Although, for single equations, diffusion can be viewed as a smoothing and trivializing process, the situation becomes drastically different when we come to systems of diffusion equations. For example, in a system of equations modeling two interactive substances, Turing had already observed that different diffusion rates could lead to nonhomogeneous distribution of such reactants. In fact, one distinctive characteristic of many such systems/models is that solutions are often highly concentrated in small areas and thereby display striking patterns. In this talk, I will use an activator-inhibitor system due to Gierer and Meinhardt (in their modeling of the regeneration phenomena of hydra) to describe the current mathematical research on some of those highly concentrated solutions, namely, those solutions whose graphs display narrow peaks or spikes -- also known as point-condensation solutions or spike-layers. Special attention will be paid to the stability/instability properties of the spike-layers, as well as their profiles and locations. Moreover, I will try to discuss other types of highly concentrated solutions which may be useful in other situations.

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1998-1999 Mathematics in Biology