Global bifurcation structure of stationary patterns arising in the SKT model with nonlinear diffusion

2018 Fiscal Year Final Research Report

• Project/Area Number: 15K04948
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Mathematical analysis
• Research Institution: The University of Electro-Communications
• Principal Investigator: Kuto Kousuke 電気通信大学, 大学院情報理工学研究科, 教授 (40386602)
• Research Collaborator: YOTSUTANI shoji
• Project Period (FY): 2015-04-01 – 2019-03-31
• Keywords: 反応拡散系 / 拡散の相互作用 / 分岐 / 極限系 / 競争モデル

Outline of Final Research Achievements

In this research, we studied a reaction-diffusion system that describes the spatiotemporal population dynamics of two competitive species. Individually, for the Lotka-Volterra system with a non-linear diffusion term called the cross diffusion term (the Shigesada-Kawasaki-Teramoto model, 1979), the asymptotic behavior of the steady-state solutions was analyzed as a coefficient of the cross diffusion term tends to infinity. We clarified the global bifurcation structure of the solution for the approximation problem called "the second limit system," for which there had been not many results. This result revealed that the steady-state solution set of the Shigesada-Kawasaki-Teramoto model forms a hook-like curve called a saddle-node bifurcation curve.

Free Research Field

非線形偏微分方程式

Academic Significance and Societal Importance of the Research Achievements

本研究課題では、1979年に重定・川崎・寺本によって提唱されて以来、今もって未解決である交差拡散項を伴うロトカ・ボルテラ系（重定・川崎・寺本モデル,1979）の定常解の大域分岐構造の解明に向けて、進展を得ている。具体的には、従来の研究が乏しかった「第２極限系」とよばれる第２極限系の解構造の概要を得ることで、懸案であったSKTモデルの定常解の大域分岐構造がサドルノード分岐曲線を描くことの数学的メカニズムを明確にすることが出来た。