### Steady State Solutions to General Competition and Cooperation Models

#### Abstract

Two species of animals are competing or cooperating 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? We investigate this phenomena in mathematical point of view.

In this paper we concentrate on coexistence solutions of the competition or cooperation model\[\left\{\begin{array}{l} \begin{array}{l}\Delta u + u(g_{1}(u) + h_{1}(v)) = 0\\\Delta v + v(g_{2}(u) + h_{2}(v)) = 0\end{array}\;\;\mbox{in}\;\;\Omega,\\u|_{\partial\Omega} = v|_{\partial\Omega} = 0. \end{array} \right.\]This system is the general model for the steady state of a competitive or cooperative interacting system depending on growth conditions for $g_{1}$, $g_{2}$, $h_{1}$ and $h_{2}$. The techniques used in this paper are elliptic theory, super-sub solutions, maximum principles, and spectrum estimates. The arguments also rely on some detailed properties of the solution of logistic equations.

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