### STEADY STATE WITH SMALL CHANGE OF REPRODUCTION AND SELF-LIMITATION

#### Abstract

In ~\cite{ko02}, we established some sufficient condition for the uniqueness of the positive solution to the general elliptic system for two competing species of animals$$\left\{ \begin{array}{l}\left.\begin{array}{l}\Delta u + u(g_{1}(u) - g_{2}(v)) = 0\\[12pt]\Delta v + v(h_{1}(v) - h_{2}(u)) = 0\end{array} \right.\;\;\mbox{in}\;\;\Omega,\\[12pt]u|_{\partial\Omega} = v|_{\partial\Omega} = 0.\end{array} \right.$$In this paper, we try to extend the uniqueness result by perturbing the reproduction and self-limitation functions $g_{1}, h_{1}$ of the above model. The techniques used in this paper are super-sub solutions, maximum principles and spectrum estimates. The arguments also rely on some detailed properties for the solution of logistic equations.

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