title: Reaction, diffusion and non-local interaction.
abstract: 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.
authors.affiliation: School of Interdisciplinary Mathematical Sciences, Meiji University, 4-21-1 Nakano, Nakano-ku, Tokyo, 164-8525, Japan. hirokazu.ninomiya@gmail.com.
authors.forename: Hirokazu
authors.identifier: http://orcid.org/0000-0001-7081-6564
authors.initials: H
authors.lastname: Ninomiya
authors.affiliation: Department of Mathematics, Faculty of Science, Hokkaido University, Kita 10, Nishi 8, Kita-ku, Sapporo, Hokkaido, 060-0810, Japan.
authors.forename: Yoshitaro
authors.initials: Y
authors.lastname: Tanaka
authors.affiliation: Organization for the Strategic Coordination of Research and Intellectual Properties, Meiji University, 4-21-1 Nakano, Nakano-ku, Tokyo, 164-8525, Japan.
authors.forename: Hiroko
authors.initials: H
authors.lastname: Yamamoto
mesh_terms: D000818:Animals; D019521:Body Patterning / Q000502:physiology; D003198:Computer Simulation; D004058:Diffusion; D006801:Humans; D055641:Mathematical Concepts; D008954:Models, Biological
keywords: Instability; Non-local interaction; Reaction–diffusion system