2001 Fiscal Year Final Research Report Summary
THE STRUCURE OF ATTRACTORS AND TRANSIENT DYNAMICS OF MULTI-COMPONETS REACTION-DIFFUSION SYSTEMS
| Project/Area Number |
10640141
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Kyoto Sangyou University |
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Principal Investigator |
HOSONO Yuzo KYOTO SANGYO UNIVERSITY, DEPARTMENT OF INFORMATION AND COMMUNICATION SCIENCES, PROFESSOR, 工学部, 教授 (50008877)
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| Co-Investigator(Kenkyū-buntansha) |
TSUJI Yoshiki KYOTO SANGYO UNIVERSITY, DEPARTMENT OF COMPUTER SCIENCES, PROFESSOR, 理学部, 教授 (90065871)
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| Project Period (FY) |
1998 – 2001
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| Keywords | LOTKA-VOLTERRA SYSTEM / TRAVELLING WAVES / THE MINIMAL WAVE SPEED / AUTOCATALYTIG REACTION SYSTEM / PUSHED AND PULLED FRONTS / LINEAR CONJECTURE / HETEROCLINIC ORBIT / PHASE PLANE ANALYSIS |
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| Research Abstract |
(1) We investigated the multi-species Lotka-Volterra competition model and predator-prey model. Our main concern is travelling waves and its propagation speeds which describe the situation of the invasion of the external species.
(i) For the two species Lotka-Volterra competition model, we proved that there exist two types of travelling waves, that is, pushed and pulled fronts, and discussed the parameter dependence of wave speeds analytically and numerically.
(ii) For the two species Lotka-Volterra predator-prey model, we obtained the numerically the condition which enable both of predator and prey to invade into an open space. This result gives us an important information on further analytical studies
(iii) We confirm numerically the chaotic behavior of solutions of the three species Lotka-Volterra competition system and the two species predator-prey system, the obtains results help our mathematical understanding of the nonlinear structure which assures the similar complex behavior of solutions of both systems.
(2) We studies the pushed and pulled fronts of the two components autocatalytic reaction models.
(i) For the mixed second and third order autocatalytic reaction, we proved the existence of traveling waves when the reactant cannot diffuse, and gave an estimate of the minimal wave speeds which depends on the ratio of the mixed reactions.
(ii) For the pure higher order autocatalytic reaction, we consider the two cases that the reactant and the autocatalysis have the same diffusivity and that the reactant does not diffuse. For both cases, we proved the existence of pulled fronts and obtained the estimate of the minimal wave speeds which showed that the minimal wave speed tends to zero as the order of autocatalysis goes to infinity.
(iii) The similar results was proved for the higher order autocatalytic reaction with decay of autocatalysis whose order of decay is less of 1 than the order of autocatalysis.
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