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elliptic theory; maximum principles
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.
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