### A predator-prey model with combined death and competition terms

The existence of a positive solution for the generalized predator-prey model for two species $$\begin{array}{c}\Delta u+u(a+g(u,v\left)\right)=0\phantom{\rule{1.0em}{0ex}}\text{in}\phantom{\rule{4pt}{0ex}}\Omega ,\\ \Delta v+v(d+h(u,v\left)\right)=0\phantom{\rule{1.0em}{0ex}}\text{in}\phantom{\rule{4pt}{0ex}}\Omega ,\\ u=v=0\phantom{\rule{1.0em}{0ex}}\text{on}\phantom{\rule{4pt}{0ex}}\partial \Omega ,\end{array}$$ 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.