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Nonlocal interactions, which have attracted attention in various fields, result from the integration of microscopic information such as a transition possibility, molecular events, and signaling networks of living creatures. Nonlocal interactions are useful to reproduce various patterns corresponding to such detailed microscopic information. However, the approach is inconvenient for observing the specific mechanisms behind the target phenomena because of the compression of the information. Therefore, we previously proposed a method capable of approximating any nonlocal interactions by a reaction-diffusion system with auxiliary factors (Ninomiya et al., J Math Biol 75:1203-1233, 2017). In this paper, we provide an explicit method for determining the parameters of the reaction-diffusion system for the given kernel shape by using Jacobi polynomials under appropriate assumptions. We additionally introduce a numerical method to specify the parameters of the reaction-diffusion system with the general diffusion coefficients by the Tikhonov regularization. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]