Computation, analysis and control of biological pattern formation

具有相互作用组件的复杂系统经常产生“涌现特性”，或模式形成现象。模式形成的模型在生物学中无处不在，是一个密集的研究领域。一类重要的模型是反应扩散方程（RDEs）。RDEs已被用于模拟许多生物模式现象，如浮游生物动力学、神经冲动、趋化性、心律失常、流行病学和形态发生，仅举几例（参见J.D. Murray的经典文本“数学生物学”）。我从广义上解释模式，包括时空动力学和由扩散引起的不稳定性（图灵模式）引起的固定解决方案。研究人员越来越意识到需要将空间结构纳入生物学和流行病学模型，并确定空间尺度。空间结构与环境的物理特征或生态过程和现象的内在特征相对应。大多数现实模型都是非线性的，因此数值方法在研究复杂系统中调节自组织的关键机制方面起着至关重要的作用。然而，实际应用的rde的数值分析滞后于建模的进展。因此，开发高效和精确的数值方法来逼近空间显式生物模型是一个肥沃和不断发展的研究领域。另一个不发达的领域是利用最优控制理论的数学学科来控制图案的形成。在大多数情况下，在建立数学模型之后，控制系统演化的需求自然会出现。最优控制理论也可用于识别生物系统中的关键参数。我在提案中使用的方法需要数值分析，应用数学分析和科学计算之间的交叉施肥，以研究RDEs模型的生物模式形成。我的建议侧重于模拟生物模式形成的非线性RDEs的数值逼近和控制，应用于流行病学，自然资源管理，生态学和各种生物医学领域。

Complex systems with interacting components frequently give rise to `emergent properties', or pattern formation phenomena. Models for pattern formation are ubiquitous in biology and are an intensive area of research. An important class of models are reaction-diffusion equations (RDEs). RDEs have been used to model many biological pattern phenomena arising in, plankton dynamics, nerve impulses, chemotaxis, cardiac arrhythmias, epidemiology, and morphogenesis, to name a few (see J.D. Murray's classic text 'Mathematical Biology'). I interpret patterning in a broad sense to include both spatio-temporal dynamics and stationary solutions arising due to diffusion induced instability (Turing patterns). Researchers are becoming more aware of the need to incorporate spatial structure in biological and epidemiological models, and to identify spatial scales. Spatial structures correspond to physical features of the environment, or to intrinsic characteristics of ecological processes and phenomena. Most realistic models are nonlinear, and thus numerical methods have a vital role to play in investigating the key mechanisms that regulate self organization in complex systems. However, the numerical analysis of RDEs with real applications lags behind advances in modeling. Therefore, the development of efficient and accurate numerical methods for the approximation of spatially explicit biological models is a fertile and growing research field. Another area that is underdeveloped is the control of pattern formation using the mathematical discipline of optimal control theory. In most situations, the need to control the evolution of a system comes naturally after establishing the mathematical model. Optimal control theory can also be used to identify key parameters in a biological system. The methodology I use in my proposal entails the cross-fertilization between numerical analysis, applied mathematical analysis, and scientific computing to study biological pattern formation modeled by RDEs. My proposal focuses on the numerical approximation and control of nonlinear RDEs modeling biological pattern formation, with applications in epidemiology, natural resource management, ecology, and various biomedical areas.