Research on propagation and interface behavior generated by reaction-diffusion system

2023 Fiscal Year Final Research Report

• Project/Area Number: 19K14602
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 12040:Applied mathematics and statistics-related
• Research Institution: Kanto Gakuin University (2023); Japan Women's University (2019-2022)
• Principal Investigator: Kaneko Yuki 関東学院大学, 理工学部, 講師 (40773916)
• Project Period (FY): 2019-04-01 – 2024-03-31
• Keywords: 反応拡散方程式 / 自由境界問題 / 伝播現象 / 界面運動 / 漸近挙動 / 正値双安定 / テラス型分布 / 進行波

Outline of Final Research Achievements

We studied a free boundary problem for a reaction-diffusion equation which models the spreading of biological species. When the reaction term is positive bistable, we showed big spreading and small spreading corresponding to the stable equilibrium points, obtained asymptotic profiles of solutions and determined the different spreading speeds. We also showed that the big spreading forms a propagating terrace under some condition. These results were extended to a multi-dimensional free boundary problem. In conclusion, the factor that causes the spreading behaviors and the propagating terrace to single reaction-diffusion equations derives from positive bistable nonlinearity.

Free Research Field

非線形偏微分方程式　拡散現象

Academic Significance and Societal Importance of the Research Achievements

侵入現象は外来種問題や生物多様性の観点から非常に重要な問題である．生物種の個体数密度と侵入前線を未知関数とするとき，侵入現象は反応拡散方程式の自由境界問題として定式化できる．この問題を解くことによって，生物種がどのように分布し侵入前線がどれほどの速度でどこまで拡がるのかということについて，理論的に深く理解することができる．特に本研究で示されたテラス型分布は，生物種の小集団が先に定着した後，大集団が遅れて押し寄せるような伝播形態が存在することを示唆している．