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The existence of a positive solution for the generalized predator-prey model for two species $$\begin{array}{c}\Delta u+u(a+g(u,v\left)\right)=0\text{in}\Omega ,\\ \Delta v+v(d+h(u,v\left)\right)=0\text{in}\Omega ,\\ u=v=0\text{on}\partial \Omega ,\end{array}$$ are investigated. The techniques used in the paper are the elliptic theory, upper-lower solutions, maximum principles and spectrum estimates. The arguments also rely on some detailed properties of the solution of logistic equations.

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...