Steady state coexistence solutions of reaction-diffusion competition models.

*(English)*Zbl 1164.35351Summary: Two species of animals are competing in the same environment. Under which conditions do they coexist peacefully? Or under which conditions does either one of the two species become extinct, i.e. is 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. In other words, they can survive if they interact strongly among themselves and weakly with others. We investigate this phenomenon from a mathematical point of view. We concentrate on coexistence solutions of the competition model
\[
\begin{cases} \Delta u + u(a - g(u,v)) = 0,\\ \Delta v + v(d - h(u,v)) = 0& \text{in} \;\Omega ,\\ u| _{\partial \Omega } = v| _{\partial \Omega } = 0. \end{cases}
\]
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, the implicit function theorem and spectrum estimates. The arguments also rely on some detailed properties of the solution of logistic equations.

##### MSC:

35J55 | Systems of elliptic equations, boundary value problems (MSC2000) |

35J60 | Nonlinear elliptic equations |

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\textit{J. H. Kang} and \textit{J. Lee}, Czech. Math. J. 56, No. 4, 1165--1183 (2006; Zbl 1164.35351)

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