This presentation deals with reaction-diffusion equations with nonlocal effect described by the convolution in the spatial direction of the unknown function and the integral kernel. Such nonlocal effects are often used to describe nonlocal diffusion effects of substances, cell-cell interactions, etc. Other studies have shown that reaction-diffusion equations with nonlocal effects can be derived from certain classes of reaction-diffusion systems and can also be derived as approximate equations for spatially discrete models.
To consider pattern formation in reaction-diffusion equations with nonlocal effect, we present results of mathematical analysis on the dynamics of localized patterns. We mainly introduce results on the interaction between localized patterns and the behavior of phase-separated patterns when the nonlocal effect is sufficiently weak. The influence of the integral kernel shapes in these situations will be reviewed, and the influence of nonlocal effects on pattern formation will be discussed.