Global solution structure in a system of two competing species with nonlinear diffusion effect

2019 Fiscal Year Final Research Report

• Project/Area Number: 16K05233
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Mathematical analysis
• Research Institution: Ehime University
• Principal Investigator: Kan-on Yukio 愛媛大学, 教育学部, 教授 (00177776)
• Co-Investigator(Kenkyū-buntansha): 桑村 雅隆 神戸大学, 人間発達環境学研究科, 教授 (30270333)
• Project Period (FY): 2016-04-01 – 2020-03-31
• Keywords: ２種競争系 / 極限系 / 解構造

Outline of Final Research Achievements

In 1979, Shigesada, Kawasaki and Teramoto proposed a reaction-diffusion system (SKT) with nonlinear diffusion effect, in order to study the coexistence and segregation in a two competing species community. Although the system (SKT) is comparatively simple, open problems on the existence and nonexistence, the spatial profile, and the stability and so on have been still remained for the solution of the system (SKT). In this research, we studied a continuous deformation with respect to the cross-diffusion rate and the inter-specific competition rate, from the classical competition-diffusion system to the system (SKT). We derived some limiting systems from the system (SKT) when the cross-diffusion rate or the inter-specific competition rate is very large, and investigated the solution structure for their limiting systems.

Free Research Field

非線形解析

Academic Significance and Societal Importance of the Research Achievements

本研究で得られた知見は，系(SKT)において交差拡散係数や種間競争係数があまり大きくない場合についての研究を進める際の重要な手がかりになると期待している．また，研究の過程で得られた極限系には周期解が存在することが数値的に確認されているため，この縮約系の解析により，系(SKT)の複雑な解構造の一部が解明できると思われる．さらに，交叉拡散効果の影響を大きくしていくと，系(SKT)は２種競争系の枠組みから外れ，一般の反応拡散系へと変化していくことを考えると，一般の反応拡散系へ研究を進める際の重要な手がかりになるものと期待している．