Pattern formation in interacting and diffusing systems in population biology.

In this article, we have been mainly concerned with spatially non-uniform stationary states and their stability, motivated by pattern formation arising in population biology. The discussions are restricted to one-dimensional space, though real systems are always distributed in at least two-dimensional space. Even if we limit ourselves to small-amplitude solutions, it seems difficult to discuss the bifurcation problems in a manner similar to that for one-dimensional space. One of the reasons is that the bifurcation points are not easily found. However, some general theories have nearly been completed. There are a variety of phenomena of other patterns such as wave trains, wave fronts, pulse waves, target patterns, and rotating patterns in equations of reaction and diffusion. We have not discussed these here. Moreover, we emphasize that there are a lot of nonlinear diffusion problems which are different from the ones that were dealt with here. The book of Fife (1), for example, provides a good exposition on these problems.