Study on free boundary problems and reaction-diffusion equations arising in mathematical ecology

2018 Fiscal Year Final Research Report

• Project/Area Number: 16K05244
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Mathematical analysis
• Research Institution: Waseda University
• Principal Investigator: YAMADA Yoshio 早稲田大学, 理工学術院, 教授 (20111825)
• Research Collaborator: KANEKO Yuki ; WAKASA Tohru ; HOSHINO Hiroki ; MATSUZAWA Hiroshi ; ENDO Maho
• Project Period (FY): 2016-04-01 – 2019-03-31
• Keywords: 自由境界問題 / 反応拡散方程式 / 漸近挙動 / 比較定理 / 数理生態学

Outline of Final Research Achievements

Our research handles a free boundary problem for reaction-diffusion equations. This problem models spreading and migrating phenomena of a biological species and the main issue is to know asymptotic behaviors of the population density of the species and the boundary (free boundary) of its habitat. It is assumed that the density is governed by a reaction-diffusion equation and that the dynamics of the free boundary is determined by the Stefan condition. Since these free boundary problems were proposed by Du and Lin in 2010, a lot of people have studied them. We have discussed a reaction-diffusion equation with positive bistable reaction term which has two positive and stable equilibrium points and found out that a new type of asymptotic behaviors occurs for the free boundary problem. Moreover, we also have succeeded in getting theoretical understanding how the population density and habitat of the species change according as time goes on.

Free Research Field

非線形解析学

Academic Significance and Societal Importance of the Research Achievements

反応拡散方程式に対する自由境界問題は外来生物の侵入現象や、環境破壊により従来の生息域を離れ、新しい生息環境を求めて生物が移動する現象を数理モデル化したものであり、生態学的にも現実的な重要問題である。生物種の新しい生息域への展開について、成功や失敗の条件・状況を知ることは数理科学的にも生態学的にも興味あるテーマである。本研究により、生息域がある一定の範囲を超せば必ず無限に拡大すること、拡大するケースでは密度関数は一定の形状を保ちながら波のように進むこと、生息域の境界（自由境界）の拡大速度は一定の値であることなど、多くのに重要な成果を得ることができた。