Dynamical behavior of mathematical models and its applications to threshold principle

2023 Fiscal Year Final Research Report

• Project/Area Number: 19K14598
• 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: Tokyo University of Science
• Principal Investigator: Enatsu Yoichi 東京理科大学, 教養教育研究院北海道・長万部キャンパス教養部, 講師 (90726910)
• Project Period (FY): 2019-04-01 – 2024-03-31
• Keywords: 微分方程式 / 感染症モデル / 被食者―捕食者モデル / 安定性 / 人口モデル

Outline of Final Research Achievements

We investigated the dynamics of free boundary problems describing the prevalence of infectious disease in space. For a diffusive SI epidemic model, the conditions for the existence and nonexistence of semi wave solutions is offered when there is no diffusion term for susceptible individuals. We proposed a multi-regional Leslie matrix model and developed a representation method of the reproductive value and stable age distribution. We also investigated the dynamics of a prey-predator model with cooperative hunting among specialist predators in predator growth. The conditions for the existence of coexistence equilibria and stability are given. Analysis for the occurrence of bifurcation including Hopf bifurcation, transcritical bifurcation, saddle-node bifurcation is carried out.

Free Research Field

数理生物モデルの定性的解析

Academic Significance and Societal Importance of the Research Achievements

自由境界をもつ感染症モデルでは，進行波の存在・非存在条件を導出した．移動境界以外の未知関数が複数個ある場合の解析が難しい中，感染者の生息領域の境界の移動速度を決める係数が存在・非存在条件での重要なパラメーターとなることを明らかにした．人口モデルでは，地域ごとの出生率や移動といった日本の人口減少の国内要因を定量的に評価できた．被食者－捕食者モデルで考慮した採餌などでの捕食者間の相互協力は，食虫性の熱帯鳥を含む多くの種で見られる．この協力が捕食者・被食者数の増減へ及ぼす影響の解析は数学的にも非自明で，捕食者・被食者数の周期変動や捕食者の絶滅可能性に対する協力行為の持つ役割を明らかにした．