Steady state coexistence solutions of reaction-diffusion competition models

Joon Hyuk Kang; Jungho Lee

Czechoslovak Mathematical Journal (2006)

  • Volume: 56, Issue: 4, page 1165-1183
  • ISSN: 0011-4642

Abstract

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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 Δ u + u ( a - g ( u , v ) ) = 0 , Δ v + v ( d - h ( u , v ) ) = 0 in Ω , u | ∂ Ω = v | ∂ Ω = 0 . 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.

How to cite

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Kang, Joon Hyuk, and Lee, Jungho. "Steady state coexistence solutions of reaction-diffusion competition models." Czechoslovak Mathematical Journal 56.4 (2006): 1165-1183. <http://eudml.org/doc/31097>.

@article{Kang2006,
abstract = {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.},
author = {Kang, Joon Hyuk, Lee, Jungho},
journal = {Czechoslovak Mathematical Journal},
keywords = {elliptic theory; maximum principles; elliptic theory; maximum principles},
language = {eng},
number = {4},
pages = {1165-1183},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Steady state coexistence solutions of reaction-diffusion competition models},
url = {http://eudml.org/doc/31097},
volume = {56},
year = {2006},
}

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
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 56
IS - 4
SP - 1165
EP - 1183
AB - 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.
LA - eng
KW - elliptic theory; maximum principles; elliptic theory; maximum principles
UR - http://eudml.org/doc/31097
ER -

References

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