Approximation theory and continuation method of nonlocal interactions and its applications

2023 Fiscal Year Final Research Report

• Project/Area Number: 20K14364
• 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: Future University-Hakodate
• Principal Investigator: Tanaka Yoshitaro 公立はこだて未来大学, システム情報科学部, 准教授 (80783977)
• Project Period (FY): 2020-04-01 – 2024-03-31
• Keywords: 非局所発展方程式 / 反応拡散系 / 数理モデリング / 空間離散モデル / 空間連続化

Outline of Final Research Achievements

The interactions which can affect the distant objects globally in space, called nonlocal interactions, are observed in the fields of cell biology, neuroscience and so on. This interaction can be modeled by the convolution with a suitable integral kernel. Many mathematical models with the nonlocal interactions have been proposed. Motivated by the advocate of the application of the nonlocal interactions, this project studied the continuation method with nonlocal interactions conserving the discrete structure and reduction method for the network with spatial interactions. The former continuation method can convert the spatially discretized model into continuous one by using the characteristic function and convolution; this renders both models point-wisely equivalent. The latter method can reduce the network system with arbitrary number of factors into the mathematical model with nonlocal interactions by considering the eigenvalue problems. We published these results as papers.

Free Research Field

現象数理学

Academic Significance and Societal Importance of the Research Achievements

積分付き相互作用は，脳の神経の発火現象や動物の表皮に見られる色素パターン，細胞運動を記述することが知られていた．これに加え上記の方法論2つから，多細胞生物の発生現象など空間離散的な構造上で時間発展する現象や，シグナル伝達系や代謝系などの大規模なネットワーク系も積分付き相互作用をもつ数理モデルで記述できることがわかった．さまざまな現象を統一的な視点で扱える枠組みを与えた点において，学術的な意義があると考えている．今後積分付き相互作用をもつ数理モデルの解析を行うことで，さまざまな現象を制御できるようになり，社会的な意義を見出すことができると考えている．