# American Institute of Mathematical Sciences

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

 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.
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
##### References:
 [1] A. Coddington and N. Levinson, "Theory of ordinary differential equations,'', McGraw-Hill, (1955). [2] J. K. Hale, "Asymptotic behavior of dissipative systems,'', Mathematical Surveys and Monographs \textbf{25}, 25 (1988). [3] Y. Kan-on, Existence of standing waves for competition-diffusion equations,, Japan J. Indust. Appl. Math., 13 (1996), 117. [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. [5] H. Kokubu, Homoclinic and heteroclinic bifurcations of vector fields,, Japan J. Appl. Math., 5 (1988), 455.

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##### References:
 [1] A. Coddington and N. Levinson, "Theory of ordinary differential equations,'', McGraw-Hill, (1955). [2] J. K. Hale, "Asymptotic behavior of dissipative systems,'', Mathematical Surveys and Monographs \textbf{25}, 25 (1988). [3] Y. Kan-on, Existence of standing waves for competition-diffusion equations,, Japan J. Indust. Appl. Math., 13 (1996), 117. [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. [5] H. Kokubu, Homoclinic and heteroclinic bifurcations of vector fields,, Japan J. Appl. Math., 5 (1988), 455.
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