Geodesic
Steady state coexistence solutions of reaction-diffusion competition models
Czechoslovak Mathematical Journal, Tome 56 (2006) no. 4, pp. 1165-1183 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

Two species of animals are competing in the same environment. Under what conditions do they coexist peacefully? Or under what conditions does either one of the two species become extinct, that is, is either one of the two species excluded by the other? It is natural to say that they can coexist peacefully if their rates of reproduction and self-limitation are relatively larger than those of competition rates. In other words, they can survive if they interact strongly among themselves and weakly with others. We investigate this phenomena in mathematical point of view. In this paper we concentrate on coexistence solutions of the competition model \[ \left\rbrace \begin{array}{ll}\Delta u + u(a - g(u,v)) = 0, \Delta v + v(d - h(u,v)) = 0 \text{in} \ \Omega , u|_{\partial \Omega } = v|_{\partial \Omega } = 0. \end{array}\right.\] This system is the general model for the steady state of a competitive interacting system. The techniques used in this paper are elliptic theory, super-sub solutions, maximum principles, implicit function theorem and spectrum estimates. The arguments also rely on some detailed properties of the solution of logistic equations.
Two species of animals are competing in the same environment. Under what conditions do they coexist peacefully? Or under what conditions does either one of the two species become extinct, that is, is either one of the two species excluded by the other? It is natural to say that they can coexist peacefully if their rates of reproduction and self-limitation are relatively larger than those of competition rates. In other words, they can survive if they interact strongly among themselves and weakly with others. We investigate this phenomena in mathematical point of view. In this paper we concentrate on coexistence solutions of the competition model \[ \left\rbrace \begin{array}{ll}\Delta u + u(a - g(u,v)) = 0, \Delta v + v(d - h(u,v)) = 0 \text{in} \ \Omega , u|_{\partial \Omega } = v|_{\partial \Omega } = 0. \end{array}\right.\] This system is the general model for the steady state of a competitive interacting system. The techniques used in this paper are elliptic theory, super-sub solutions, maximum principles, implicit function theorem and spectrum estimates. The arguments also rely on some detailed properties of the solution of logistic equations.
Classification : 35J55, 35J60, 92D25
Keywords: elliptic theory; maximum principles 
@article{CMJ_2006_56_4_a6,
     author = {Kang, Joon Hyuk and Lee, Jungho},
     title = {Steady state coexistence solutions of reaction-diffusion competition models},
     journal = {Czechoslovak Mathematical Journal},
     pages = {1165--1183},
     year = {2006},
     volume = {56},
     number = {4},
     mrnumber = {2280801},
     zbl = {1164.35351},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2006_56_4_a6/}
}
TY  - JOUR
AU  - Kang, Joon Hyuk
AU  - Lee, Jungho
TI  - Steady state coexistence solutions of reaction-diffusion competition models
JO  - Czechoslovak Mathematical Journal
PY  - 2006
SP  - 1165
EP  - 1183
VL  - 56
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/CMJ_2006_56_4_a6/
LA  - en
ID  - CMJ_2006_56_4_a6
ER  - 
%0 Journal Article
%A Kang, Joon Hyuk
%A Lee, Jungho
%T Steady state coexistence solutions of reaction-diffusion competition models
%J Czechoslovak Mathematical Journal
%D 2006
%P 1165-1183
%V 56
%N 4
%U http://geodesic.mathdoc.fr/item/CMJ_2006_56_4_a6/
%G en
%F CMJ_2006_56_4_a6
Kang, Joon Hyuk; Lee, Jungho. Steady state coexistence solutions of reaction-diffusion competition models. Czechoslovak Mathematical Journal, Tome 56 (2006) no. 4, pp. 1165-1183. http://geodesic.mathdoc.fr/item/CMJ_2006_56_4_a6/

[1] R. S. Cantrell and C. Cosner: On the steady-state problem for the Volterra-Lotka competition model with diffusion. Houston Journal of mathematics 13 (1987), 337–352. | MR

[2] R. S. Cantrell and C. Cosner: On the uniqueness and stability of positive solutions in the Volterra-Lotka competition model with diffusion. Houston J. Math. 15 (1989), 341–361. | MR

[3] C. Cosner and A. C. Lazer: Stable coexistence states in the Volterra-Lotka competition model with diffusion. Siam J. Appl. Math. 44 (1984), 1112–1132. | DOI | MR

[4] D. Dunninger: Lecture note for applied analysis at Michigan State University.

[5] R. Courant and D. Hilbert: Methods of Mathematical Physics, Vol. 1. Interscience, New York, 1961.

[6] C. Gui and Y. Lou: Uniqueness and nonuniqueness of coexistence states in the Lotka-Volterra competition model. Comm. Pure and Appl. Math. 12 (1994), 1571–1594. | MR

[7] J. L. Gomez and J. P. Pardo: Existence and uniqueness for some competition models with diffusion. C.R. Acad. Sci. Paris, 313 Série 1 (1991), 933–938. | MR

[8] P. Hess: On uniqueness of positive solutions of nonlinear elliptic boundary value problems. Math. Z. 165 (1977), 17–18. | MR | Zbl

[9] L. Li and R. Logan: Positive solutions to general elliptic competition models. Differential and Integral Equations 4 (1991), 817–834. | MR

[10] A. Leung: Equilibria and stabilities for competing-species, reaction-diffusion equations with Dirichlet boundary data. J. Math. Anal. Appl. 73 (1980), 204–218. | DOI | MR | Zbl

[11] M. H. Protter and H. F. Weinberger: Maximum Principles in Differential Equations. Prentice Hall, Englewood Cliffs, N. J., 1967. | MR

[12] I. Stakgold and L. E. Payne: Nonlinear Problems in Nuclear Reactor Analysis. In nonlinear Problems in the Physical Sciences and Biology, Lecture notes in Mathematics 322, Springer, Berlin, 1973, pp. 298–307.