2005 Fiscal Year Final Research Report Summary
Mathematical analysis of emergence of a rich variety of solutions in reaction-diffusion systems
| Project/Area Number |
14540143
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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 Sangyo University |
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Principal Investigator |
HOSONO Yuzo Kyoto Sangyo University, Faculty of Engineering, Professor, 工学部, 教授 (50008877)
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| Co-Investigator(Kenkyū-buntansha) |
TSUJII Yoshiki Kyoto Sangyo University, Faculty of Sciences, Professor, 理学部, 教授 (90065871)
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| Project Period (FY) |
2002 – 2005
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| Keywords | reaction-diffusion system / traveling wave / propagation speed / linear conjecture / competition model / autocatalytic reaction / epidemic / phase plane analysis |
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| Research Abstract |
(1)The Lotka-Volterra competition models
We consider the May-Leonard 3-species competition system with a singular perturbation parameter. We first discuss the complete picture of the stability of the coexistence steady state of this system, and show the occurrence of the subcritical Hopf bifurcation with the help of AUTO. We further investigate the relation between the 3-species Lotka-Volterra competition model and the 2-species predator-prey model by the formal singular perturbation analysis and the numerical simulations.
(2)Traveling waves for the higher order autocatalytic reaction-diffusion systems
We prove the existence of traveling waves for the two component higher order autocatalytic reaction-diffusion systems for any diffusion coefficients. We further discuss the existence problem for the system with higher order decay when the reactant does not diffuse. Our analysis of the vector fields in the phase space gives the estimate of the minimal propagation speeds in terms of the order of autocatalysis and the diffusion coefficients.
(3)The reaction-diffusion systems related to epidemic modeling
We discuss the relation between the reaction-diffusion models and the integral equation models in the study of the spatial spread of epidemic and to discuss the speeds of spatial spread of an infectious disease through the reaction-diffusion models. The speeds of the epidemic waves are often derived heuristically from the linearization of the model equations, which is called ‘the linear conjecture.' For the diffusive Kermack-McKendrick model, we show the validity of the linear conjecture by the use of the existence results of traveling wave solutions. Then, we give the examples of the reaction-diffusion epidemic models which have traveling wave solutions with the speed greater than the predicted values of the speed by the linear conjecture.
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Research Products
(28 results)
