉扩散可以影响其演化速度; 对于时间依赖性, 交叉扩散可以激发随时间周期变化的斑图产生, 并可将此类斑图转换为随时间相对稳定的斑图. 因此, 交叉扩散对于Schnakenberg系统的斑图产生、蜕变、演化速度及时间依赖性都起着至关重要的作用.
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关键词:
- 交叉扩散 /
- Schnakenberg系统 /
- 时空斑图 /
- 稳定性 /
- 环波
Abstract: Schnakenberg system is a typical chemical reaction-diffusion control system. Currently, research both domestically and internationally is limited to Turing instability and bifurcation of such system, with relatively fewer reports on the evolution mechanism of its chemical patterns. Pattern mechanism analysis can accurately reveal the generation of self-organization phenomena and the evolution of spatial patterns in chemical reactions. This paper investigates the structural transformation, evolution speed, and time-dependence of patterns of Schnakenberg system under cross-diffusion driving, and focuses on exploring the response mechanism of cross-diffusion to its kinetics and pattern evolution. The findings indicate that when a system induced by self-diffusion is stable, cross-diffusion can trigger pattern formation. When a system induced by self-diffusion is unstable, cross-diffusion can lead to the transformation of pattern structures. For circular wave structures, different components of cross-diffusion can influence their evolution speed. In the context of temporal dependence, cross-diffusion can trigger the generation of patterns that vary periodically with time and can transform such patterns into relatively stable ones over time. Therefore, cross-diffusion plays a crucial role in the pattern generation, transformation, evolution speed, and temporal dependence of Schnakenberg system.-
Key words:
- Cross-diffusion /
- Schnakenberg system /
- spatiotemporal patterns /
- stability /
- circular wave
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图 1 系统(1)色散关系曲线($ a = 0.1,\ b = 1, $ $\ {{D}_{u}} = 1.0,\ {{D}_{v}} = 0.1$)
Fig. 1 Dispersion relation curve of system (1)($ a = 0.1,\ b = 1,\ {{D}_{u}} = 1.0,\ {{D}_{v}} = 0.1$)
图 2 系统(1)稳定的斑图($ {{D}_{uv}} = {{D}_{vu}} = 0,\ {t} = 1\ 000$)
Fig. 2 Stable pattern in the system (1)($ {{D}_{uv}} = {{D}_{vu}} = 0,\ {t} = 1\ 000$)
图 3 点-稀疏条混合斑图(${{D}_{uv}} = 1.5, $ $ \ {{D}_{vu}} = -0.1,\ {t} = 1\ 000$)
Fig. 3 Mixed pattern of points and sparse Stripes ($ {{D}_{uv}} = 1.5,\ {{D}_{vu}} = -0.1,\ {t} = 1\ 000$)
图 4 不同交叉扩散系数取值的色散关系曲线及斑图((a)色散关系曲线; (b)点状斑图; (c)点条混合斑图; (d)条状斑图; (e)六边形斑图; (f)星星状斑图)
Fig. 4 Dispersion relation curves and patterns with different cross-diffusion coefficient values ((a) Dispersion relation curves; (b) Point pattern; (c) Mixed point and stripe pattern; (d) Stripe pattern; (e) Hexagonal pattern; (f) Star pattern)
图 6 环波形成过程((a)$t = 50$; (b)$t = 75$; (c)$t = 100$)
Fig. 6 Formation process of circular wave((a)$t = 50$; (b)$t = 75$; (c)$t = 100$)
图 7 引入交叉扩散的环波结构((a)${{D}_{uv}} = -0.15,\ {{D}_{vu}} = 0$; (b)${{D}_{uv}} = 0.15,\ {{D}_{vu}} = 0$; (c)${{D}_{uv}} = 0,\ {{D}_{vu}} = -0.1$; (d)${{D}_{uv}} = 0,\ {{D}_{vu}} = 0.15$)
Fig. 7 Introduction of circular wave structure with cross-diffusion ((a)${{D}_{uv}} = -0.15,\ {{D}_{vu}} = 0$; (b)${{D}_{uv}} = 0.15,\ {{D}_{vu}} = 0$; (c)${{D}_{uv}} = 0,\ {{D}_{vu}} = -0.1$; (d)${{D}_{uv}} = 0,\ {{D}_{vu}} = 0.15$)
图 8 随时间周期变化的时空结构((a)${t} = 506$; (b)${t} = 510$; (c)${t} = 520$; (d)${t} = 530$; (e)${t} = 536.5$; (f)${t} = 540.5$; (g)${t} = 550.5$; (h)${t} = 560.5$)
Fig. 8 Spatio-temporal structure with time cycle((a)${t} = 506$; (b)${t} = 510$; (c)${t} = 520$; (d)${t} = 530$; (e)${t} = 536.5$; (f)${t} = 540.5$; (g)${t} = 550.5$; (h)${t} = 560.5$)
图 11 系统(1)中变量$u(100,100,t)$的波形图($a = 0.05,\ b = 0.2625,\ D_u = D_v = 0.1$) ((a)${{D}_{uv}} = -0.1$, ${{D}_{vu}} = 0$; (b)${{D}_{uv}} = {{D}_{vu}} = 0.05$)
Fig. 11 Waveform of variables $u(100,100,t)$ in system (1)($a = 0.05,\ b = 0.2625,\ D_u = D_v = 0.1$)((a)${{D}_{uv}} = -0.1$, ${{D}_{vu}} = 0$; (b)${{D}_{uv}} = {{D}_{vu}} = 0.05$)
表 1 不同交叉扩散系数取值的12环成型时间 ($a = 0.1$, $b = 1.0$, $D_u = 0.1$, $D_v = 2.0$)
Table 1 different formation times of 12-ring patterns with various cross-diffusion coefficient values ($a = 0.1$, $b = 1.0$, $D_u = 0.1$, $D_v = 2.0$)
$D_{uv}$ $D_{vu}$ $t=75$时环数 12环成型时间 0 0 9 100 −0.15 0 6 155 0.15 0 11 85 0 −0.10 10 95 0 0.15 7 125 
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