## Outline of survey

In the latter half of the 20^{th} century, a tremendous scientific paradigm shift was experienced. Academically speaking, this shift is a historic change from a static viewpoint of equilibrium thermodynamics to a dynamic viewpoint of nonlinear-nonequilibrium thermodynamics. Unfortunately, mathematics at that time was unable to keep up with this change. However, since reaction-diffusion equations have appeared as mathematical models to describe various phenomena arising in nonlinear-nonequilibrium sciences, the realm of mathematics, especially the field of nonlinear analysis, has inspired change. Hence, new nonlinear analysis (more widely speaking, a group of mathematical sciences) in our country has steadily grown in an effort to actively participate in studying these new science disciplines, which has led us to propose the study of mathematical theory of nonlinear-nonequilibrium reaction-diffusion systems. A characteristic of our research is that mathematical sciences, especially nonlinear analysis, can actively contribute to theoretical studies of phenomena in nonlinear-nonequilibrium sciences, suggesting that mathematics will soon be conducting interdisciplinary collaborations with other scientific fields.

## Expected outcomes

One challenge of mathematical sciences, especially mathematics toward understand phenomena in nonlinear-nonequilibrium science, is that this is our country’s first attempt in using this unique approach. Therefore, the significance of this proposal is that it should allow the field of mathematical analysis to widely develop so that mathematics becomes intertwined with natural sciences. Hence, mathematics, which has yet to make a marked contribution to the development of theoretical studies, will be able to actively collaborate with several natural science fields.

## Reference articles by principal investigators

- M. Mimura: Pattern Formation in Consumer-Finite Resource Reaction-Diffusion S Publ. RIMS, Kyoto Univ., 40, 413-1431, (2004)
- S.-I. Ei, M. Mimura and M. Nagayama: Interacting Spots in Reaction-Diffusion Systems, J. Discrete and Continuous Dynamical Systems, Series A, 14 31-62 (2006)