Approximations of systems of partial differential equations for nonlocal interactions

Motivated by pattern formations and cell movements, many evolution equations incorporating spatial convolution with suitable integral kernel have been proposed. Numerical simulations of these nonlocal evolution equations can reproduce various patterns depending on the shape and form of integral kernel.The solutions to nonlocal evolution equations are similar to the patterns obtained by reaction-diffusion system and Keller-Segel system models. In this talk, we classify nonlocal interactions into two types, and investigate their relationship with reaction-diffusion systems and Keller-Segel systems, respectively. In these partial differential equation systems, we introduce multiple auxiliary diffusive substances and consider the singular limit of the quasi-steady state to approximate nonlocal interactions. In particular, we introduce how the parameters of the partial differential equation system are determined by the given integral kernel. These analyses demonstrate that, under certain conditions, nonlocal interactions and partial differential equation systems can be treated within a unified framework.  
This talk is based on collaborations with Hiroshi Ishii of Hokkaido University and Hideki Murakawa of Ryukoku University.