John J. Tyson
In a 1952 paper, Alan Turing showed that spatially distributed chemical reactions evolving in time by local kinetic rate laws and in space by unbiased molecular diffusion can develop a stable, time-independent, spatial pattern from a spatially homogeneous, steady state solution subjected to small, spatially periodic perturbations over a critical range of wavelengths. He proposed this mechanism as a potential chemical basis for biological morphogenesis. Although his proposal was initially ignored and remains controversial, Turing's idea still plays a major role in any discussion of spontaneous pattern formation in biological and chemical systems. Nevertheless, it is safe to say that few people have carefully studied his 1952 paper, which is notoriously difficult to read. For this reason, I am 'revisiting' Turing's paper to help new investigators to understand and appreciate his remarkable contribution to mathematical biology. Along the way, we shall resolve several 'peculiarities' of Turing's reaction mechanisms and numerical simulations, and place his work in context with later textbook examples. The relation of stationary Turing patterns to time-dependent traveling waves of chemical activity is also described.