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2013, 2013(special): 427-436. doi: 10.3934/proc.2013.2013.427

Structure on the set of radially symmetric positive stationary solutions for a competition-diffusion system

Yukio Kan-On 1,

1. 

Department of Mathematics, Faculty of Education, Ehime University, Matsuyama, 790-8577

Received  August 2012 Revised  April 2013 Published  November 2013

In this paper, we consider a reaction-diffusion system which describes the dynamics of population density for a two competing species community, and discuss the structure on the set of radially symmetric positive stationary solutions for the system by assuming the habitat of the community to be a ball. To do this, we shall treat the dimension of the habitat and the diffusion rates of the system as bifurcation parameters, and employ the comparison principle and the implicit function theorem.
Keywords: Competition-diffusion system, comparison principle, exponential dichotomy..
Mathematics Subject Classification: Primary: 35B32; Secondary: 35B6.
Citation: Yukio Kan-On. Structure on the set of radially symmetric positive stationary solutions for a competition-diffusion system. Conference Publications, 2013, 2013 (special) : 427-436. doi: 10.3934/proc.2013.2013.427
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A. Coddington and N. Levinson, "Theory of ordinary differential equations,'' McGraw-Hill, New York-Toronto-London, 1955.  Google Scholar

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J. K. Hale, "Asymptotic behavior of dissipative systems,'' Mathematical Surveys and Monographs 25, American Mathematical Society, Providence, RI, 1988.  Google Scholar

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Y. Kan-on, Existence of standing waves for competition-diffusion equations, Japan J. Indust. Appl. Math., 13 (1996), 117-133.  Google Scholar

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Y. Kan-on, Existence of positive travelling waves for generic Lotka-Volterra competition model with diffusion, Dynam. Contin. Discrete Impuls. Systems, 6 (1999), 345-365.  Google Scholar

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H. Kokubu, Homoclinic and heteroclinic bifurcations of vector fields, Japan J. Appl. Math., 5 (1988), 455-501.  Google Scholar

show all references

References:
[1]

A. Coddington and N. Levinson, "Theory of ordinary differential equations,'' McGraw-Hill, New York-Toronto-London, 1955.  Google Scholar

[2]

J. K. Hale, "Asymptotic behavior of dissipative systems,'' Mathematical Surveys and Monographs 25, American Mathematical Society, Providence, RI, 1988.  Google Scholar

[3]

Y. Kan-on, Existence of standing waves for competition-diffusion equations, Japan J. Indust. Appl. Math., 13 (1996), 117-133.  Google Scholar

[4]

Y. Kan-on, Existence of positive travelling waves for generic Lotka-Volterra competition model with diffusion, Dynam. Contin. Discrete Impuls. Systems, 6 (1999), 345-365.  Google Scholar

[5]

H. Kokubu, Homoclinic and heteroclinic bifurcations of vector fields, Japan J. Appl. Math., 5 (1988), 455-501.  Google Scholar

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