Existence of steady-state solutions of predator-prey equations in a heterogeneous environment (Q1824104)

The authors prove the existence of a classical positive steady-state solution for a weakly coupled semilinear reaction-diffusion system with spatially inhomogeneous reaction terms; the system represents a two- species predator-prey model with ``heterogeneous environment''. The system admits an invariant rectangle. A steady-state solution is found inside that rectangle by a variational method (iterative minimization of appropriate energy functionals for both single equations, followed by an application of Schauder's fixed point theorem). Although the existence result can be obtained with less effort by different methods, the approach of this paper is interesting in that it provides a variational characterization of the solution.