Turing theory plays an important role in real biological pattern formation problems, such as solid tumor growth and animal coat patterns. To understand how patterns form and develop over time due to growth, we consider spatiotemporal patterns, in particular Turing patterns, for reaction diffusion systems on growing surfaces with curvature. Of particular interest is isotropic growth of the sphere, where growth of the domain occurs in the same proportion in all directions. Applying a modified linear stability analysis and a separation of timescales argument, we derive the necessary and sufficient conditions for a diffusion driven instability of the steady state and for the emergence of spatial patterns. Finally, we explore these results using numerical simulations.

本研究提出一個兩階段的時空模型來分析空氣污染與健康的關係。我們選取在台灣的49個有設置空氣品質監測站的鄉鎮市區做為研究地區。資料包含這些小地區中1997-2001年的各地區每日因呼吸道疾病而就醫的門診人數與空氣污染物濃度與氣象監測資料。在第一階段中，對每一個月所有地區的每日因呼吸道疾病而就醫的門診人數與空氣污染配適時空模型，並利用氣象條件等因素做調整。在第二階段裡，利用線性混合效果模型將第一階段所獲得的60 個月空氣污染物係數估計值來獲得代表這五年全國整體污染物係數的估計。本文利用模擬研究來探討當季節因素與不可解釋的因素，例如像流行性感冒等存在時會對文獻上其他時空模型中參數的估計所造成的影響，同時與我們所提出的方法作一比較。 / We proposed a spatiotemporal model to investigate the association between the acute health effects and daily numbers of clinic visits for respiratory illness. The data include clinic records due to respiratory illness and environmental variables from air quality monitoring stations in Taiwan during 1997-2001. A small-area design and two-phase modeling were used for the analysis. In the first phase, we constructed a Poisson regression with autogressive residual process and spatial correlation to obtain the pollution coefficient of each single month. In the second phase, we combined the information from phase one model to improve estimates of the pollution coefficients of each month and to obtain an overall pollution coefficient across the temporal course. Simulation study was used to illustrate the bias of estimation when there are seasonal, spatial and the unexplained effects in the data.