Reaction, diffusion and non-local interaction.

Recent years have seen the introduction of non-local interactions in various fields. A typical example of a non-local interaction is where the convolution kernel incorporates short-range activation and long-range inhibition. This paper presents the relationship between non-local interactions and reaction-diffusion systems in the following sense: (a) the relationship between the instability induced by non-local interaction and diffusion-driven instability; (b) the realization of non-local interactions by reaction-diffusion systems. More precisely, it is shown that the non-local interaction of a Mexican-hat kernel destabilizes the stable homogeneous state and that this instability is related to diffusion-driven instability. Furthermore, a reaction-diffusion system that approximates the non-local interaction system with any even convolution kernel is shown to exist.