Recent experimental studies on primary hair follicle formation and feather bud morphogenesis indicate a coupling between Turing-type diffusion driven instability and chemotactic patterning. Inspired by these findings we develop and analyse a mathematical model that couples chemotaxis to a reaction–diffusion system exhibiting diffusion–driven (Turing) instability. While both systems, reaction–diffusion systems and chemotaxis, can independently generate spatial patterns, we were interested in how the coupling impacts the stability of the system, parameter region for patterning, pattern geometry, as well as the dynamics of pattern formation. We conduct a classical linear stability analysis for different model structures, and confirm our results by numerical analysis of the system. Our results show that the coupling generally increases the robustness of the patterning process by enlarging the pattern region in the parameter space. Concerning time scale and pattern regularity, we find that an increase in the chemosensitivity can speed up the patterning process for parameters inside and outside of the Turing space, but generally reduces spatial regularity of the pattern. Interestingly, our analysis indicates that pattern formation can also occur when neither the Turing nor the chemotaxis system can independently generate pattern. On the other hand, for some parameter settings, the coupling of the two processes can extinguish the pattern formation, rather than reinforce it. These theoretical findings can be used to corroborate the biological findings on morphogenesis and guide future experimental studies. From a mathematical point of view, this work sheds a light on coupling classical pattern formation systems from the parameter space perspective.

最近对初级毛囊形成和羽芽形态发生的实验研究表明图灵型扩散驱动的不稳定性和趋化图案之间存在耦合。受这些发现的启发，我们开发并分析了一个数学模型，该模型将趋化性与表现出扩散驱动（图灵）不稳定性的反应扩散系统耦合起来。虽然反应扩散系统和趋化性这两个系统都可以独立生成空间图案，但我们感兴趣的是耦合如何影响系统的稳定性、图案化的参数区域、图案几何形状以及图案形成的动力学。我们对不同的模型结构进行了经典的线性稳定性分析，并通过系统的数值分析证实了我们的结果。我们的结果表明，耦合通常通过扩大参数空间中的图案区域来提高图案化过程的鲁棒性。关于时间尺度和图案规律性，我们发现化学敏感性的增加可以加速图灵空间内外参数的图案化过程，但通常会降低图案的空间规律性。有趣的是，我们的分析表明，当图灵和趋化系统都不能独立生成模式时，也会发生模式形成。另一方面，对于某些参数设置，两个过程的耦合可能会消除而不是强化模式的形成。这些理论发现可用于证实形态发生的生物学发现并指导未来的实验研究。从数学的角度来看，这项工作揭示了从参数空间角度耦合经典模式形成系统。