图灵系统近平衡分析的现代观点
Modern Perspectives on Near-Equilibrium Analysis of Turing Systems
摘要：
自艾伦·图灵的形态发生研究发表以来的近70年里，人们在理解他提出的反应扩散理论的数学和生物学方面都取得了巨大的进展。其中一些发展是在图灵的论文中出现的，另一些则是由于现代数学技术的新见解，数值模拟的进步和广泛的生物实验。尽管取得了这些进展，但理论和实验之间仍然存在重大差距，许多生物模式的例子中，潜在的机制仍然不清楚。在这里，我们回顾了图灵开创的数学理论的现代发展，展示了他的方法是如何被推广到经典的两种反应扩散框架之外的一系列环境中的，包括演化和复杂的流形，在空间和时间上异质的系统，以及更一般的反应输运方程。虽然在理解这些更复杂的模型方面取得了实质性进展，但我们始终强调仍存在许多挑战。我们专注于数学理论，特别是线性稳定性分析的“平凡”的基本状态。我们强调重要的开放式问题，在进一步发展这一理论，并讨论在使用这些技术来了解生物现实的障碍。
英文摘要：
In the nearly seven decades since the publication of Alan Turing's work on morphogenesis, enormous progress has been made in understanding both the mathematical and biological aspects of his proposed reaction-diffusion theory. Some of these developments were nascent in Turing's paper, and others have been due to new insights from modern mathematical techniques, advances in numerical simulations, and extensive biological experiments. Despite such progress, there are still important gaps between theory and experiment, with many examples of biological patterning where the underlying mechanisms are still unclear. Here we review modern developments in the mathematical theory pioneered by Turing, showing how his approach has been generalized to a range of settings beyond the classical two-species reaction-diffusion framework, including evolving and complex manifolds, systems heterogeneous in space and time, and more general reaction-transport equations. While substantial progress has been made in understanding these more complicated models, there are many remaining challenges that we highlight throughout. We focus on the mathematical theory, and in particular linear stability analysis of `trivial' base states. We emphasise important open questions in developing this theory further, and discuss obstacles in using these techniques to understand biological reality.
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