Stochastic Pattern Formation in Mathematical Biology

许多不同的生物系统都表现出静态和动态的视觉冲击模式，从动物皮肤的色素到植被生长的空间分布。通过扩散驱动的“图灵不稳定性”机制，确定性连续统（偏微分方程）模型已被广泛用于分析和成功预测这些模式的各种特征。图灵不稳定性最初是作为胚胎发生的解释性模型提出的，它描述了通过局部“反应”和空间扩散的耦合动力学，从简单的均匀初始条件自发出现空间不均匀结构。有时候，模式不稳定性的连续性是可能的，并且模型不能预测哪个模式将随着时间的推移而持续。此外，对于现实的反应速率，预测的不稳定性阈值通常需要一个数量级（或更大）的物种扩散率，这是很少观察到在现实世界systems.Stochastic模型可以避免这些问题，同时允许分析噪声和（微观尺度上）离散的图案形成的影响。这种模型还扩大了图灵机制，使之包括随机波动驱动的“随机图灵模式”。隔室为基础的（介观）随机模型提供了一个分析易处理的方法来分析的时空特性和统计图案的不稳定性，扩散驱动和波动驱动。该项目的目标首先是开发一个有凝聚力的分析框架，用于识别，分类和预测自发模式化不稳定性;从基于隔室的模型开始，但最终跨越一系列离散的随机反应扩散设置，包括确定性和随机图灵模式。其次，这个项目将应用这个框架来分析最近开发的随机反应扩散模型，如体积排阻过程的图案不稳定性。有了这个，我们的目标是通过比较不同的模型和类型的不稳定性，以更好地理解驱动图案形成的潜在物理机制，进一步分析图案不稳定性。

Many varied biological systems exhibit visually striking patterns both static and dynamic, from the pigmentations of animal skins to the spatial distribution of vegetation growth. Deterministic continuum (partial differential equation) models have been widely used to analyse and successfully predict various characteristics of these patterns via the diffusion-driven `Turing instability' mechanism. Initially proposed as an explanatory model for embryogenesis, the Turing instability describes the spontaneous emergence of spatially inhomogeneous structure from simple homogeneous initial conditions through the coupled dynamics of localised `reactions' and spatial diffusion.Such PDE models often suffer from issues of robustness and parameter tuning. Sometimes a continuum of pattern instabilities is possible, and the model cannot predict which pattern will persist over time. Further, for realistic reaction rates, the predicted instability thresholds typically require an order of magnitude (or greater) disparity in species diffusivities, which is rarely observed in real-world systems.Stochastic models can avoid these problems while simultaneously allowing analysis of the impacts of noise and (on microscopic scales) discreteness on the pattern formation. Such models have also expanded the Turing mechanism to include random fluctuation-driven `stochastic Turing patterns'. Compartment-based (mesoscopic) stochastic models offer an analytically tractable approach to analysing the spatiotemporal characteristics and statistics of patterning instabilities, both diffusion-driven and fluctuation-driven. The aims of this project are first and foremost to develop a cohesive analytical framework for the identification, classification, and prediction of spontaneous patterning instabilities; starting with compartment-based models, but eventually spanning a range of discrete stochastic reaction-diffusion set-ups, encompassing both deterministic and stochastic Turing patterns. Secondly, this project will apply this framework to analysing patterning instabilities in more recently developed stochastic reaction-diffusion models, such as volume exclusion processes. With this, we aim to further the analysis of patterning instabilities by enabling comparison between different models and types of instability, to better understand the underlying physical mechanisms driving pattern formation.