2022 Fiscal Year Final Research Report
Clarification of the multi-layered structure of stationary solutions induced by the cross-diffusion limit in the Lotka-Volterra system
Project/Area Number |
19K03581
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Research Category |
Grant-in-Aid for Scientific Research (C)
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Allocation Type | Multi-year Fund |
Section | 一般 |
Review Section |
Basic Section 12020:Mathematical analysis-related
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Research Institution | Waseda University |
Principal Investigator |
Kousuke Kuto 早稲田大学, 理工学術院, 教授 (40386602)
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Project Period (FY) |
2019-04-01 – 2023-03-31
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Keywords | 交差拡散 / ロトカ・ボルテラ系 / 定常解 / 極限系 / 分岐 / 数理生物学モデル |
Outline of Final Research Achievements |
The asymptotic behavior of stationary solutions for the Lotka-Volterra system, which describes the population density of two competing species competing for territory in a bounded region, is clarified as cross-diffusion coefficients of both species tend to infinity. To elucidate the asymptotic behavior, we first gave a priori estimate that the height of any stationary solution (the maximum norm) is suppressed by a positive constant independent of both cross-diffusion coefficients. Next, it was shown that stationary solutions converge to solutions of limit systems as cross-diffusion coefficients of both species tend to infinity, and the global bifurcation structure of solutions of the limit systems was determined for the respective boundary condition cases of the Neumann and Dirichlet types. The results show that species segregation phenomena can be reproduced at the level of stationary solutions by the cross-diffusion effects of both competing species.
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Free Research Field |
反応拡散方程式
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Academic Significance and Societal Importance of the Research Achievements |
競争種同士が空間的なに反発を記述する交差拡散項に対しては、数学的な解析の難しさから、ロトカ・ボルテラ系においては、片方のみの交差拡散係数を無限大にする操作しか研究されていなかった。本研究成果の最たる学術的意義は、定常問題において両方の交差拡散係数を無限大にする数学的処方(両方交差拡散極限)を提案し、拡散の相互作用の効果を定常解の分岐構造の見地から明示したことである。ロトカ・ボルテラ系以外にも交差拡散を伴う数理モデルが存在することから、両方交差拡散極限の他のモデルへの応用も期待出来る。
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