In this work, we show theoretically and numerically that a one-dimensional reaction-diffusion system, near the Turing bifurcation, produces different number of stripes when, in addition to random noise, the Fourier mode of a prepattern used to initialize the system changes. We also show that the Fourier modes that persist are inside the Eckhaus stability regions, while those outside this region follow a wave number selection process not predicted by the linear analysis. To test our results, we use the Brusselator reaction-diffusion system obtaining an excellent agreement between the weakly nonlinear predictions of the real Ginzburg-Landau equations and the numerical solutions near the bifurcation. Although the persistence of patterns is not relevant as a simple generating mechanism of self-organization, it is crucial to understand the formation of patterns that occurs in multiple stages. In this work, we discuss the relevance of our results on the robustness and diversity of solutions in multiple-steps mechanisms of biological pattern formation and auto-organization in growing domains.

中文翻译：

---

Eckhaus选择：反应扩散系统中模式持久性的机制

在这项工作中，我们在理论上和数字上表明，当除随机噪声外，用于初始化系统的预图案的傅立叶模式发生变化时，图灵分叉附近的一维反应扩散系统会产生不同数量的条纹。我们还表明，持久的傅立叶模式在Eckhaus稳定区域内，而在该区域之外的模式遵循的是线性分析未预测的波数选择过程。为了测试我们的结果，我们使用Brusselator反应扩散系统获得了真实的Ginzburg-Landau方程的弱非线性预测与分支附近的数值解之间的极好的一致性。尽管模式的持久性与作为自组织的简单生成机制并不相关，了解在多个阶段中发生的模式的形成至关重要。在这项工作中，我们讨论了在增长领域中生物模式形成和自动组织的多步机制中，解决方案的健壮性和多样性与结果的相关性。