二维领域中延迟图案形成的延迟
Delayed Pattern Formation in Two-Dimensional Domains
AI总结：
本研究探讨了二维领域中基因表达时间延迟与领域大小对时空图案形成的影响，通过扩展Schnakenberg模型揭示了非单调相关性。
摘要：
本研究探讨了基因表达时间延迟与领域大小之间的相互作用如何控制反应-扩散系统中的时空图案形成。为了研究这些现象，我们利用了一种修改版的Schnakenberg模型，称为配体内化（LI）模型。在一维领域中，已观察到基因表达时间延迟与图案形成时间之间存在线性关系。我们将模型扩展到二维领域，并确认在那里也存在类似的关系。然而，我们的探索揭示了领域大小与图案出现时间之间非单调的相关性。为了揭示这些动态，我们考虑了各种初始条件，包括空间均匀稳态的随机扰动以及其不稳定流形的初始条件。我们计算了图案相对于时间延迟和领域大小的二维参数图。
英文摘要：
This study investigates how the interaction between gene expression time delay and domain size governs spatio-temporal pattern formation in a reaction-diffusion system. To investigate these phenomena, we utilize a modified version of the Schnakenberg model called the ligand internalisation (LI) model. In a one-dimensional domain, a linear relationship has been observed between the gene expression time delay and the time it takes for patterns to form. We extend the model to the two-dimensional domain and confirm that a similar relationship holds there as well. However, our exploration reveals a non-monotonic correlation between domain size and the time required for pattern emergence. To unravel these dynamics, we consider a range of initial conditions, including random perturbations of the spatially homogeneous steady state and initial conditions from its unstable manifold. We compute a two-parameter chart of patterns with respect to time delay and domain size.
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